text:
\chapter{Introduction}

Many symmetric monoidal categories $\cat{C}$ have the property that each object $c\in\cat{C}$ is equipped with a certain algebraic structure---say that of a monoid or a comonoid---in a way that is compatible with $\cat{C}$'s monoidal structure.

For example, consider the category $\Cat{Rel}$ of relations between sets. It has a symmetric monoidal structure $(I, \otimes, \gamma)$ coming from the cartesian monoidal structure of $\Cat{Set}$. This is not a cartesian monoidal structure on $\Cat{Rel}$: indeed, $I$ is not terminal. And yet each object $r\in\Cat{Rel}$ is equipped with morphisms $\delta_r\colon r\to r\otimes r$ and $\epsilon_r\colon r\to I$, which satisfy the same \emph{algebraic} properties that a diagonal and a terminal morphism do. Namely, the diagrams expressing commutativity, unitality, and associativity commute:
\[
\begin{tikzcd}[column sep=small]
	&r\ar[dl, "\delta"']\ar[dr, "\delta"]\\
	r\otimes r\ar[rr, "\gamma"']&&
	r\otimes r
\end{tikzcd}
\qquad
\begin{tikzcd}[column sep=small]
	r\ar[rr, equal]\ar[dr, "\delta"']&&
	r\\&
	r\otimes r\ar[ur, "\epsilon\otimes r"']
\end{tikzcd}
\qquad
\begin{tikzcd}
	r\ar[r, "\delta"]\ar[d, "\delta"']&
	r\otimes r\ar[d, "r\otimes\delta"]\\
	r\otimes r\ar[r, "\delta\otimes r"']&
	r\otimes r\otimes r
\end{tikzcd}
\]
In string diagrams, the maps $\delta$ and $\epsilon$ can be drawn as:
\[
\begin{tikzpicture}[WD]
	\node[bb={1}{0}] (eta') {$\epsilon$};
	\draw (eta'_in1) to +(-.8,0);
	\node[bb={1}{2}, right=4 of eta'] (mu') {$\delta$};
	\draw (mu'_in1) -- +(-.8,0);
	\draw (mu'_out1) -- +(.8,0);
	\draw (mu'_out2) -- +(.8,0);
\end{tikzpicture}
\]
and the equations can be drawn as:
\[
\begin{tikzpicture}
	\node (Q11) {
	\begin{tikzpicture}[WD]
		\node[bb={1}{2}] (a) {};
		\coordinate (a1) at ($(a_out1)+(1,0)$);
		\coordinate (a2) at ($(a_out2)+(1,0)$);
		\draw (a_out2) to[out=0, in=180] (a1);
		\draw (a_out1) to[out=0, in=180] (a2);
		\draw (a_in1) -- +(-.5,0);
	\end{tikzpicture}
	};
	\node (Q12) [right=.8 of Q11] {
	\begin{tikzpicture}[WD]
		\node[bb={1}{2}] (a) {};
		\draw (a_out1) -- +(.5,0);
		\draw (a_out2) -- +(.5,0);
		\draw (a_in1) -- +(-.5,0);
	\end{tikzpicture}
	};
	\node[label=above:{\tiny commutative}] at ($(Q11.east)!.5!(Q12.west)$) {$=$};
	\node (Q21) [right=1 of Q12] {
  \begin{tikzpicture}[WD]
  	\node[bb={1}{2}] (a1) {};
  	\node[bb={1}{0}, right=.5 of a1_out1] (a2) {};
  	\draw (a2_in1) -- (a1_out1);
  	\draw (a1_out2) -- +(2,0);
  	\draw (a1_in1) -- +(-.5,0);
	\end{tikzpicture}
	};
	\node (Q22) [right=.8 of Q21] {
	\begin{tikzpicture}[WD]
		\draw (0,0) -- (2,0);
	\end{tikzpicture}
	};
	\node[label=above:{\tiny unital}] at ($(Q21.east)!.5!(Q22.west)$) {$=$};
	\node (Q31) [right=1 of Q22] {
	\begin{tikzpicture}[WD]
		\node[bb={1}{2}] (a1) {};
		\node[bb={1}{2}, minimum height=1ex, right=.5 of a1_out1] (a2) {};
		\draw (a2_in1) -- (a1_out1);
		\draw (a1_out2) -- +(2,0);
		\draw (a2_out1) -- +(.5,0);
		\draw (a2_out2) -- +(.5,0);
		\draw (a1_in1) -- +(-.5,0);
	\end{tikzpicture}
	};
	\node (Q32) [right=.8 of Q31] {
	\begin{tikzpicture}[WD]
		\node[bb={1}{2}] (a1) {};
		\node[bb={1}{2}, minimum height=1ex, right=.5 of a1_out2] (a2) {};
		\draw (a2_in1) -- (a1_out2);
		\draw (a1_out1) -- +(2,0);
		\draw (a2_out1) -- +(.5,0);
		\draw (a2_out2) -- +(.5,0);
		\draw (a1_in1) -- +(-.5,0);
	\end{tikzpicture}
	};
	\node[label=above:{\tiny associative}] at ($(Q31.east)!.5!(Q32.west)$) {$=$};
\end{tikzpicture}
\]
Not only is every object $r\in\Cat{Rel}$ equipped with these operations $\epsilon_r,\delta_r$, but they are coherent with respect to $\Cat{Rel}$'s monoidal structure. By this we mean first that the operations assigned to the monoidal unit are coherence isomorphisms: $\epsilon_I=\id_I$ and $\delta_I=\rho_I=\lambda_I$, where $\rho$ and $\lambda$ are the right and left unitors. Second, for any $r,s\in\Cat{Rel}$, the operations interact appropriately with the monoidal product:
\[
  \epsilon_{r\otimes s}=\epsilon_r\otimes\epsilon_s
  \qquad\text{and}\qquad
  \delta_{r\otimes s}=(\delta_r\otimes\delta_s)\mathbin{\fatsemi}(\id_r\otimes\gamma_{r,s}\otimes\id_s).
\]
One notices immediately the need for a symmetry isomorphism $\gamma_{r,s}$ in the second equation, which is there so that the codomains agree ($r\otimes s\otimes r\otimes s$). In pictures:
\begin{equation} \label{eqn.compat_tensor}
\begin{tikzpicture}[baseline=(current bounding box.center)]
	\node (P1) {
	\begin{tikzpicture}[WD]
		\node[bb={2}{0}] (rs) {$\epsilon_{r\otimes s}$};
		\draw (rs_in1) to node[above, font=\tiny] {$r$} +(-1, 0);
		\draw (rs_in2) to node[below, font=\tiny] {$s$} +(-1, 0);
	\end{tikzpicture}
	};
	\node (P2) [right=.7 of P1] {
	\begin{tikzpicture}[WD]
		\node[bb={1}{0}] (r) {$\epsilon_r$};
		\node[bb={1}{0}, below=.5 of r] (s) {$\epsilon_s$};
		\draw (r_in1) to node[above, font=\tiny] {$r$} +(-1, 0);
		\draw (s_in1) to node[below, font=\tiny] {$s$} +(-1, 0);
	\end{tikzpicture}
	};
	\node at ($(P1.east)!.5!(P2.west)$) {$=$};
	\node (P3) [right=2 of P2] {
	\begin{tikzpicture}[WD, bb port sep=.7]
		\node[bb={2}{4}] (rs) {$\delta_{r\otimes s}$};
		\begin{scope}[font=\tiny]
  		\draw (rs_in1) to node[above] {$r$} +(-1, 0);
  		\draw (rs_in2) to node[below] {$s$} +(-1, 0);
  		\draw (rs_out1) to node[above=-1pt] {$r$} +(1, 0);
  		\draw (rs_out2) to node[above=-1pt] {$s$} +(1, 0);
  		\draw (rs_out3) to node[below=-2pt] {$r$} +(1, 0);
  		\draw (rs_out4) to node[below=-2pt] {$s$} +(1, 0);
		\end{scope}
	\end{tikzpicture}
	};
	\node (P4) [right=.7 of P3] {
	\begin{tikzpicture}[WD, bb port sep=.7]
		\node[bb={1}{2}] (r) {$\delta_r$};
		\node[bb={1}{2}, below=.5 of r] (s) {$\delta_s$};
		\coordinate (r2) at ($(s_out1)+(2,0)$);
		\coordinate (s1) at ($(r_out2)+(2,0)$);
		\begin{scope}[font=\tiny]
			\draw (r_in1) to node[above] {$r$} +(-1, 0);
			\draw (s_in1) to node[below] {$s$} +(-1, 0);
			\draw (r_out1) -- +(2.5, 0) to node[above=-1pt] {$r$} +(.5, 0);
			\draw (r_out2) -- +(1, 0) to[out=0, in=180] (r2) to node[below=-1pt] {$r$} +(.5, 0);
			\draw (s_out1) -- +(1, 0) to[out=0, in=180] (s1) to node[above=-2pt] {$s$} +(.5, 0);
			\draw (s_out2) -- +(2.5, 0) to node[below=-2pt] {$s$} +(.5, 0);
    \end{scope}
	\end{tikzpicture}
	};
	\node at ($(P3.east)!.5!(P4.west)$) {$=$};
\end{tikzpicture}
\end{equation}
The point is that every object in $\Cat{Rel}$ has this commutative comonoid structure, and the operations are coherent with $I$ and $\otimes$. In this situation we will say that $\Cat{Rel}$ \emph{supplies commutative comonoids}.

In general, we may talk of algebraic structures on a object being defined by a prop $\mathbb{P}$. A prop is a strict symmetric monoidal category whose monoid of objects is $(\mathbb{N},0,+)$; in other words, a prop is a single-sorted symmetric monoidal theory. In the case of commutative comonoids, the relevant prop is the skeleton of $\Cat{FinSet}^{\tn{op}}$. Indeed, $\epsilon$ and $\delta$ represent the (opposites of) the unique functions $\varnothing\to\{1\}$ and $\{1,2\}\to\{1\}$, respectively.

We say that a symmetric monoidal category $\cat{C}$ \emph{supplies} $\mathbb{P}$ if every object of $\cat{C}$ is equipped with the structure of $\mathbb{P}$ in a way compatible with $\cat{C}$'s monoidal structure. This notion appears frequently in recent literature. One reason for this is that the compatibility with the monoidal product is a useful and intuitive feature when adding extra icons---``bells and whistles''---into the standard monoidal category string diagram language, yielding equations such as those in \cref{eqn.compat_tensor}.

Examples abound. To list a few: categories supplying comonoids figure strongly in categorical approaches to probability theory \cite{fong2012causal,fritz2019synthetic,cho2019disintegration}; categories supplying frobenius monoids---known as hypergraph categories---are important in networks and wiring-diagram languages \cite{Carboni:1991a,fong2019hypergraph}; categories supplying bimonoids underlie a categorical perspective on differentiation \cite{blute2009cartesian}; categories supplying so-called adjoint frobenius monoids and abelian relations underlie alternative approaches to regular and abelian categories \cite{fong2019regular,fong2019abelian}, and so on. Indeed, if we add an extra condition, which we call \emph{homomorphic supply}, categories homomorphically supplying commutative comonoids are simply cartesian monoidal categories \cite{fox1976coalgebras}, and categories homomorphically supplying bimonoids are those where the product is a biproduct.

Yet despite this plethora of examples, a general definition of supply has not yet been given: to now do so is the first goal of this article. If $\mathbb{P}$ is a prop and $\cat{C}$ is a symmetric monoidal category, we define what it means for $\cat{C}$ to supply the algebraic structure encoded in $\mathbb{P}$. We also define what it means for a strong monoidal functor $\cat{C}\to\cat{D}$ to \emph{preserve supplies}, i.e.\ to send a given supply of $\mathbb{P}$ in $\cat{C}$ to a given supply of $\mathbb{P}$ in $\cat{D}$. We give a number of examples both of supply and supply preservation.

The second goal is to provide some basic theory of supply. For example, given a supply of $\mathbb{P}$ in $\cat{C}$ and a prop functor $\mathbb{P}'\to\mathbb{P}$, one obtains a supply of $\mathbb{P}'$ in $\cat{C}$. We show that if $\cat{C}$ and $\cat{D}$ both supply $\mathbb{P}$ then so does their biproduct $\cat{C}\oplus\cat{D}$, and that the projections and coprojections preserve supplies. Finally, any supply of $\mathbb{P}$ in $\cat{C}$ induces a supply of $\mathbb{P}$ in the strictification $\strict{\cat{C}}$, and it is preserved by the equivalence $\strict{\cat{C}}\to\cat{C}$.

