Connected category - Citizendia

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

$f_i : X_i \to X_{i+1}$

$f_i : X_{i+1} \to X_i$

for each 0 ≤ i < n (both directions are allowed in the same sequence). In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple Equivalently, a category J is connected if each functor from J to a discrete category is constant. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, especially Category theory, a discrete category is a category whose only Morphisms are the Identity morphisms It is the simplest 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. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships ↔ In Mathematics and Computer science, connectivity is one of the basic concepts of Graph theory.

Each category J can be written as a disjoint union (or coproduct) of a connected categories, which are called the connected components of J. In Category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological Each connected component is a full subcategory of J. In Mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in

References

Mac Lane, Saunders (1998). Saunders Mac Lane ( 4 August 1909, Taftville, Connecticut – 14 April 2005, San Francisco) was an American Categories for the Working Mathematician, (2nd ed. Categories for the Working Mathematician is a textbook in Category theory written by American Mathematician Saunders Mac Lane, who ), Graduate Texts in Mathematics 5, Springer-Verlag. ISBN 0-387-98403-8.

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