+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Category theory +-- {: .hide} [[!include category theory - contents]] =-- =-- =-- # Contents * table of contents {: toc} ## Definition A [[category]] consists of a collection of **objects** together with [[morphisms]] between these objects. Thus, naively, we may think of objects as the '[[elements]]' of a category. More generally, in [[higher category theory]] the objects of an $(n,r)$-[[(n,r)-category|category]] are the $0$-dimensional cells of that structure, the $0$-[[0-morphism|morphisms]]. ## Examples * If a [[set]] is regarded as a [[discrete category]] (with no nontrivial morphisms) then the objects of that category are precisely the elements of the set. * In the [[fundamental groupoid]] $\Pi_1(X)$ of a [[topological space]] $X$, the objects are the points of $X$. * In the category [[Set]], the objects are sets; in [[Vect]] the objects are [[vector space]]s; in [[Top]] the objects are [[topological space]]s, etc. * If a [[simplicial set]] that is a [[Kan complex]] is regarded as an $\infty$-[[infinity-groupoid|groupoid]], then its vertices are the objects of that $\infty$-groupoid. * Similarly if a simplicial set that is a [[quasi-category]] is regarded as an $(\infty,1)$-[[(∞,1)-category|category]], then its vertices are the objects of that $(\infty,1)$-category. * If a [[globular set]] is equipped with the structure of a [[strict ∞-category]], then its $0$-cells are the objects of that [[∞-category]]. [[!redirects object]] [[!redirects objects]] [[!redirects Object]] [[!redirects Objects]]