We also discuss what it means for various maps in $(\cat{C},I,\otimes)$ to be \emph{homomorphisms} for the supplied structure. For example, it is well-known that if $\cat{C}$ is cartesian monoidal (i.e.\ if the monoidal product is given by the categorical product) then it supplies comonoids and every morphism $f\colon c\to d$ is a comonoid homomorphism, in the sense that the following diagrams commute:
\[
\begin{tikzcd}[column sep=small]
	c\ar[rr, "f"]\ar[rd, "\epsilon_c"']&&
	d\ar[dl, "\epsilon_d"]\\&
	I
\end{tikzcd}
\hspace{.7in}
\begin{tikzcd}
	c\ar[r, "f"]\ar[d, "\delta_c"']&
	d\ar[d, "\delta_d"]\\
	c\otimes c\ar[r, "f\otimes f"']&
	d\otimes d
\end{tikzcd}
\]
Again in pictures:
\begin{equation}\label{eqn.homo}
\begin{tikzpicture}[baseline=(P1)]
	\node (P1) {
	\begin{tikzpicture}[WD]
		\node[bb={1}{0}] (e) {$\epsilon_c$};
		\draw (e_in1) -- +(-1, 0);
	\end{tikzpicture}
	};
	\node (P2) [right=.7 of P1] {
	\begin{tikzpicture}[WD]
		\node[bb={1}{1}] (f) {$f$};
		\node[bb={1}{0}, right=.5 of f] (e) {$\epsilon_d$};
		\draw (f_in1) -- +(-1, 0);
		\draw (f_out1) -- (e_in1);
	\end{tikzpicture}
	};
	\node at ($(P1.east)!.5!(P2.west)$) {$=$};
	\node (P3) [right=2 of P2] {
	\begin{tikzpicture}[WD]
		\node[bb={1}{1}] (f) {$f$};
		\node[bb={1}{2}, right=.5 of f] (d) {$\delta_d$};
		\draw (f_in1) -- +(-1, 0);
		\draw (f_out1) -- (d_in1);
		\draw (d_out1) -- +(1, 0);
		\draw (d_out2) -- +(1, 0);
  \end{tikzpicture}
	};
	\node (P4) [right=.7 of P3] {
	\begin{tikzpicture}[WD]
		\node[bb port sep=1.2, bb={1}{2}] (d) {$\delta_c$};
		\node[bb={1}{1}, right=.5 of d_out1, font=\tiny] (f1) {$f$};
		\node[bb={1}{1}, right=.5 of d_out2, font=\tiny] (f2) {$f$};
		\draw (d_in1) -- +(-1, 0);
		\draw (d_out1) -- (f1_in1);
		\draw (d_out2) -- (f2_in1);
		\draw (f1_out1) -- +(1, 0);
		\draw (f2_out1) -- +(1, 0);
	\end{tikzpicture}
	};
	\node at ($(P3.east)!.5!(P4.west)$) {$=$};
\end{tikzpicture}
\end{equation}
These equations hold in any cartesian monoidal category, e.g.\ $\Cat{Set}$, but they \emph{do not hold} in $\Cat{Rel}$. (As an example, the first equation in \eqref{eqn.homo} does not hold in $\Cat{Rel}$. Take $c=d=1$, take $f\coloneqq\varnothing\ss 1\times 1$ to be the empty relation, and note that $\epsilon_c \neq (f\mathbin{\fatsemi}\epsilon_d)$.) We will show that the morphisms in $\cat{C}$ that are homomorphisms for the $\mathbb{P}$-structure always form a monoidal subcategory. The above notion of homomorphic supply simply refers to the case that this subcategory is all of $\cat{C}$.

The main theorems of this paper are that supply and supply preservation are well-behaved with respect to coherence isomorphisms. In \cref{thm.coherence_isos_homos} we show that every associator and unitor in $\cat{C}$ is automatically a homomorphism for any supply. In \cref{thm.pres_supp_strongators_homo} we show that the coherence isomorphisms for strong monoidal functors $F\colon\cat{C}\to\cat{D}$ are automatically homomorphisms whenever $F$ preserves the supply. We give strictification theorems \cref{thm.supply_strictification,cor.strict_equiv_pres_supply}.

\paragraph{Acknowledgements.}
We thank Dmitry Vagner for interesting and useful conversations. We also thank Tobias Fritz for catching an error in a draft and for suggesting the strictification theorem (\cref{thm.supply_strictification}), which he had previously proven independently in a special case and shared with us; see \cite{fritz2019synthetic} forthcoming. We acknowledge support from Honeywell Inc., and from AFOSR grants FA9550-17-1-0058 and FA9550-19-1-0113.

\chapter{Notation and background}

\paragraph{Basic notation.}
For a natural number $n\in\mathbb{N}$ we denote the corresponding ordinal by $\ord{n}=\{1,\ldots,n\}\in\Cat{Set}$.
We denote composition of $f\colon a\to b$ and $g\colon b\to c$ by $(f\mathbin{\fatsemi} g)\colon a\to c$, i.e.\ we use diagrammatic order. When $c$ is an object we denote the identity morphism on it either by $c$ or by $\id_c$.

\paragraph{Symmetric monoidal categories and coherence.}
Suppose $(\cat{C}, I, \otimes)$ is a symmetric monoidal category, $m\in\mathbb{N}$ is a natural number, and $c\colon\ord{m}\to\cat{C}$ is a family of objects in $\cat{C}$. We denote
\begin{equation}\label{eqn.big_mon_prod}
	\bigotimes_{i\in\ord{m}}c(i)\coloneqq
  \big((c(1)\otimes c(2))\cdots\big)\otimes c(m)
\end{equation}
with the convention that when $m=0$ and $!\colon\ord{0}\to \cat{C}$ is the unique function, we put $\bigotimes !\coloneqq I$. We take this to be the canonical bracketing, so $c\otimes d\otimes e$ denotes $(c\otimes d)\otimes e$. If there exists $b\in\cat{C}$ such that $b=c(i)$ for all $i\in\ord{m}$, we denote the monoidal product in \eqref{eqn.big_mon_prod} by $b\tpow{m}\coloneqq\bigotimes_{i\in\ord{m}}b$.

If $m,n\in\mathbb{N}$ are natural numbers, and $c\colon \ord{m}\times \ord{n}\to\cat{C}$ is a family of objects in $\cat{C}$, we also have a natural isomorphism
\begin{equation}\label{eqn.symmetry}
\sigma\colon
\bigotimes_{i\in\ord{m}}\bigotimes_{j\in\ord{n}}c(i,j)\Too{\cong}
\bigotimes_{j\in\ord{n}}\bigotimes_{i\in\ord{m}}c(i,j).
\end{equation}
We refer to $\sigma$ as the \emph{symmetry} isomorphism, though note that it involves associators and unitors too, not just the symmetric braiding. We will be interested in two particular cases of the symmetry isomorphism \cref{eqn.symmetry}, namely for $m=2$ and $m=0$ and any $n\in\mathbb{N}$:
\[\sigma\colon c_1\tpow{n}\otimes c_2\tpow{n}\Too{\cong}(c_1\otimes c_2)\tpow{n}
\qquad\text{and}\qquad
\sigma\colon I\Too{\cong} I\tpow{n}.
\]

Many of our results will rely on Mac Lane's coherence theorem for symmetric monoidal categories \cite[Theorem XI.1]{MacLane:1998a}, which says the following. For any two ways to arrange brackets and monoidal units into a word with $n$ placeholders for objects in $\cat{C}$, and for each permutation of $n$ letters, there is an associated natural isomorphism, which Mac Lane calls the \emph{canonical isomorphism}, between the resulting functors $\cat{C}^n\to\cat{C}$. Moreover, composites and tensor products of canonical isomorphisms are again canonical. For example, everything we called a symmetry isomorphisms $\sigma$ in \cref{eqn.symmetry} is one of these canonical isomorphisms.

\paragraph{The 2-category $\mathbb{S}\Cat{MC}$.}
Recall that a strong monoidal functor $(F,\varphi)\colon\cat{C}\to\cat{D}$ consists of a functor $F$ and natural isomorphisms
\[
	\varphi\colon I_{\cat{D}}\To{\cong} F(I_{\cat{C}})
	\qquad\text{and}\qquad
	\varphi_{c,c'}\colon F(c)\otimes_{\cat{D}} F(c')\To{\cong} F(c\otimes_{\cat{C}} c').\]
	We refer to these isomorphism as the \emph{strongators} for $F$. A strong monoidal functor is \emph{strict} if all strongators are identities.

\begin{definition}\label{def.smf}
Let $\cat{C}$ and $\cat{D}$ be symmetric monoidal categories. Define $\Cat{SMF}(\cat{C},\cat{D})$ to be the symmetric monoidal category whose objects are strong monoidal functors $(F,\varphi)\colon\cat{C}\to\cat{D}$, whose morphisms are monoidal natural transformations, and whose symmetric monoidal structure is given pointwise.

Define $\CCat{SMC}$ to be the 2-category whose objects are symmetric monoidal categories and whose hom-categories are given by $\Cat{SMF}$.
\end{definition}

The pointwise condition in \cref{def.smf} means that the monoidal unit in $\Cat{SMF}(\cat{C},\cat{D})$ is given by the constant functor at the monoidal unit of $\cat{D}$ and that the monoidal product is given by $(F\otimes G)(c)\coloneqq F(c)\otimes G(c).$ The strongator of $F\otimes G$ for any $c,c'\in\cat{C}$ is given by the symmetry isomorphism
\[
	\sigma\colon
	\big(F(c)\otimes G(c)\big)\otimes\big(F(c')\otimes G(c')\big)
	\To{\cong}
	\big(F(c)\otimes F(c')\big)\otimes\big(G(c')\otimes G(c')\big).
\]

\cref{thm.smc_biprod,prop.biprod_smf_strict,thm.smc_all_prod_coprod} are not necessary for the main thrust of this note, so we will save their proofs for later; see \cref{chap.proofs}.
However, they seem important to us, and not sufficiently well known.

\begin{theorem}\label{thm.smc_all_prod_coprod}
The 2-category $\CCat{SMC}$ has all small products and coproducts, and products are strict.
\end{theorem}

In fact, finite products and coproducts coincide in $\CCat{SMC}$.

\begin{theorem}\label{thm.smc_biprod}
The 2-category $\CCat{SMC}$ of symmetric monoidal categories, strong monoidal functors, and monoidal natural transformations has 2-categorical biproducts.
\end{theorem}

We denote the biproduct of symmetric monoidal categories $\cat{C}$ and $\cat{D}$ by $\cat{C}\oplus\cat{D}$.

\begin{proposition}\label{prop.biprod_smf_strict}
Let $\cat{C}_1,\cat{C}_2,\cat{D}_1,\cat{D}_2$ be symmetric monoidal categories. The functor
\[
  \oplus\colon
  \Cat{SMF}(\cat{C}_1,\cat{D}_1)\times\Cat{SMF}(\cat{C}_2,\cat{D}_2)
  \to
  \Cat{SMF}(\cat{C}_1\oplus\cat{C}_2,\cat{D}_1\oplus\cat{D}_2)
\]
is strict monoidal.
\end{proposition}

\chapter{Supply}

In \cref{sec.supply} we define supply and give some first examples. \cref{sec.main_thm} then proves our main theorem, \cref{thm.coherence_isos_homos}: coherence isomorphisms are supply homomorphisms. We also provide a more compact definition of supply in \cref{thm.supply_v2}. In \cref{sec.further_theory}, we record some useful ways to construct new supplies from old.

\section{Definition of supply}\label{sec.supply}

Recall that a prop $\mathbb{P}$ is a symmetric strict monoidal category whose monoid of objects is $(\mathbb{N},0,+)$. We denote its objects by $m$, $n$, etc.

\begin{definition}[Supply]\label{def.supply}
Let $\mathbb{P}$ be a prop and $\cat{C}$ a symmetric monoidal category. A \emph{supply of $\mathbb{P}$ in $\cat{C}$} consists of a strong monoidal functor $s_c\colon\mathbb{P}\to\cat{C}$ for each object $c\in\cat{C}$, such that
\begin{enumerate}[label=(\roman*)]
	\item $s_c(m)=c\tpow{m}$ for each $m\in\mathbb{N}$,
	\item the strongator $c\tpow{m}\otimes c\tpow{n}\to c\tpow{(m+n)}$ is the unique coherence isomorphism for each $m,n\in\mathbb{N}$, and
	\item the following diagrams commute for every $c,d\in\cat{C}$ and $\mu\colon m\to n$ in $\mathbb{P}$:
\begin{equation}\label{eqn.supply_commute_tensors}
\begin{tikzcd}[column sep=55pt]
	c\tpow{m}\otimes d\tpow{m}\ar[r, "s_c(\mu)\otimes s_d(\mu)"]\ar[d, "\sigma"']&
	c\tpow{n}\otimes d\tpow{n}\ar[d, "\sigma"]\\
	(c\otimes d)\tpow{m}\ar[r, "s_{c\otimes d}(\mu)"']&
	(c\otimes d)\tpow{n}
\end{tikzcd}
\hspace{.7in}
\begin{tikzcd}
	I\ar[r, equal]\ar[d, "\sigma"']&
	I\ar[d, "\sigma"]\\
	I\tpow{m}\ar[r, "s_I(\mu)"']&
	I\tpow{n}
\end{tikzcd}
\end{equation}
where the $\sigma$'s are the symmetry isomorphisms from \cref{eqn.symmetry}.
\end{enumerate}
We further say that $f\colon c\to d$ is an \emph{$s$-homomorphism} if the following diagram commutes for all $\mu\colon m\to n$ in $\mathbb{P}$:
\begin{equation}\label{eqn.nat_means_homo}
\begin{tikzcd}
	c\tpow{m}\ar[r, "s_c(\mu)"]\ar[d, "f\tpow{m}"']&
	c\tpow{n}\ar[d, "f\tpow{n}"]\\
	d\tpow{m}\ar[r, "s_d(\mu)"']&
	d\tpow{n}
\end{tikzcd}
\end{equation}
If every morphism in $\cat{C}$ is an $s$-homomorphism, we say that $s$ is a \emph{homomorphic supply}.
\end{definition}

\begin{remark}\label{rem.strict}
Note that if $s_c$ is strict, then conditions (i) and (ii) can be replaced by the condition $s_c(1)=c$. Moreover, if $\cat{C}$ is strict, then each $s_c$ must be too.
\end{remark}

\begin{example}\label{ex.supply_ids}
Let $\bb$ denote the initial prop: $\bb(m,n)$ is the set of bijections $\ord{m}\cong\ord{n}$. For any monoidal category $\cat{C}$ and object $c\in\cat{C}$, there is a unique strong monoidal functor $s_c\colon\bb\to\cat{C}$ satisfying conditions (i) and (ii) of \cref{def.supply}. The only morphisms in $\bb$ are the symmetries, and a computation shows that condition (iii) holds. Every morphism in $\cat{C}$ is an $s$-homomorphism.

