Connected category
Connected category
- A category J is connected if, for any two objects X and Y, there is a finite sequence of objects X = X0, X1, ..., Xn = Y where for each i there is a morphism Xi → Xi+1 or Xi+1 → Xi. In other words, you can connect any two objects by a chain of morphisms, allowing moves in either direction.
- Equivalently, every functor from J to a discrete category must send all objects to the same value (it must be constant on objects).
- Some people do not count the empty category as connected.
- A stronger condition would require that for every X, Y there is at least one morphism X → Y (or Y → X). Any category with this property is connected in the above sense.
- A small category is connected exactly when its underlying graph is weakly connected (ignoring arrow directions).
- Every category J can be written as a disjoint union (coproduct) of connected categories, called the connected components of J. Each connected component is a full subcategory of J.
This page was last edited on 3 February 2026, at 04:49 (CET).