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 , … , X n − 1 , X n = Y {\displaystyle X=X_{0},X_{1},\ldots ,X_{n-1},X_{n}=Y} with morphisms f i : X i → X i + 1 {\displaystyle f_{i}:X_{i}\to X_{i+1}} or f i : X i + 1 → X i {\displaystyle 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.