One might thus say that every symmetric monoidal category $\cat{C}$ uniquely \emph{supplies symmetries}, and every morphism in $\cat{C}$ is a homomorphism for symmetries.

Note that condition (ii) of \cref{def.supply} is necessary for the supply of $\bb$ to be unique.
\end{example}

\begin{remark}
One could give an alternative definition of supply by dropping conditions (i) and (ii) from \cref{def.supply}. This relaxed definition should be equivalent to ours in an appropriate sense. However we made the choice we did in order to cut down on the number of equivalent supplies. For example, we appreciate the fact from \cref{ex.supply_ids} that there is a unique supply of symmetries in any symmetric monoidal category. Props are syntactic in nature (a monoidal category equivalent to a prop is not generally a prop), and our choice was made in order to match that syntactic aesthetic.
\end{remark}

\begin{example}\label{ex.terminal_supply}
Let $\cat{I}=\{*\}$ denote the zero object in $\CCat{SMC}$ (see \cref{thm.smc_biprod}). For any prop $\mathbb{P}$ there is a unique supply of $\mathbb{P}$ in $\cat{I}$.
\end{example}

\begin{example}[Involutions]\label{ex.supply_involutions}
Consider the prop $\ccat{I}$ whose morphisms are given as follows:
\[
  \ccat{I}(m,n)=
  \begin{cases}
  	\emptyset&\tn{ if }m\neq n\\
		\{\id_m, i_m\}&\tn{ if }m=n
  \end{cases}
 \]
 with $i_m\mathbin{\fatsemi} i_m=\id_m$ and $i_m+i_n=i_{m+n}$. If $\cat{C}$ supplies $\ccat{I}$, we say it \emph{supplies involutions}. That means that every object $c\in\cat{C}$ is equipped with an involution $i_c\colon c\to c$, compatible with tensor products in the sense that $i_{c\otimes d}=i_c\otimes i_d$ and $i_I=\id_I$.

For a morphism $f\colon c\to d$ to be an involution-homomorphisms just means that $f$ commutes with the chosen involutions, i.e.\ $f\mathbin{\fatsemi} i_d=i_c\mathbin{\fatsemi} f$.
\end{example}

\begin{example}[Initial objects] \label{ex.term_init}
	Let $\mathbb{P}$ be the prop generated by a unique map $\eta\colon 0 \to 1$. The monoidal unit of a symmetric monoidal category is an initial object iff the category homomorphically supplies $\mathbb{P}$. Dually, the monoidal unit is an terminal object iff the category supplies $\mathbb{P}^{\tn{op}}$.
\end{example}

\begin{example}[Monoids]\label{def.prop_monoids}
The \emph{prop for commutative monoids} is given by two generators
\begin{equation}\label{eqn.gen_monoid}
\begin{tikzpicture}[WD, baseline=(eta)]
	\node[bb={0}{1}, fill=white] (eta) {$\eta$};
	\draw (eta_out1) -- +(.8,0);
	\node[bb={2}{1}, fill=white, right=4 of eta] (mu) {$\mu$};
	\draw (mu_out1) -- +(.8,0);
	\draw (mu_in1) -- +(-.8,0);
	\draw (mu_in2) -- +(-.8,0);
\end{tikzpicture}
\end{equation}
and three equations:
\begin{equation}\label{eqn.rel_monoid}
\begin{tikzpicture}[baseline=(a)]
	\node (P11) {
	\begin{tikzpicture}[WD]
		\node[bb={2}{1}, fill=white] (a) {};
		\coordinate (a1) at ($(a_in1)-(1,0)$);
		\coordinate (a2) at ($(a_in2)-(1,0)$);
		\draw (a1) to[in=180, out=0] (a_in2);
		\draw (a2) to[in=180, out=0] (a_in1);
		\draw (a_out1) -- +(.5,0);
	\end{tikzpicture}
	};
	\node (P12) [right=.8 of P11] {
	\begin{tikzpicture}[WD]
		\node[bb={2}{1}, fill=white] (a) {};
		\draw (a_in1) -- +(-.5,0);
		\draw (a_in2) -- +(-.5,0);
		\draw (a_out1) -- +(.5,0);
	\end{tikzpicture}
	};
	\node at ($(P11.east)!.5!(P12.west)$) {$=$};
	\node (P21) [right=1 of P12] {
  \begin{tikzpicture}[WD]
  	\node[bb={2}{1}, fill=white] (a1) {};
  	\node[bb={0}{1}, fill=white, left=.5 of a1_in1] (a2) {};
  	\draw (a2_out1) -- (a1_in1);
  	\draw (a1_in2) -- +(-2,0);
  	\draw (a1_out1) -- +(.5,0);
	\end{tikzpicture}
	};
	\node (P22) [right=.8 of P21] {
	\begin{tikzpicture}[WD]
		\draw (0,0) -- (2,0);
	\end{tikzpicture}
	};
	\node at ($(P21.east)!.5!(P22.west)$) {$=$};
	\node (P31) [right=1 of P22] {
	\begin{tikzpicture}[WD]
		\node[bb={2}{1}, fill=white] (a1) {};
		\node[bb={2}{1}, fill=white, minimum height=1ex, left=.5 of a1_in1] (a2) {};
		\draw (a2_out1) -- (a1_in1);
		\draw (a1_in2) -- +(-2,0);
		\draw (a2_in1) -- +(-.5,0);
		\draw (a2_in2) -- +(-.5,0);
		\draw (a1_out1) -- +(.5,0);
	\end{tikzpicture}
	};
	\node (P32) [right=.8 of P31] {
	\begin{tikzpicture}[WD]
		\node[bb={2}{1}, fill=white] (a1) {};
		\node[bb={2}{1}, fill=white, minimum height=1ex, left=.5 of a1_in2] (a2) {};
		\draw (a2_out1) -- (a1_in2);
		\draw (a1_in1) -- +(-2,0);
		\draw (a2_in1) -- +(-.5,0);
		\draw (a2_in2) -- +(-.5,0);
		\draw (a1_out1) -- +(.5,0);
	\end{tikzpicture}
	};
	\node at ($(P31.east)!.5!(P32.west)$) {$=$};
\end{tikzpicture}
\end{equation}
It is equivalent to the skeleton of $\Cat{FinSet}$, i.e. with $\Hom(m,n)\coloneqq\Cat{Set}(\ord{m},\ord{n})$. For example, the generators shown in \cref{eqn.gen_monoid} correspond to the unique functions $\ord{0}\to\ord{1}$ and $\ord{2}\to\ord{1}$ respectively.

A supply of commutative monoids in $\cat{C}$ gives a map $\mu_c\colon c\otimes c\to c$ and $\eta_c\colon I\to c$ for each object $c$, compatible with tensor product in $\cat{C}$ and satisfying the usual monoid laws. A morphism $f\colon c\to d$ is a monoid-homomorphism in the sense of \cref{def.supply} iff it is in the usual sense: $\mu_c\mathbin{\fatsemi} f=(f\otimes f)\mathbin{\fatsemi} \mu_d$ and $\eta_c\mathbin{\fatsemi} f=\eta_d$.

Similarly, to supply commutative comonoids means to supply the prop given by the skeleton of $\Cat{FinSet}^{\tn{op}}$.
\end{example}

\begin{example}[Cartesian categories]\label{ex.cart_grant_comonoids}
A symmetric monoidal category has finite products iff it homomorphically supplies commutative comonoids. In this case, the categorical product coincides with the monoidal product. This was shown in \cite{fox1976coalgebras}.
\end{example}

\begin{example}[Compact closed categories]\label{ex.self_duals}
The \emph{prop $\mathbb{D}$ for self-duals} has two generators
\[
\begin{tikzpicture}[WD, font=\tiny, light gray nodes]
	\node[bb={2}{0}] (a) {};
	\node[bb={0}{2}, right=5 of a] (b) {};
	\draw (a_in1) -- +(-.5, 0);
	\draw (a_in2) -- +(-.5, 0);
	\draw (b_out1) -- +(.5, 0);
	\draw (b_out2) -- +(.5, 0);
	\node[fill=white, inner sep=1pt, right=10pt of b] {};
	\node[fill=white, font=\normalsize] at ($(a.east)!.5!(b.west)$) {and};
\end{tikzpicture}
\]
and four equations
\begin{equation}\label{eqn.self_duals}
\begin{tikzpicture}
	\node (Q1) {
\begin{tikzpicture}[WD, font=\tiny, bb port length=.5, light gray nodes]
	\node[bb={2}{0}] (a) {};
\end{tikzpicture}
	};
	\node (Q2) [right=.5 of Q1] {
\begin{tikzpicture}[WD, font=\tiny, bb port length=.2, light gray nodes]
	\node[bb={2}{0}] (a) {};
	\coordinate (end) at ($(a_in1)+(-1,0)$);
	\draw (a_in1) to[out=180, in=0] (a_in2-|end);
	\draw (a_in2) to[out=180, in=0] (a_in1-|end);
\end{tikzpicture}
	};
	\node (Q3) [right=.9 of Q2]{
\begin{tikzpicture}[WD, font=\tiny, bb port length=.5, light gray nodes]
	\node[bb={0}{2}] (a) {};
\end{tikzpicture}
	};
	\node (Q4) [right=.5 of Q3] {
\begin{tikzpicture}[WD, font=\tiny, bb port length=.2, light gray nodes]
	\node[bb={0}{2}] (a) {};
	\coordinate (end) at ($(a_out1)+(1,0)$);
	\draw (a_out1) to[out=0, in=180] (a_out2-|end);
	\draw (a_out2) to[out=0, in=180] (a_out1-|end);
\end{tikzpicture}
	};
  \node at ($(Q1.east)!.5!(Q2.west)$) {$=$};
  \node at ($(Q3.east)!.5!(Q4.west)$) {$=$};
	\node (P1) [right=.9 of Q4] {
  \begin{tikzpicture}[WD, light gray nodes]
  	\node[bb={0}{2}] (a) {};
  	\node[bb={2}{0}, above right=-.75 and .5 of a] (b) {};
  	\draw (a_out1) -- (b_in2);
  	\draw (a_out2) -- +(2, 0);
  	\draw (b_in1) -- +(-2, 0);
  \end{tikzpicture}
  };
  \node (P2) [right=.5 of P1] {
  \begin{tikzpicture}[WD]
  	\draw (0,0) -- (1.5,0);
  \end{tikzpicture}
  };
  \node at ($(P1.east)!.5!(P2.west)$) {$=$};
	\node (P3) [right=.9 of P2]{
  \begin{tikzpicture}[WD, light gray nodes]
  	\node[bb={0}{2}] (a) {};
  	\node[bb={2}{0}, below right=-.75 and .5 of a] (b) {};
  	\draw (a_out2) -- (b_in1);
  	\draw (a_out1) -- +(2, 0);
  	\draw (b_in2) -- +(-2, 0);
  \end{tikzpicture}
  };
  \node (P4) [right=.5 of P3] {
  \begin{tikzpicture}[WD]
  	\draw (0,0) -- (1.5,0);
  \end{tikzpicture}
  };
  \node at ($(P3.east)!.5!(P4.west)$) {$=$};
\end{tikzpicture}
\end{equation}
$\mathbb{D}$ is equivalent to the symmetric monoidal category of unoriented 1-cobordisms, though we will not need that fact.

A category suppling self-duals is called a \emph{self-dual compact closed} category.
\end{example}

\begin{example}\label{ex.mat}
  Let $\Cat{Mat}$ be the symmetric monoidal category with finite-dimensional real vector spaces $\mathbb{R}^n$ as objects, matrices between them as morphisms, Kronecker product $\otimes$ of matrices as tensor product. Then $\Cat{Mat}$ supplies self-duals, and the supply homomorphisms are orthogonal matrices.
\end{example}

\begin{example}\label{ex.frob_mon}
The prop for (special, commutative) frobenius monoids is $\Cat{Cospan}$, the category of cospans in $\Cat{FinSet}$. A category supplying frobenius monoids is called a \emph{hypergraph} category; see \cite{fong2019hypergraph}.
\end{example}

\begin{proposition}\label{prop.p_supplies_itself}
Let $\mathbb{P}$ be a prop. Then there is a supply of $\mathbb{P}$ in $\mathbb{P}$.
\end{proposition}
\begin{proof}
The monoidal product in a prop is denoted $+$; we denote the $n$-fold monoidal product of some $k$ by $k\cdot n\coloneqq k+\Cdots{n}+k$.

