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.
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.
connected, category, category, theory, branch, mathematics, connected, category, category, which, every, objects, there, finite, sequence, objects, displaystyle, ldots, with, morphisms, displaystyle, displaystyle, each, both, directions, allowed, same, sequenc. 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 lt 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 disregards 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 References editMac Lane Saunders 1998 Categories for the Working Mathematician Graduate Texts in Mathematics 5 2nd ed Springer Verlag ISBN 0 387 98403 8 nbsp This category theory related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Connected category amp oldid 1207786126, wikipedia, wiki, book, books, library,