Wiedemann–Franz law

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. Clearly, any category which this property is connected in the above sense.

A small category is connected if and only if its underlying graph is weakly connected.

Each category J can be written as a disjoint union (or coproduct) of a connected categories, which are called the connected components of J. Each connected component is a full subcategory of J.

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534