For any $k\in\mathbb{P}$ let $s_k\colon\mathbb{P}\to\mathbb{P}$ act on objects by $s_k(n)=k\cdot n$; this is strict because $s_k(m+n)=k\cdot(m+n)=(k\cdot m)+(k\cdot n)$. Given $\mu\colon m\to n$ in $\mathbb{P}$, define $s_k(\mu)\colon k\cdot m\to k\cdot n$ by conjugating with the symmetries and applying $\mu$, on each of the $k$ factors:
\begin{equation}\label{eqn.conjugation}
	k\cdot m\To{\sigma_{k,m}}
	m\cdot k\To{\mu\cdot k}
	n\cdot k\To{\sigma_{n,k}}
	k\cdot n.
\end{equation}
This is functorial because $\sigma_{n,k}\mathbin{\fatsemi}\sigma_{k,n}=\id_{n\cdot k}$. It is an easy exercise to show that the diagrams in \cref{eqn.supply_commute_tensors} commute for any $k,\ell\in\mathbb{P}$.
\end{proof}

Recall that the 2-category $\CCat{SMC}$ has coproducts (\cref{thm.smc_all_prod_coprod}). It is sometimes useful to note the following basic fact, which follows immediately from the definition of supply (\cref{def.supply}) and the universal property of coproducts.

\begin{proposition}
A supply $s$ of $\mathbb{P}$ in $\cat{C}$ induces a strong monoidal functor $\bigsqcup_{c\in\ob(\cat{C})}\mathbb{P}\to\cat{C}$ that is surjective on objects.
\end{proposition}

\section{An equivalent definition}\label{sec.main_thm}

In this section we prove our main theorem, \cref{thm.coherence_isos_homos}, which says that all coherence isomorphisms---associators, unitors, and braiding---are homomorphisms for any supply. We use it to provide a slightly more compact definition of supply in \cref{thm.supply_v2}.

\begin{theorem}\label{thm.coherence_isos_homos}
Suppose $s$ is a supply of $\mathbb{P}$ in $\cat{C}$. All the coherence isomorphisms in $\cat{C}$ (associators, unitors, and braiding) are $s$-homomorphisms.
\end{theorem}
\begin{proof}
Choose any $\mu\colon m\to n$ in $\mathbb{P}$. We need to show that whenever $f\colon x\to y$ is an associator, a unitor, or a braiding, the following diagram commutes:
\[
 \begin{tikzcd}
	x\tpow{m}\ar[d, "s_x(\mu)"']\ar[r, "f\tpow{m}"]&
	y\tpow{m}\ar[d, "s_y(\mu)"]\\
	x\tpow{n}\ar[r, "f\tpow{n}"']&
	y\tpow{n}
\end{tikzcd}
\]
When $f$ is the associator $(a\otimes b)\otimes c\to a\otimes (b\otimes c)$ we consider the following diagram:
\[
\begin{tikzcd}[row sep=24pt, font=\small, column sep=25pt]
  \big((a\otimes b)\otimes c\big)\tpow{m}\ar[d, "s(\mu)_{(a\otimes b)\otimes c}" description]&
  (a\tpow{m}\otimes b\tpow{m})\otimes c\tpow{m}\ar[l, "\sigma"']\ar[r, "\alpha"]\ar[d, "(s(\mu)_a\otimes s(\mu)_b)\otimes s(\mu)_c" description]&
  a\tpow{m}\otimes (b\tpow{m}\otimes c\tpow{m})\ar[r, "\sigma"]\ar[d, "s(\mu)_a\otimes (s(\mu)_b\otimes s(\mu)_c)" description]&
  \big(a\otimes (b\otimes c)\big)\tpow{m}\ar[d, "s(\mu)_{a\otimes (b\otimes c)}" description]
  \\
  \big((a\otimes b)\otimes c\big)\tpow{n}&
  (a\tpow{n}\otimes b\tpow{n})\otimes c\tpow{n}\ar[l, "\sigma"]\ar[r, "\alpha"']&
  a\tpow{n}\otimes (b\tpow{n}\otimes c\tpow{n})\ar[r, "\sigma"']&
  \big(a\otimes (b\otimes c)\big)\tpow{n}
\end{tikzcd}
\]
The left- and right-hand squares commute by two applications of the left-hand diagram in \cref{eqn.supply_commute_tensors}, while the center square is just the naturality of the associator. Replacing the leftward horizontal maps by their inverses, Mac Lane's coherence theorem implies the composite horizontal maps are simply the relevant tensor powers of associators. Moreover, the diagram still commutes, and hence associators are $s$-homomorphisms.

The argument that braidings are homomorphisms is strictly analogous to the above. The argument that unitors are homomorphisms is almost analogous, but the proof requires also the commutativity of the right-hand diagram in \cref{eqn.supply_commute_tensors}. Indeed, consider the following diagram:
\[
\begin{tikzcd}[row sep=30pt]
	(a\otimes I)\tpow{m}\ar[d, "s(\mu)_{a\otimes I}"']&
	a\tpow{m}\otimes I\tpow{m}\ar[d, "s(\mu)_a\otimes s(\mu)_I" description]\ar[l, "\sigma"']&[10pt]
	a\tpow{m}\otimes I\ar[d, "s(\mu)_a\otimes I" description]\ar[r, "\rho"]\ar[l, "a\tpow{m}\otimes\sigma"']&
	a\tpow{m}\ar[d, "s(\mu)_a"]\\
	(a\otimes I)\tpow{n}&
	a\tpow{n}\otimes I\tpow{n}\ar[l, "\sigma"]&
	a\tpow{n}\otimes I\ar[l, "a\tpow{n}\otimes\sigma"]\ar[r, "\rho"']&
	a\tpow{n}
\end{tikzcd}
\]
Its left-hand and middle diagrams commute by \cref{eqn.supply_commute_tensors} and the right-hand diagram commutes by the unitor axiom.
\end{proof}

We can use \cref{thm.coherence_isos_homos} to provide a more compact definition of supply. To do so, we need the following definition, which puts all the coherence isomorphisms in $\cat{C}$ into a single monoidal subcategory, denoted $\mob{\cat{C}}$.

\begin{definition}\label{def.mob}
For any symmetric monoidal category $\cat{C}$, let $\mob{\cat{C}}\ss\cat{C}$ denote the smallest subcategory containing
\begin{enumerate}[label=(\roman*)]
	\item all objects of $\cat{C}$ (and identity morphisms), and
	\item all coherence isomorphisms---unitors, associators, braiding, and their inverses---from $\cat{C}$.
\end{enumerate}
Thus $\mob{\cat{C}}$ inherits a symmetric monoidal structure, and we refer to it as the \emph{symmetric monoidal category of $\cat{C}$-objects}. There is an identity-on-objects inclusion $\inc\colon\mob{\cat{C}}\to\cat{C}$.
\end{definition}

\begin{example}
When $\cat{C}$ is strict monoidal, $\mob{\cat{C}}=\ob(\cat{C})$ is discrete.
\end{example}

The reader may find it useful to consider the meaning of $\inc\tpow{m}\in\Cat{SMF}(\mob{\cat{C}},\cat{C})$ for $m\in\mathbb{N}$. In particular it sends $c\mapsto c\tpow{m}=((c\otimes c)\otimes\cdots\otimes c)\otimes c$ and its strongators are the symmetry isomorphisms; see \cref{def.smf}.

\begin{theorem}\label{thm.supply_v2}
There is a one-to-one correspondence between supplies $s$ of $\mathbb{P}$ in $\cat{C}$ and strong monoidal functors $\tilde{s}\colon\mathbb{P}\to\Cat{SMF}(\mob{\cat{C}},\cat{C})$ such that
\begin{enumerate}[label=(\roman*)]
	\item $m\mapsto\inc\tpow{m}$ for each $m\in\mathbb{N}$, and
	\item the strongator $\tilde{s}(m)\otimes \tilde{s}(n)\to \tilde{s}(m+n)$ is the unique coherence map for each $m,n \in \mathbb{N}$.
\end{enumerate}
\end{theorem}\begin{proof}
A strong monoidal functor $\tilde{s}$ obeying (i) and (ii) is simply a supply $s$ of $\mathbb{P}$ in $\cat{C}$ such that the coherence maps are $s$-homomorphisms. But \cref{thm.coherence_isos_homos} shows that every supply has this property, and so the two notions coincide. We explain this in detail.

Let $\tilde{s}$ be a strong monoidal functor obeying (i) and (ii). Note that (i) defines $\tilde{s}$ on objects. On morphisms, each $\mu\colon m \to n$ defines a monoidal natural transformation $\tilde{s}(\mu)\colon \inc\tpow{m} \Rightarrow \inc\tpow{n}$. Explicitly, this is, for each object $c \in \cat{C}$, a morphism $\tilde{s}(\mu)_c \colon c\tpow{m} \to c \tpow{n}$ obeying naturality and monoidality conditions. Naturality requires
\begin{equation}\label{eqn.C_0_homos}
\begin{tikzcd}
	c\tpow{m}\ar[d, "s_c(\mu)"']\ar[r, "f\tpow{m}"]&
	d\tpow{m}\ar[d, "s_d(\mu)"]\\
	c\tpow{n}\ar[r, "f\tpow{n}"']&
	d\tpow{n}
\end{tikzcd}
\end{equation}
to commute for all maps $f\colon c \to d$ in $\mob{\cat{C}}$---that is, all coherence maps of $\cat{C}$---while monoidality requires the diagrams
\begin{equation}\label{eqn.supply_commute_tensors_v2}
\begin{tikzcd}[column sep=55pt]
	c\tpow{m}\otimes d\tpow{m}\ar[r, "\tilde{s}(\mu)_c\otimes \tilde{s}(\mu)_d"]\ar[d, "\sigma"']&
	c\tpow{n}\otimes d\tpow{n}\ar[d, "\sigma"]\\
	(c\otimes d)\tpow{m}\ar[r, "\tilde{s}(\mu)_{c\otimes d}"']&
	(c\otimes d)\tpow{n}
\end{tikzcd}
\hspace{.7in}
\begin{tikzcd}
	I\ar[r, equal]\ar[d, "\sigma"']&
	I\ar[d, "\sigma"]\\
	I\tpow{m}\ar[r, "\tilde{s}(\mu)_I"']&
	I\tpow{n}
\end{tikzcd}
\end{equation}
commute for all $m,n \in \mathbb{N}$.

The functoriality of $\tilde{s}$ requires that for all $\mu\colon m \to n$ and $\nu\colon n\to p$ we have
\[
  \tilde{s}(\mu\mathbin{\fatsemi}\nu)_c=\tilde{s}(\mu)_c\mathbin{\fatsemi} \tilde{s}(\nu)_c
\]
while the monoidality of $\tilde{s}$ with respect to the strongators given in condition (ii) imply that for all $\mu\colon m \to n $ and $\mu'\colon m' \to n'$ we have
\[
\begin{tikzcd}[column sep=55pt]
	c\tpow{m}\otimes c\tpow{m'}\ar[r, "\tilde{s}(\mu)_c\otimes \tilde{s}(\mu')_c"]\ar[d, "\alpha"']&
	c\tpow{n}\otimes c\tpow{n'}\ar[d, "\alpha"]\\
	c\tpow{m+m'}\ar[r, "\tilde{s}(\mu+\mu')_{c}"']&
	c\tpow{n+n'}.
\end{tikzcd}
\]

It is now straightforward to see [1] that the functoriality and monoidality of $\tilde{s}$ with respect to the strongators of condition (ii) states exactly that for each $c \in \cat{C}$ the component $\tilde{s}(-)_c$ defines a strong monoidal functor $\mathbb{P} \to \cat{C}$ obeying conditions (i) and (ii) of \cref{def.supply}, [2] that the monoidality diagrams \cref{eqn.supply_commute_tensors_v2} of each natural transformation $\tilde{s}(\mu)$ are exactly the diagrams \cref{eqn.supply_commute_tensors} of condition (iii) in \cref{def.supply}, and [3] that the naturality of each $\tilde{s}(\mu)$ with respect to $\mob{\cat{C}}$ is exactly the homomorphism property of \cref{thm.coherence_isos_homos}. This proves the theorem.
\end{proof}

\begin{corollary}\label{cor.strict_supply_v2}
Let $\cat{C}$ be a symmetric strict monoidal category. There is a one-to-one correspondence between supplies $s$ of $\mathbb{P}$ in $\cat{C}$ and strict monoidal functors $\tilde{s}\colon\mathbb{P}\to\Cat{SMF}(\mob{\cat{C}},\cat{C})$ such that $1 \mapsto \inc$.
\end{corollary}

Because of the one-to-one correspondence \cref{thm.supply_v2}, we often elide the difference between the supply $s$ and the strong monoidal functor $\tilde{s}$.

\begin{theorem}\label{thm.homos_form_subcat}
Let $s$ be a supply of $\mathbb{P}$ in $\cat{C}$. Then the collection of $s$-homomorphisms forms a monoidal subcategory $\mob{\cat{C}}\ss\cat{C}_s\ss\cat{C}$, and the functor $s\colon\mathbb{P}\to\Cat{SMF}(\mob{\cat{C}},\cat{C})$ factors through a strong monoidal functor
\[s\colon\mathbb{P}\to\Cat{SMF}(\cat{C}_s,\cat{C})\]
satisfying the two conditions of \cref{thm.supply_v2}.
\end{theorem}
\begin{proof}
We showed in \cref{thm.coherence_isos_homos} that every coherence isomorphism in $\cat{C}$ is an $s$-homomorphism. It is obvious that if $f\colon c\to d$ and $g\colon d\to e$ are $s$-homomorphisms then so is $f\mathbin{\fatsemi} g$. Moreover, if $f_1\colon c_1\to d_1$ and $f_2\colon c_2\to d_2$ are $s$-homomorphisms then so is $(f_1\otimes f_2)$; this follows from \cref{eqn.supply_commute_tensors_v2}. Thus $\cat{C}_s$ forms a monoidal subcategory of $\cat{C}$, and $\mob{\cat{C}}\ss\cat{C}_s$. The factoring of $s$ through $\Cat{SMF}(\cat{C}_s,\cat{C})$ is just a repackaging of the statement that every morphism in $\cat{C}_s$ is an $s$-homomorphism.
\end{proof}

\section{Transfer of supply}\label{sec.further_theory}

In this section we present a number of propositions that describe how new supplies may be constructed from old: a supply of $\mathbb{Q}$ in $\cat{C}$ induces a supply of $\mathbb{P}$ in $\cat{C}$ for any prop functor $\mathbb{P} \to \mathbb{Q}$; if $\cat{C}$ and $\cat{D}$ supply $\mathbb{P}$ then so does their biproduct $\cat{C}\oplus\cat{D}$; a supply transfers along an essentially surjective, strict monoidal functor $\cat{C} \to \cat{D}$; and a supply on $\cat{C}$ induces a supply on its strictification $\strict{C}$.

