  Let $\mathcal{C}$ be a category with pullbacks. We define a $\textit{Beck
torsor}$ in $\mathcal{C}$ as a morphism $Z\to Y$ in $\mathcal{C}$ which is a
torsor for a Beck module over $Y$. We say that an object $X$ of $\mathcal{C}$
is $\textit{formally unramified}$ if, for every Beck torsor $Z\to Y$ in
$\mathcal{C}$, the canonical map $\text{Hom}_{\mathcal{C}}(X, Z)\to
\text{Hom}_{\mathcal{C}}(X, Y)$ is injective. If $A$ is a commutative ring with
identity, then an $A$-algebra $B$ is formally unramified in the category of
$A$-algebras if and only if the ring homomorphism $A\to B$ is formally
unramified. Given that $A\to B$ is formally unramified if and only if
$\Omega_{B/A} = 0$, we seek a similar classification for general formally
unramified objects. We say that $\mathcal{C}$ has $\textit{K\"ahler
differentials}$ if, for each object $X$ of $\mathcal{C}$, the forgetful functor
$\text{Ab}(\mathcal{C}/X)\to \mathcal{C}/X$ from the category of Beck modules
over $X$ has a left adjoint $\Omega: \mathcal{C}/X\to
\text{Ab}(\mathcal{C}/X)$. Our main result is that if $\mathcal{C}$ has
K\"ahler differentials, then an object $X$ of $\mathcal{C}$ is formally
unramified if and only if $\Omega_X$ is a zero object in
$\text{Ab}(\mathcal{C}/X)$.

Category Theory