title: Connected category
text: # Connected category
  
  In category theory, a branch of mathematics, a connected category is a category in which, for every two objects X and Y there is a finite sequence of objects
  
      $X=X_{0},X_{1},\ldots ,X_{n-1},X_{n}=Y$
  
  with morphisms
  
      $f_{i}:X_{i}\to X_{i+1}$
  
  or
  
      $f_{i}:X_{i+1}\to X_{i}$
  
  for each 0 ≤ i < n (both directions are allowed in the same sequence). Equivalently, a category J is connected if each functor from J to a discrete category is constant. In some cases it is convenient to not consider the empty category to be connected.
  
  A stronger notion of connectivity would be to require at least one morphism f between any pair of objects X and Y. Any category with this property is connected in the above sense.
  
  A small category is connected if and only if its underlying graph is weakly connected, meaning that it is connected if one disregard the direction of the arrows.
  
  Each category J can be written as a disjoint union (or coproduct) of a collection of connected categories, which are called the connected components of J. Each connected component is a full subcategory of J.
  
  