\begin{proposition}\label{cor.change_of_supply}
Let $F\colon\mathbb{P}\to\mathbb{Q}$ be a prop functor. For any supply $s$ of $\mathbb{Q}$ in $\cat{C}$, we have a supply $(F\mathbin{\fatsemi} s)$ of $\mathbb{P}$ in $\cat{C}$.
\end{proposition}
\begin{proof}
Given a strong monoidal functor $s\colon\mathbb{Q}\to\Cat{SMF}(\mob{\cat{C}},\cat{C})$, we compose it with $F$ (which is strict and sends $1\mapsto 1$) to get the required supply of $\mathbb{P}$; see \cref{thm.supply_v2}.
\end{proof}

\begin{example}
The prop $\mathbb{D}$ for self-duals was given in \cref{ex.self_duals} and that for frobenius monoids was given in \cref{ex.frob_mon}; it is $\Cat{Cospan}$. There is a prop functor $\mathbb{D}\to\Cat{Cospan}$ sending the generators\;
$
\begin{tikzpicture}[WD, font=\tiny, light gray nodes, bb port sep=.3]
	\node[bb={2}{0}] (a) {};
	\draw (a_in1) -- +(-.5, 0);
	\draw (a_in2) -- +(-.5, 0);
\end{tikzpicture}
$
\;and\;
$
\begin{tikzpicture}[WD, font=\tiny, light gray nodes, bb port sep=.3]
	\node[bb={0}{2}, right=2 of a] (b) {};
	\draw (b_out1) -- +(.5, 0);
	\draw (b_out2) -- +(.5, 0);
\end{tikzpicture}
$\;
to the cospans $2\to 1\leftarrow 0$ and $0\to 1\leftarrow 2$. It is easy to check that the equations \cref{eqn.self_duals} hold in $\Cat{Cospan}$, i.e.\ the composites
\[
\begin{tikzpicture}[x=.3cm, y=.4cm]
  \begin{scope}
  	[every node/.style={circle, fill=black, inner sep=1pt}]
  	\node (a) {};
  	\node[below right=1.25 and 1 of a] (b1) {};
  	\node[right=1 of b1] (b2) {};
  	\node at (a-|b2) (c1) {};
  	\node[right=1 of c1] (c2) {};
  	\node[right=1 of c2] (c3) {};
  	\node at (c3|-b1) (d1) {};
  	\node[right=1 of d1] (d2) {};
  	\node[right=1] at (d2|-a) (e) {};
  \end{scope}
	\node[inner sep=2.5pt, draw, fit=(a)] {};
	\node[inner sep=2.5pt, draw, fit=(b1) (b2)] {};
	\node[inner sep=2.5pt, draw, fit=(c1) (c3)] {};
	\node[inner sep=2.5pt, draw, fit=(d1) (d2)] {};
	\node[inner sep=2.5pt, draw, fit=(e)] {};
	\draw (a) -- (b1);
	\draw (c1) -- (b1);
	\draw (c2) -- (b2);
	\draw (c3) -- (b2);
	\draw (c1) -- (d1);
	\draw (c2) -- (d1);
	\draw (c3) -- (d2);
	\draw (e) -- (d2);
  \begin{scope}
  	[every node/.style={circle, fill=black, inner sep=1pt}]
  	\node[right=2cm of e] (a) {};
  	\node[below right=1.25 and 1 of a] (b1) {};
  	\node[right=1 of b1] (b2) {};
  	\node at (a-|b2) (c1) {};
  	\node[right=1 of c1] (c2) {};
  	\node[right=1 of c2] (c3) {};
  	\node at (c3|-b1) (d1) {};
  	\node[right=1 of d1] (d2) {};
  	\node[right=1] at (d2|-a) (e) {};
  \end{scope}
	\node[inner sep=2.5pt, draw, fit=(a)] {};
	\node[inner sep=2.5pt, draw, fit=(b1) (b2)] {};
	\node[inner sep=2.5pt, draw, fit=(c1) (c3)] {};
	\node[inner sep=2.5pt, draw, fit=(d1) (d2)] {};
	\node[inner sep=2.5pt, draw, fit=(e)] {};
	\draw (a) -- (b1);
	\draw (c1) -- (b1);
	\draw (c2) -- (b1);
	\draw (c3) -- (b2);
	\draw (c1) -- (d1);
	\draw (c2) -- (d2);
	\draw (c3) -- (d2);
	\draw (e) -- (d2);
\end{tikzpicture}
\]
are both the identity cospan $1=1=1$. Thus by \cref{cor.change_of_supply}, every hypergraph category is a self-dual compact closed category.
\end{example}

Recall from \cref{thm.smc_biprod} that the 2-category of symmetric monoidal categories has biproducts.

\begin{proposition}
If $\cat{C}$ and $\cat{D}$ both supply $\mathbb{P}$ then so does their biproduct $\cat{C}\oplus\cat{D}$.
\end{proposition}
\begin{proof}
Noting that $\mob{(\cat{C}\oplus \cat{D})} =\mob{\cat{C}} \oplus \mob{\cat{D}}$, this follows from \cref{prop.biprod_smf_strict,thm.supply_v2}.
\end{proof}

We next prove that supplies transfer along strict monoidal, essentially surjective functors. Note that this assumes the axiom of choice, i.e.\ that fully faithful essentially surjective functors have inverses.

\begin{proposition}\label{cor.strong_bo}
Suppose $F\colon\cat{C}\to\cat{D}$ is a strict symmetric monoidal, essentially surjective functor. If $\cat{C}$ supplies $\mathbb{P}$ then so does $\cat{D}$.
\end{proposition}
\begin{proof}
Any strict symmetric monoidal functor $F\colon\cat{C}\to\cat{D}$ induces a strict symmetric monoidal functor $\mob{F}\colon\mob{\cat{C}}\to\mob{\cat{D}}$, in fact a fully faithful functor, commuting with the inclusions. (Note that this requires \emph{strictness}; it does not in general hold for strong monoidal functors.)

Since in this case $F$ is essentially surjective, the symmetric monoidal functor $\mob{F}$ is an equivalence, thus so is $\Cat{SMF}(\mob{F},\cat{D})\colon\Cat{SMF}(\mob{\cat{D}},\cat{D})\to\Cat{SMF}(\mob{\cat{C}},\cat{D})$. Now given a supply $s$ as in the following diagram (see \cref{thm.supply_v2}), one simply defines $t$ using the inverse of the equivalence $\Cat{SMF}(\mob{F},\cat{D})$:
\[
\begin{tikzcd}[column sep=50pt,baseline=(W.base)]
	\mathbb{P}\ar[r, "s"]\ar[d, dashed, "t"']&
	\Cat{SMF}(\mob{\cat{C}},\cat{C})\ar[d, "{\Cat{SMF}(\mob{\cat{C}},F)}"]\\
	\Cat{SMF}(\mob{\cat{D}},\cat{D})\ar[r, "\cong"']& |[alias=W]|
	\Cat{SMF}(\mob{\cat{C}},\cat{D})
\end{tikzcd}
\qedhere
\]
\end{proof}

\begin{example}
Using \cref{cor.strong_bo}, a supply of comonoids on $\Cat{Rel}$ can be obtained from the one on $\Cat{Set}$ via the bijective-on-objects (hence strict) monoidal inclusion $\Cat{Set}\to\Cat{Rel}$. The supply homomorphisms are precisely the functional relations \cite{fong2019regular}.
\end{example}

\begin{remark}
  Supplies do not in general transfer along equivalences of categories. For an example, see \cite[Example 2.20]{fong2019hypergraph}, which gives an equivalence of symmetric monoidal categories together with a hypergraph structure on one that cannot be transferred to the other.
\end{remark}

The failure of supplies to in general transfer along equivalences notwithstanding, we close this section by proving that if $\cat{C}$ supplies $\mathbb{P}$, then so does its Mac Lane strictification.

\begin{lemma}\label{lemma.strictification}
Let $\cat{C}$ a symmetric monoidal category, $\strict{\cat{C}}$ its strictification, and $\bigotimes\colon\strict{\cat{C}}\to\cat{C}$ the strong monoidal equivalence. For any prop $\mathbb{P}$, a strong monoidal functor $F\colon\mathbb{P}\to\cat{C}$ factors as $\mathbb{P}\To{\strict{F}}\strict{\cat{C}}\To{\bigotimes}\cat{C}$, for some strict monoidal functor $\strict{F}$ iff
\begin{enumerate}[label=(\roman*)]
	\item $F(m)=F(1)\tpow{m}$ for all $m\in\mathbb{N}$, and
	\item the strongator $F(m)\otimes F(n)\to F(m+n)$ is the unique coherence map for all $m,n \in \mathbb{N}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Clearly if $F$ factors as $\strict{F}\mathbin{\fatsemi}\bigotimes$ then it satisfies the two conditions. Conversely, if $F$ satisfies the two conditions, define $\strict{F}$ on objects by $\strict{F}(m)\coloneqq[F(1),\Ldots{m},F(1)]$; note that $\strict{F}(m)\mathbin{\fatsemi}\bigotimes=F(m)$. On morphisms define $\strict{F}$ to be the composite
\[\mathbb{P}(m,n)\To{F}\cat{C}(F(m),F(n))\To{\cong}\strict{\cat{C}}(\strict{F}(m),\strict{F}(n)),\]
where the second map is the isomorphism coming from the fact that $\bigotimes$ is fully faithful. It is clear both that $\strict{F}$ is strict and that its composite with $\bigotimes$ is $F$.
\end{proof}

\begin{proposition}\label{thm.supply_strictification}
For any supply on $\cat{C}$, there is an induced supply on its strictification $\strict{\cat{C}}$.
\end{proposition}
\begin{proof}
Let $\mathbb{P}$ be a prop, and suppose $s$ is a supply of $\mathbb{P}$ in $\cat{C}$. For each $c\in\cat{C}$, \cref{lemma.strictification} says that the map $s_c\colon\mathbb{P}\to\cat{C}$ factors through a strict monoidal functor $s_{[c]}\colon\mathbb{P}\to\strict{\cat{C}}$ sending $1\mapsto[c]$. It immediately satisfies conditions (i) and (ii) of \cref{def.supply}, and it satisfies condition (iii) because $\bigotimes\colon\strict{\cat{C}}\to\cat{C}$ is faithful.

For an arbitrary object $c=[c_1,\ldots,c_k]\in\strict{C}$, define $\strict{s}_c\colon\mathbb{P}\to\strict{\cat{C}}$ on each object $m\in\mathbb{N}$ by $\strict{s}_c(m)\coloneqq[c,\Ldots{m},c]$ and on each morphism $\mu$ by conjugating with the symmetries:
\[
\begin{tikzcd}[column sep=90pt]
	{[c_1,\Ldots{m},c_1] \cdot \ldots \cdot [c_k,\Ldots{m}, c_k]}
		\ar[r, "{s_{[c_1]}(\mu) \cdot \ldots \cdot s_{[c_k]}(\mu)}"]&
	{[c_1,\Ldots{n},c_1] \cdot \ldots\cdot [c_k,\Ldots{n}, c_k]}
		\ar[d, "\sigma"]\\
	{[c_1,\ldots, c_k] \cdot \Ldots{m} \cdot [c_1,\ldots,c_k]}
		\ar[u, "\sigma"]\ar[r, dashed, "\strict{s}_c(\mu)"']&
	{[c_1,\ldots, c_k] \cdot \Ldots{n} \cdot [c_1,\ldots,c_k]}
\end{tikzcd}
\]
where we have written $\cdot$ for the monoidal product in $\strict{\cat{C}}$, namely list concatenation.
With this assignment, $\strict{s}$ is easily seen to be a supply of $\mathbb{P}$ in $\strict{\cat{C}}$.
\end{proof}
Once we have defined preservation of supply in \cref{sec.pres_supply}, we will see immediately that the equivalence $\strict{\cat{C}}\to\cat{C}$ is supply-preserving; see \cref{cor.strict_equiv_pres_supply}.

\chapter{Preservation of supply}\label{sec.pres_supply}

In \cref{sec.pres_supply} we define preservation of supply---i.e.\ the notion of \emph{homomorphism} between categories equipped with supply---and give some basic examples. In \cref{sec.preserve_thms} we prove some useful properties of supply-preserving functors. Of these, the most important is \cref{thm.pres_supp_strongators_homo}, which says that for any strong monoidal functor preserving supply, the strongators are homomorphisms.

\section{Definition and examples}\label{sec.pres_supply}

\begin{definition}[Preserves supply]\label{def.preserve_supply}
Let $\mathbb{P}$ be a prop, $\cat{C}$ and $\cat{D}$ symmetric monoidal categories, and suppose $s$ is a supply of $\mathbb{P}$ in $\cat{C}$ and $t$ is a supply of $\mathbb{P}$ in $\cat{D}$. We say that a strong symmetric monoidal functor $(F,\varphi)\colon\cat{C}\to\cat{D}$ \emph{preserves the supply} if the strongators $\varphi$ provide a natural isomorphism $t_{Fc}\cong (s_c\mathbin{\fatsemi} F)$ of functors $\mathbb{P}\to\cat{D}$ for each $c\in\cat{C}$.
\end{definition}

Unpacking, a strong monoidal functor $(F,\varphi)$ preserves the supply iff the diagram
\begin{equation}\label{eqn.unpack_preserve_supply}
	\begin{tikzcd}[column sep=large]
  	F(c)\tpow{m}\ar[r, "t_{F(c)}(\mu)"]\ar[d, "\varphi"', "\cong"]&
  	F(c)\tpow{n}\ar[d, "\varphi", "\cong"']\\
  	F(c\tpow{m})\ar[r, "F(s_c(\mu))"']&
  	F(c\tpow{n})
  \end{tikzcd}
\end{equation}
commutes for each morphism $\mu\colon m\to n$ in $\mathbb{P}$ and object $c\in\cat{C}$.

\begin{example}
Taking $\mathbb{P}=\bb$ as in \cref{ex.supply_ids} we see that every strong monoidal functor $\cat{C}\to\cat{D}$ preserves the supply of symmetries.
\end{example}

\begin{example}
Let $s$ be a supply of $\mathbb{P}$ in $\cat{C}$. Recall that there is a unique supply of $\mathbb{P}$ on $\cat{I}$ by \cref{ex.terminal_supply}. It follows from the second diagram in \cref{eqn.supply_commute_tensors} that the unique monoidal functor $\cat{I}\to\cat{C}$ preserves the $\mathbb{P}$-supply (and clearly so does $\cat{C}\to\cat{I}$).
\end{example}

\begin{example}\label{ex.preserve_involutions}
Suppose we have a supply $s$ of involutions in $\cat{C}$ and a supply $t$ of involutions in $\cat{D}$. As we saw in \cref{ex.supply_involutions} this just means that every object $x$ is equipped with an involution $i_x\colon x\cong x$. A symmetric monoidal functor $F\colon\cat{C}\to\cat{D}$ preserves the supply iff $F(i_x)=i_{F(x)}$.
\end{example}

\begin{example}
  A hypergraph functor is defined to be a strong symmetric monoidal functor between hypergraph categories that preserves the supply of frobenius monoids.
\end{example}

\begin{remark}
  Note that \cref{def.preserve_supply} permits a straightforward generalization to lax monoidal functors. We use this stronger definition as all the examples we are aware of use strong monoidal functors, and because this structure is used in the results below.
\end{remark}

\section{Basic theory of preservation}\label{sec.preserve_thms}

\begin{theorem}\label{thm.pres_supp_strongators_homo}
Let $s$ be a supply of $\mathbb{P}$ in $\cat{C}$ and let $t$ be a supply of $\mathbb{P}$ in $\cat{D}$, and suppose that $(F,\varphi)\colon\cat{C}\to\cat{D}$ is a strong monoidal functor preserving supply. Then the strongators $\varphi$ are $t$-homomorphisms, i.e.\ the following diagrams commute for each morphism $\mu\colon m\to n$ in $\mathbb{P}$ and objects $c,c'\in\cat{C}$:
\[
\begin{tikzcd}[column sep=50pt]
	(Fc\otimes Fc')\tpow{m}
		\ar[r, "t_{Fc\otimes Fc'}(\mu)"]\ar[d, "(\varphi_{c,c'})\tpow{m}"']&
 	(Fc\otimes Fc')\tpow{n}\ar[d, "(\varphi_{c,c'})\tpow{n}"]\\
  F(c\otimes c')\tpow{m}\ar[r, "t_{F(c\otimes c')}(\mu)"']&
  F(c\otimes c')\tpow{n}
\end{tikzcd}
\hspace{.7in}
\begin{tikzcd}[column sep=35pt]
	I\tpow{m}
		\ar[r, "t_I(\mu)"]\ar[d, "\varphi\tpow{m}"']&
 	I\tpow{n}\ar[d, "\varphi\tpow{n}"]\\
  F(I)\tpow{m}\ar[r, "t_{F(I)}(\mu)"']&
  F(I)\tpow{n}
\end{tikzcd}
\]
\end{theorem}
\begin{proof}
Each of these is proved by a diagram chase. Indeed, consider the diagram:
\[
\begin{tikzcd}[column sep=6pt]
	(Fc\otimes Fc')\tpow{m}
		\ar[rrrr, "t_{Fc\otimes Fc'}(\mu)"]
		\ar[ddddd, "\varphi\tpow{m}"']
		\ar[from=dr, "\sigma"]&&[18pt]
		\ar[d, phantom, near start,  "\textsc{\tiny (t)}"]&[18pt]&
	(Fc\otimes Fc')\tpow{n}
		\ar[ddddd, "\varphi\tpow{n}"]
		\ar[from=dl, "\sigma"']\\&
	(Fc)\tpow{m}\otimes(Fc')\tpow{m}
		\ar[d, "\varphi"']
		\ar[rr, "t_{Fc}(\mu)\otimes t_{Fc'}(\mu)"]&
		\ar[d, phantom, near start, "\textsc{\tiny (Fp)}"]&
	(Fc)\tpow{n}\otimes(Fc')\tpow{n}
		\ar[d, "\varphi"]\\&
	F(c\tpow{m})\otimes F(c'\,\tpow{m})
		\ar[rr, "Fs_c(\mu)\otimes Fs_{c'}(\mu)"]
		\ar[d, "\varphi"']&
		\ar[d, phantom, near start, "\textsc{\tiny (Fm)}"]&
	F(c\tpow{n})\otimes F(c'\,\tpow{n})
		\ar[d, "\varphi"]\\&
	F(c\tpow{m}\otimes c'\,\tpow{m})
		\ar[rr, "F(s_c(\mu)\otimes s_{c'}(\mu))"]&
		\ar[d, phantom, near start, "\textsc{\tiny (s)}"]&
	F(c\tpow{n}\otimes c'\,\tpow{n})\\&
	F((c\otimes c')\tpow{m})
		\ar[u, "\sigma"]
		\ar[rr, "F(s_{c\otimes c'}(\mu))"]&
		\ar[d, phantom, near start, "\textsc{\tiny (Fp)}"]&
	F((c\otimes c')\tpow{n})
		\ar[u, "\sigma"']\\
	F(c\otimes c')\tpow{m}
		\ar[rrrr, "t_{F(c\otimes c')}(\mu)"]
		\ar[ur, "\varphi"]&&~&&
	F(c\otimes c')\tpow{n}
		\ar[ul, "\varphi"']
\end{tikzcd}
\]
Every vertical or diagonal morphism is an isomorphism, and the (unlabeled) side diagrams commute because symmetries commute with strongators. Diagrams \textsc{(s)} and \textsc{(t)} commute because $s$ and $t$ are supplies (see \cref{eqn.supply_commute_tensors}); diagrams \textsc{(Fp)} commute because $F$ preserves the supply (see \cref{eqn.unpack_preserve_supply}); and \textsc{(Fm)} commutes because $F$ is monoidal.

The proof for the unit is similar, except that squares \textsc{(Fp)} are not present.
\end{proof}

Recall from \cref{thm.homos_form_subcat} that for any supply $s$ in $\cat{C}$ there is a symmetric monoidal subcategory $\cat{C}_s\ss\cat{C}$ of $s$-homomorphisms.

\begin{proposition}\label{prop.preserve_homs}
If $F$ preserves supply, it sends homomorphisms to homomorphisms, i.e.\ it restricts to a strong monoidal functor $F_{s,t}\colon\cat{C}_s\to\cat{D}_t$.
\end{proposition}
\begin{proof}
Choose $\mu\colon m\to n$ in $\mathbb{P}$ and $f\colon c\to d$ in $\cat{C}$, and consider the diagram below:
\[
\begin{tikzcd}[column sep=6pt]
	(Fc)\tpow{m}
		\ar[rrrr, "t_{Fc}(\mu)"]
		\ar[dddd, "(Ff)\tpow{m}"']
		\ar[dr, "\varphi"]
		&
    &[23pt]
    \ar[d, phantom, near start,  "\textsc{\tiny (Fp)}"]
		&[23pt]
    &
  (Fc)\tpow{n}
		\ar[dddd, "(Ff)\tpow{n}"]
		\ar[dl, "\varphi"']
		\\
		&
	F(c\tpow{m})
		\ar[dd, "F(f\tpow{m})"]
    \ar[rr, "F(s_{c}(\mu))"]
    &
    ~
    &
	F(c\tpow{n})
    \ar[dd, "F(f\tpow{n})"']
    \\
    \ar[r, phantom,"\textsc{\tiny (Fm)}"]
    &~&
    \textsc{\tiny (s)}
    &~&
    \ar[l, phantom,"\textsc{\tiny (Fm)}"]
		\\
		&
	F(d\tpow{m})
		\ar[rr, "F(s_d(\mu))"']
    \ar[dl, "\varphi"']
    &
    \ar[d, phantom, pos=.6, "\textsc{\tiny (Fp)}"]
    &
	F(d\tpow{n})
    \ar[dr, "\varphi"]
    \\
	(Fd)\tpow{m}
		\ar[rrrr, "t_{Fd}(\mu)"']
		&&~&&
	(Fd)\tpow{n}
\end{tikzcd}
\]
The diagrams \textsc{(Fp)} commute because $F$ preserves supply, while the diagrams \textsc{(Fm)} commute because $F$ is monoidal. Thus whenever $f$ is an $s$-homomorphism, the functoriality of $F$ implies \textsc{(s)} commutes, and hence that $F(f)$ is a $t$-homomorphism.
\end{proof}

\begin{proposition}\label{prop.easy_pres_supply}
Let $s$ and $t$ be supplies of $\mathbb{P}$ in $\cat{C}$ and $\cat{D}$ respectively. Then $F\colon\cat{C}\to\cat{D}$ preserves the supply iff
\begin{itemize}
	\item $F$ sends coherence maps to $t$-homomorphisms, i.e.\ it factors through some $F_{0,t}\colon\cat{C}_0\to\cat{D}_t$
	\item the strongators $\varphi_c\colon F(c)\tpow{m}\to F(c\tpow{m})$ define a natural isomorphism:
\begin{equation}\label{eqn.pres_supply}
\begin{tikzcd}[column sep=55pt]
	\mathbb{P}\ar[r, "s"]\ar[d, "t"']&
	\Cat{SMF}(\cat{C}_0,\cat{C})\ar[d, "{\Cat{SMF}(\cat{C}_0,F)}"]\\
	\Cat{SMF}(\cat{D}_t,\cat{D})\ar[r, "{\Cat{SMF}(F_{0,t},\cat{D})}"']&
	\Cat{SMF}(\cat{C}_0,\cat{D})\ar[ul, phantom, "\overset{\varphi}{\cong}"]
\end{tikzcd}
\end{equation}
\end{itemize}
\end{proposition}
\begin{proof}
By \cref{prop.preserve_homs}, if $F$ preserves the supply then it sends coherence maps (in fact all $s$-homomorphisms) to $t$-homomorphisms. So we may assume we have $F_{0,t}$ and prove that $F$ preserves the supply iff diagram \eqref{eqn.pres_supply} commutes.

Consider an object $m\in\mathbb{P}$. Along the top-right, it is sent to the functor $c\mapsto F(c\tpow{m})$, and along the left-bottom, it is sent to the functor $c\mapsto F(c)\tpow{m}$. The strongators for $F$ provide the component isomorphisms $\varphi_c\colon F(c)\tpow{m}\to F(c\tpow{m})$ natural in $c\in\cat{C}$ (and hence in $c\in\cat{C}_0$). For these $\varphi$ to be natural in $\mathbb{P}$ works out to be exactly the condition that  \cref{eqn.unpack_preserve_supply} commutes for any morphism $\mu\colon m\to n$ in $\mathbb{P}$.
\end{proof}

\cref{cor.strict_equiv_pres_supply} follows easily from the proof of \cref{thm.supply_strictification}.

\begin{corollary}\label{cor.strict_equiv_pres_supply}
Let $s$ be a supply of $\mathbb{P}$ in $\cat{C}$ and let $\strict{s}$ be the induced supply of $\mathbb{P}$ in the strictification $\strict{\cat{C}}$. Then the equivalence $\bigotimes\colon\strict{\cat{C}}\to\cat{C}$ preserves the supply.
\end{corollary}

\chapter{Outlook}

Many of the ideas in this paper should extend to the enriched setting, e.g.\ replacing props and symmetric monoidal categories with 2-props and symmetric monoidal 2-categories, etc. Indeed, in \cite{fong2019abelian}, we work out the theory for the locally posetal case. The results contained here, and their locally posetal generalizations, organize and significantly streamline key arguments in that paper.

We leave the development of the general enriched theory open for future work.

\printbibliography

\newpage

redpajama_set_name: RedPajamaArXiv


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Legal forms, documents and contracts in the UK.
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Okabe & Haushalter are cyber crime and internet crime lawyers who can defend you against state or federal charges involving crimes committed over the internet.
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text:
Q: Contador no FastReport soma mais de uma vez, como resolver? Tenho um relatorio bem complexo no fastreport em uma determinada página a estrutura é um MasterData -> DetailData e dentro deste Detail tenhos os requeridos componentes, funciona tudo 100%, acontece que em um fxtMemo preciso colocar o numero da planilha este irá numerar em sequencia cada planilha do Detail.
Fiz o seguinte, no codigo criei uma variavel global ContaPlanilha: integer = 0;
No BeforePrint do frxMemo coloco: ContaPlanilha := ContaPlanilha +1;
No text do frxMemo coloquei "Planilha número: [ContaPlanilha]"
Ele pega o valor da variavel e coloca no frxMemo como deve ser, acontece que mesmo tendo somente 1 registro teste, a variavel tem o valor setado como 2 é como se estivesse passando duas vezes no OnBeforePrint, mas como dito só tem 1 unico registro...
Estou usando "Duplo Passo" por se tratar de um relatorio complexo e preciso de alguns valores de variaveis que só estão disponiveis nessa condição
Me parece que esse "Duplo Passo" está causando esse problema, mas como criar uma variavel incremental sem que de esse problema?

A: Amigo, faça uso do if Engine.FinalPass antes de incrementar sua variável, com ele você consegue saber se está no primeiro ou segundo passo e consequentemente, pode incrementar sua variável apenas uma vez.

redpajama_set_name: RedPajamaStackExchange


Are you a budding amateur photographer or a seasoned veteran at capturing the scenic landscapes and views of this mother nature? Either way, the galvanizing experience of photography hits the rights spot for the wonder seekers, travelers, or the people picking photography as a hobby. In order to snap everything to your heart's content, you might need a substantial amount of room to accommodate all the photos which tend to easily stack up into multiple GBs before you know it. For that matter, going with a random run-of-the-mill hard drive is nothing but a dicey bet, for it is not always about the space of a drive only, you have to take other facets into your due consideration and appraisal. Fret not, we have arranged a stunning list of best external hard drives for your photography needs.
The hard drive market has been a florid niche for WD for the past years. Their Elements portable drives are renown for their compact and fast drives offering storage capacities from 1TB to 4TB. It is a 5000 rpm drive, and via USB 3.0 you are guaranteed to get a sustained real-world copy speed of more than 95MBps for writing and more than 100MBps for reading.
Out of the box, the Elements comes with a standard foot-long Micro-USB 3.0 cable and nothing else. But you don't need anything else to make it work, either. Of course, it is compatible with USB 2.0 as well.
It is tip-top and ready for all Windows versions. As for the Mac users, there have been complaints with regard to the failure of reformatting, the issue with erasing/formatting for a mac just has to do with the view you have opened in the Disk Utility. Make sure you go to View > Show All Devices, and then "erase" the top tier drive.
After reformatting you'll have 1.8 TB available at your disposal. It doesn't require any battery, If you are using Windows 10 OS all you have to do is plug it in the USB port. Follow instructions to complete the installation and it should show up as a specific drive. It certainly is an appealing catch for the value-conscious individuals and an ample investment for your journey as a photographer.
Toshiba hard drives keep WD drives in tight check with their fast and robust transfer speeds and decent consumptions. This 5200 rpm drive's write consumption is around 0.48A with write speeds of 94MB/s average (USB3.0). While the read consumption pegs at 0.22A with an impressive read speed of 101MB/s average (USB 3.0).
Portability fleshes out of the card with its lightweight and a very finely sleek body covered with a matte and smudge-resistant finish. It is USB 2.0 and USB 3.0 compatible coming with a USB 3.0 data transfer cable which is backward compatible with USB 2.0.
It holds a wide range of compatibility, it comes formatted for Microsoft Windows. It will need reformatting for Mac OS v10.12 /OS X, v10.11 / v10.10. For Mac users, reformatting can be a confounding task. For that, under applications, you go to Utilities and open up Disk Utility. With the drive plugged in, select it and go to the Erase tab and there should be a drop-down menu with Format. Select Mac OS and click erase and after that's done, the drive should be formatted for Mac.
After having it plugged into your computer and it not being in use for a certain amount of time, it will shut off automatically, pretty nifty. Lastly, it includes Toshiba 1-year standard limited warranty.
WD's new My Passport series is a new take on their hard drive game by throwing more useful utilities and premium features into the mix. It is loaded with a rich color scheme of six different colors to chose from. It is a simple plug-and-play drive and an extremely sleek body easily fitting in the palm of your hand with a playful touch of style that goes along with the modern design language.
This hard drive is USB 3.0 and USB 2.0 compatible, we were able to reach 105.44MB/s write and 109.48MB/s read speeds, while 2MB random speeds hit 66.59MB/s write and 48.94MB/s read. It has slip resistant feet on the bottom to keep undue sliding off in check.
It brings many new features to the table. Firstly, it has an auto backup with included WD Backup software so you can automatically back up photos, videos, music, and documents. Set it to your schedule. Just pick the time and frequency. You 'll get password protection with hardware encryption, built-in 256-bit AES hardware encryption with WD Security software helps keep your content private and safe.
Easily add a 'return-if-found' message as the password prompt in case your My Passport ever gets lost. It is compatible with Windows and it requires reformatting for Mac OS X operating system as well. Those who want to use this drive with their Mac and PC simultaneously, you can format it to exFAT file system to use it use it with both. It is a solid and highly secure solution for your storage needs.
This is a solid bet for travelers buckling up for a rough journey. This bad boy has all it takes to withstand the shocks by its military-grade shockproof armor and IPX4 water-resistant protection. We've noted average transfer rates of 120.5 MB/s read and 122.5 MB/s write on this device at 5400 RPMs, given that it has USB 3.1 port, which is very outstanding.
The drive is quite small and lightweight. This drive could slip easily into a slim laptop neoprene case or even a jacket pocket. The plastic casing on top and bottom is your standard ABS, hence it carries the risk of getting scratches despite their advertised context. They have added two anti-slip rubber channels on the bottom.
The A60 is Pre-formatted as FAT32, you could use it directly on Mac or Windows. However, when transferring a single file larger than 4GB, do re-format it to exFAT and make sure you back up the drive before reformat. It comes with 2 cables, the shorter one that lets you place the drive next to your computer and conveniently stores in a side slot and a longer cable that allows you to place the drive further away.
Each can plug into a USB port on the drive that has a dust/water resistant snap-in hinged cover. Interestingly, these USB cables can slip into a silicon rubber channel on the drive once you're done with them and you can pull them out whenever you need them. You'll get a 3-year manufacturer warranty on this bad boy.
Seagate drives are more compliant with numerous operating systems with impressive read and write speeds above 90MBs/s and 103MBs/s running at 5400 RPM. It is a 2.5-inch drive which offers enough data storage for up to 1,000 hours of digital video, up to 250 two-hour DVD films, up to 320,000 digital photos, up to 16,660 hours of digital music. Like the rest of entries, it is USB 2.0 and USB 3.0 compatible.
It is very neat and works right out of the box, you can drag and drop to save files to the drive. It has a thick plastic shell and a very smooth 3D texture on it which will prevent it from drifting off any surface. It is very light and you can easily slip it into your pocket and carry it around.
The drive is automatically recognized by the Windows operating system – no software to install and nothing to configure. It comes formatted for Windows which the Mac reads just fine or it can be reformatted in a couple of minutes to a native Mac OSX format.
If you are using the drive on a Mac only, you should probably reformat it – the only disadvantage of doing that is that you will lose a couple of Windows utilities that it comes preinstalled with. You'll be left with 1.84TB of space after reformatting.


text:
Q: modular java project (IntelliJ IDEA): Module 'com.test' reads package 'javax.xml.bind' from both 'java.xml.bind' and 'java.xml.bind' I'm facing a really frustrating issue related to modules. Can't figure out why I'm getting
Module 'com.test' reads package 'javax.xml.bind' from both 'java.xml.bind' and 'java.xml.bind'

How to solve it?
Module list
module com.test {
    requires java.activation;
    requires slf4j.api;
    requires spring.context;
    requires java.persistence;
    requires spring.beans;
    requires org.apache.cxf.core;
    requires java.xml;
    requires java.xml.bind;
}

My environment
IntelliJ IDEA 2020.3.1 (Community Edition)

JAVA JDK 11

Update no.1
found a post on stackoverflow.com that has a very similar issue I'm facing
Package 'com.example' reads package 'javafx.beans' from both 'javafx.base' and 'javafx.base'
Update no.2
[INFO] +- org.hibernate:hibernate-core:jar:5.4.23.Final:provided
[INFO] |  +- org.jboss.logging:jboss-logging:jar:3.4.1.Final:provided
[INFO] |  +- javax.persistence:javax.persistence-api:jar:2.2:provided
[INFO] |  +- org.javassist:javassist:jar:3.24.0-GA:provided
[INFO] |  +- net.bytebuddy:byte-buddy:jar:1.10.17:provided
[INFO] |  +- antlr:antlr:jar:2.7.7:provided
[INFO] |  +- org.jboss.spec.javax.transaction:jboss-transaction-api_1.2_spec:jar:1.1.1.Final:provided
[INFO] |  +- org.jboss:jandex:jar:2.1.3.Final:provided
[INFO] |  +- com.fasterxml:classmate:jar:1.5.1:provided
[INFO] |  +- javax.activation:javax.activation-api:jar:1.2.0:compile
[INFO] |  +- org.dom4j:dom4j:jar:2.1.3:provided
[INFO] |  +- org.hibernate.common:hibernate-commons-annotations:jar:5.1.2.Final:provided
[INFO] |  +- javax.xml.bind:jaxb-api:jar:2.3.1:compile
[INFO] |  \- org.glassfish.jaxb:jaxb-runtime:jar:2.3.1:compile
[INFO] |     +- org.glassfish.jaxb:txw2:jar:2.3.1:compile
[INFO] |     \- com.sun.istack:istack-commons-runtime:jar:3.0.7:compile
[INFO] +- org.springframework:spring-context:jar:5.3.1:provided
[INFO] |  +- org.springframework:spring-aop:jar:5.3.1:provided
[INFO] |  +- org.springframework:spring-beans:jar:5.3.1:provided
[INFO] |  +- org.springframework:spring-core:jar:5.3.1:provided
[INFO] |  |  \- org.springframework:spring-jcl:jar:5.3.1:provided
[INFO] |  \- org.springframework:spring-expression:jar:5.3.1:provided
[INFO] +- jakarta.xml.bind:jakarta.xml.bind-api:jar:2.3.3:compile
[INFO] |  \- jakarta.activation:jakarta.activation-api:jar:1.2.2:compile
[INFO] +- org.apache.cxf:cxf-core:jar:3.4.1:compile
[INFO] |  +- com.fasterxml.woodstox:woodstox-core:jar:6.2.1:compile
[INFO] |  +- org.apache.ws.xmlschema:xmlschema-core:jar:2.2.5:compile
[INFO] |  +- jakarta.annotation:jakarta.annotation-api:jar:1.3.5:compile
[INFO] |  +- jakarta.jws:jakarta.jws-api:jar:2.1.0:compile
[INFO] |  +- jakarta.xml.soap:jakarta.xml.soap-api:jar:1.4.2:compile
[INFO] |  +- com.sun.activation:jakarta.activation:jar:1.2.2:compile
[INFO] |  +- com.sun.xml.messaging.saaj:saaj-impl:jar:1.5.2:runtime
[INFO] |  +- org.apache.geronimo.specs:geronimo-jta_1.1_spec:jar:1.1.1:compile
[INFO] |  \- org.jboss.spec.javax.rmi:jboss-rmi-api_1.0_spec:jar:1.0.6.Final:compile
[INFO] +- com.sun.xml.bind:jaxb-impl:jar:2.3.3:runtime
[INFO] +- jakarta.xml.ws:jakarta.xml.ws-api:jar:2.3.3:compile
[INFO] +- com.sun.xml.ws:jaxws-rt:jar:2.3.3:compile
[INFO] |  +- com.sun.xml.ws:policy:jar:2.7.10:compile
[INFO] |  +- org.glassfish.ha:ha-api:jar:3.1.12:compile
[INFO] |  +- org.glassfish.external:management-api:jar:3.2.2:compile
[INFO] |  +- org.glassfish.gmbal:gmbal:jar:4.0.1:compile
[INFO] |  +- org.glassfish.pfl:pfl-tf:jar:4.1.0:compile
[INFO] |  +- org.glassfish.pfl:pfl-basic:jar:4.1.0:compile
[INFO] |  +- org.jvnet.staxex:stax-ex:jar:1.8.3:compile
[INFO] |  +- com.sun.xml.stream.buffer:streambuffer:jar:1.5.9:compile
[INFO] |  +- org.jvnet.mimepull:mimepull:jar:1.9.13:compile
[INFO] |  +- com.sun.xml.fastinfoset:FastInfoset:jar:1.2.18:compile
[INFO] |  \- org.codehaus.woodstox:stax2-api:jar:4.1:compile
[INFO] +- org.apache.logging.log4j:log4j-api:jar:2.13.3:compile
[INFO] +- org.apache.logging.log4j:log4j-core:jar:2.13.3:compile
[INFO] +- org.apache.logging.log4j:log4j-slf4j-impl:jar:2.13.3:compile
[INFO] |  \- org.slf4j:slf4j-api:jar:1.7.25:compile
[INFO] +- org.junit.jupiter:junit-jupiter-params:jar:5.7.0:test
[INFO] |  \- org.apiguardian:apiguardian-api:jar:1.1.0:test
[INFO] +- org.junit.jupiter:junit-jupiter-api:jar:5.7.0:test
[INFO] |  +- org.opentest4j:opentest4j:jar:1.2.0:test
[INFO] |  \- org.junit.platform:junit-platform-commons:jar:1.7.0:test
[INFO] \- org.mockito:mockito-junit-jupiter:jar:3.5.10:test
[INFO]    \- org.mockito:mockito-core:jar:3.5.10:test
[INFO]       +- net.bytebuddy:byte-buddy-agent:jar:1.10.13:test
[INFO]       \- org.objenesis:objenesis:jar:3.1:test

A: Thanks to Naman and the link provided in original post I found out that I was mixing jakarta.activation and javax.activation while using cxf.
And, because cxf-xjc-plugin:3.3.1 generates code from xsd files using javax.activation I had to exclude
everywhere in a pom jakarta.activation
Update No.1
to be precise I had to replace javax dependencies with jakarta dependencies
that have javax package naming structure (i.e javax.activation and not jakarta.activation), i.e
<has javax packaging structure>
<dependency>
    <groupId>jakarta.jws</groupId>
    <artifactId>jakarta.jws-api</artifactId>
    <version>2.1.0</version>
</dependency>

but not
<has jakarta packaging structure>
<dependency>
    <groupId>jakarta.jws</groupId>
    <artifactId>jakarta.jws-api</artifactId>
    <version>3.0.0</version>
</dependency>


What you suggest would require sellers to be realistic about pricing. $345k for that building in that location? Good luck. Just by paying that amount, the buyer will be hit with insane taxes. Combined with a mortgage and upkeep, I could not see a new owner being able to break even given what I expect the market rate of the rentals to be. The excitement over the "hot" Newburgh market is leading to some unrealistic expectations.
Oh yes, the price is incredibly high. I don't think anyone would or should buy it for that. But I present it as is listed, not what I think they should sell it for.
the value is in being the owner of one of the busiest drug dealing corners in the tristate area. PLUS there is a business that sells lotto tickets and BEER and cigarettes which in the ghetto are the cash crops!


Tuft & Needle is one of the more intriguing bedding companies to emerge in the last few years. The mattress is composed of a foam they have named simply "T&N Adaptive", This startup company out of Arizona has created a simple, no nonsense sleep solution that is delivered to your door at a reasonable price. The company's last few years have been quite eventful. Today, these beds are available direct from Tuft & Needle, Amazon, or home improvement giant, Lowes.
The merger of a Phoenix-based online mattress marketer with the nation's largest bedding manufacturer has sent shock waves through the industry. Serta Simmons is one of the world's most recognizable brand names. Tuft & Needle Co-founders JT Marino and Daehee Park have built their company into a 170 million dollar a year business in just 6 short years. Their partnership represents an industry milestone. 20% of all mattress purchases are now made online. With many of the nation's largest bedding retailers struggling to keep pace, the merger makes a lot of sense. Park and Marino will assume roles in company's management team. Their focus will be expanding ecommerce operations.
Tuft & Needle offers a solid value at a reasonable price.
The cover is a 40/60 blend of micro polyamide and polyester and polyester. This fabric is designed to prevent moisture build up and sleep cool.
T&N Adaptive: (3" of 2.9 lb density) While more supportive foams are available, this is a quality foam for the price point. T&N is polyurethane based and designed to "adapt" to different body types and sleep preferences. The objective was to produce a foam that has the bounce of latex with the pressure relief of memory foam. Meanwhile, heat wicking graphite and cooling gels help regulate sleep temperature.
Support foam: This 7" bottom layer of 1.8 lb. density foam provides stability for the comfort layer. A quality base is essential to overall support. Combined with T&N Adaptive, most sleepers will find adequate support.
We had several people of various body types lay on this mattress. While Tuft & Needle performed quite well for most, larger individuals (those over 250 lbs) tended to bottom out a bit on this mattress. Still, not bad for its 10" profile.
This mattress has a buoyant feel. It's pressure relieving but with a bit more bounce than memory foam.
Great motion transfer reduction. A plus for couples.
While movement is not as easy as some innerspring or all latex beds, it's an improvement over memory foam.
This new innovative process reacts differently than traditional foams. T&N Adaptive was created to be less heat retentive than either memory foam or latex. It's worth noting, all foams retain some body heat. Some just do a better job than others of dispersing it. Consumer feedback seems to indicate the Tuft & Needle mattress sleeps reasonably cool. That said, there are new memory foam options that sleep far cooler than years past. New technologies offset earlier heat retention issues that plagued the industry for years. While T&N Adaptive may perform as well or even better than many synthetic foams, natural latex is still the coolest sleeping foam option.
This bed is CertiPUR certified to not emit dangerous chemicals into the air. Only a slight "new" odor was detected that dissipated quickly.
We have some concerns after seeing how this mattress gave way to larger body types. However, our research was not able to uncover significant complaints as to its durability. The company is only 6 years old so additional time may be needed for a fair evaluation.
Like memory foam, this is not a strong suit for T&N.
100 night sleep trial is the norm for most online mattress companies.
Shipping is free to your door in the continental U.S.
Movement is easier than on a memory foam bed, with a bit more bounce.
This mattress is an exceptional value for the dollar. its low price make the Tuft & Needle a great buy.
Although designed to provide adequate pressure relief for side sleepers, this mattress might be a bit too firm for those who prefer a softer feel.
Most back sleeper will appreciate this bed's medium firm feel. Proper alignment is created with decent pressure relief and conformity. Heavier individuals may experience more sinkage and require more support for proper alignment.
This mattress is designed to have that universal medium firm feel. It will be soft enough for most but not all side sleepers. Those seeking more hug and body conformity will find this mattress too firm.
Stomach sleepers need a mattress that does not allow their lower backs to dip and cause lumbar pain. Those looking for a little more bounce and buoyancy opposed to sinking into their bed will appreciate Tuft & Needle's firm but not too firm feel.
Low Motion Transfer: Our evaluation found little to no motion detection. This is a solid benefit for couples.
Sleeps Cool: Cooling gels and graphite infused foam regulates heat.
Great Value: At a price of $595 for a queen mattress, Tuft & Needle is a great value.
Only One Firmness Option: This mattress will accommodate most but not all comfort preferences.
Support may not be suitable for larger people. Bigger body types may not find sufficient support.
If you want to sleep cool, heat regulating technology keeps you cool and comfortable.
If you're looking for an affordable memory foam mattress, Tuft & Needle offers good value at an affordable price.
People in search of a very soft or very firm mattress will most likely fair better with other options. This mattress falls right in the middle.
Those seeking significant edge support should stick with an innerspring mattress. Like memory foam and latex beds, Tuft & Needle's edge support is fair at best.
Larger individuals should seek more supportive options.
At 72 lbs, this mattress weighs much less than many comparable options. One person could handle this bed by themselves. However, it's still easier with 2 sets of hands to assist you.
Tuft & Needle is a well-priced mattress that delivers decent quality. For those on a budget, it's money well spent.


text:
Other surveys of state treasurer candidates
Kentucky Forward (concerning open records)
The attorney general serves as the state's chief law enforcement officer and consumer advocate, writing legal opinions to government officials and agencies, defending or initiating lawsuits involving the state, and prosecuting consumer and election fraud, among other duties.
Our current Attorney General, Andy Beshear, is running for Governor, so this race has no incumbent.
The Democratic candidate is Gregory D. Stumbo and his survey responses follow. The Republicans candidate is Daniel Cameron who did not respond to our survey.
Lea las respuestas en español de los candidatos aquí.
Gregory D. Stumbo
Political party: Democrat
Campaign website: http://www.StumboForAG.com
Campaign Facebook: Facebook
Campaign Twitter: Twitter
What is your vision for the role of the Kentucky Attorney General? How will our commonwealth be better in four years if you are elected?
I want to make sure the office stays independent and represents all Kentuckians equally. Therefore, as when I was Attorney General previously, I will reopen satellite offices across the Commonwealth to make the office more accessible to everyone.
It is my adamant belief that there is no room for partisan politics in the Office of the Attorney General, and I have and will hold everyone equally accountable to the law. I am dedicated to upholding our state and federal constitutions, and firmly believe personal opinions and politics have no place in that office.
I was the first Attorney General in the nation to sue Purdue Pharma, and I will vigorously pursue the cases pending in Kentucky's courts against others complicit in the drug epidemic that is plaguing our Commonwealth.

redpajama_set_name: RedPajamaCommonCrawl


Why Facebook's Mobile Messenger App is a big deal for Telcos.
I just finished reading a CNN article about the launch of Facebook's Mobile Messenger App. For some reason, this new development from the world's largest social network of 700 million users struck me as super significant. At this time, Facebook Messenger is only available as an iPhone and Android mobile app in a certain markets – it seems I cannot download it yet on Apple's App Store in Kenya. However, you have to consider that once its available globally, and on multiple mobile operating systems and platforms, how big could it possibly become?
The truth of the matter is that most mobile networks globally make a significant volume of their revenues via SMS messaging. It's a major part of their business and one that they have prospered on for many many years. However, courtesy of smartphones and the growing range of innovative mobile messaging apps, this revenue line is now under attack. I recently got a BlackBerry and have only started using BB Messenger. I have to say its excellent and I keep signing up more and more friends. It works and its free but unfortunately only works on BlackBerry.
In addition to the launch of Facebook Messenger, other mobile messaging apps of note are the soon to be launched Apple iMessenger in the next upgrade of iOS as well as WhatsApp Messenger which is really taking off as a multi-platform mobile messaging app. Not to be left behind, Samsung also recently announced that they plan to launch a mobile messaging service that will work across all their mobile devices in a ubiquitous manner.
Going forward, the most significant thing about Facebook Messenger is Facebook's huge user base on a global scale. Facebook Messenger has the potential to reinvent mobile messaging and communications as we know it. Take Kenya for instance – there are over 1 million Facebook users. This in itself is as large as a small mobile network's subscriber base, and its only getting bigger by the day. Facebook Messenger essentially makes Facebook a quasi-mobile network operator for free messaging.
This could indeed be a significant moment in telecoms history – the day that Facebook moved from being a social network to the "de facto" global mobile communications platform – it's certainly conceivable that this could happen. One just has to look at how successful Google's Android has been in the smartphone space to realize that technology innovation reinvents everything and that today's enabler could be tomorrow's disruptor.
[More] on social media and QR codes in Kenyan print advertising.
Mocality, Kenya's largest online and mobile business directory launches Daily Deals and Mobile Apps.
I agree with your analysis and too was shocked at lack of media attention to recent release of Facebook application for java enabled phones.
Facebook appears to understand that getting lots of users early and often penetrates user-ship into areas previously not available to the company. I guess when looking for the NEXT 1/2 billion users Africa offers growth unfound elsewhere.
The ubiquitous and free universal messaging you mention is yet to sort itself out and I question your statement that Facebook's app provides free messaging. It is my understanding that some volume of free SMS may be available upon launch but I thought Facebook was reserving right to have people pay for text messages they send using the service after initial offer ends. Am I wrong.
As Africa leap-frogs into mobile its interesting to see Facebook leap-frog others in capturing impressive uptake in this increasingly important and growing user-base.
I think a relevant point that your article leads to but does not claim is that new applications, especially p2a person2application, while increase all forms of messaging including SMS which is important part of MNOs revenue streams.


text:
Lacrosse Downs Western Connecticut To Earn Spot In LEC Tournament On Senior Day
Western Connecticut State University (10-7, 5-3) 1 4 2 3 10
UMass Boston (4-13, 3-5) 5 6 2 3 16
G: Conor Lenfest - 3
A: 4 Players (#9, #21, #41, #55) - 2
Sv: Cameron Dixson - 5
GB: 3 Players (#9, #27, #55) - 4
G: 2 Players (#21, #31) - 3
Sv: Perry Cassidy - 6
GB: Billy Youngblood - 8
BOSTON, Mass. – It was quite the senior day for Beacons lacrosse, who despite battling through a torrential down pour, a lightning delay, and with the spot in the conference tournament on the line, played perhaps their best game of the 2019 campaign as they earned a hard-fought 16-10 win over Western Connecticut State University Friday evening at James Cotter Field.
UMass Boston 16, Western Connecticut State U. 10
UMass Boston (4-13, 3-5 Little East Conference)
Western Connecticut State (10-7, 5-3 Little East Conference)
With the win the Beacons will be the sixth-seed in the Little East Tournament which begins Tuesday. The Beacons will find out who their opponent is this weekend on www.littleeast.com.
The appearance in the Little East tournament, will mark the fourth straight for the Beacons, and sixth in the last seven years.
Along with the huge win, the Beacons also saw sophomore standout Conor Lenfest (Waltham, Mass.) become the newest member of the 100-point club. The Waltham native got the five points he needed to become the third Beacon in the last three seasons to earn a spot on the historic list.
The sophomore and reigning Little East Rookie of the Year led all scorers with his five points as he registered three goals and two assists. Darragh Fahey (Walpole, Mass.) finished second on the team with a pair of goals and assists.
Knowing that all they needed to do was earn the victory, UMass Boston played their most balanced game of the 2019 season. They completely dominated the first two quarters of Friday's wet and frigid contest. In the first nine minutes of the contest, the Beacons were in complete control as they held a 5-0 lead. Derek Caruso (Port Washington, N.Y.) would score a pair of goals during the Beacons early run with Donny Bickford (Braintree, Mass.), Jeremy Girard (Hinsdale, Mass.), and Jack O'Brien (East Longmeadow, Mass.) also getting in on the scoring.
WCSU got on the board with 2:18 left in the opening stanza, and had the first two goals of the second to cut UMB's lead to two (5-3) with just over 13 minutes to play before halftime.
Completely unfazed by the Colonial's quick run, the Beacons responded with a four-goal flurry to push their lead to a game-high six goals (9-3). Lenfest got all three of his goals during that stretch. Western Connecticut managed just two more goals over the final six minutes, but it was UMB who had the final say before the break with Ned Parker (Stafford, Va.) and Fahey each scoring once over a 2:13 span.
With the rain increasing dramatically, the two rivals would each score twice in the third with Cam Hladick (Natick, Mass.) and Fahey leading the way for UMB. Jesse Weaver recorded both goals in the third for the Colonials.
In the fourth the Beacons all but secured the victory with a man-up goal by Michael Claflin (Newton, Mass.) and an unassisted by O'Brien. After the long lightning delay, WCSU got their eighth goal of the contest on a Ryan Reuling tally. Jack Riley (Wrentham, Mass.) connected with Caruso at the 3:23 mark to put the final nail in the Colonials coffin. WCSU mustered just two more scores over the final 73 seconds.
Cameron Dixson (Wayland, Mass.) who was honored prior to the start of the game alongside his classmates Jack Fournier (West Newbury, Mass.) and Matt Murphy (Watertown, Mass.) recorded just five saves. Perry Cassidy made six stops on 17 shots as he was handed the loss. Derek Strine closed out the final two stanzas and was solid allowing five goals on 10 shots, while making five saves.
Tom Redmond (Scituate, Mass.) and Jack Dobrzynski (East Greenwich, R.I.) combined for 14 face-off wins on 29 tries respectively.
Jack Ranta (Newton, Mass.) anchored the Beacons defense with his three caused turnovers, while Redmond, Fahey, and Lenfest each had a team-leading four ground balls.
All-time the Beacons are now 7-11 against WCSU, and have won a program record four straight over the Colonials
UMB's 16 goals marked the most goals since April 24, 2004 in the first-ever meeting between the two teams
The Beacons led the contest in shots 49-26, 27-15 in shots on goal
UMB also held a 36-29 advantage in ground balls and were 1-for-5 on man-up attempts; WCSU finished 0-for-4 on the night
Lenfest currently has 63 goals and 37 assists in 29 games played as a Beacon
WHO'S NEXT
The Beacons will find out who their opponent is this weekend, when the official tournamnet bracket is released on www.littleeast.com.