NationMaster - Encyclopedia: Category of elements

In category theory, for every presheaf the category of elements $mathbf{El}(P)$ of $P$ is the category defined as follows: Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...

its objects are pairs $(A, a)$ where $A$ is an object of $C$ and $ain P(A)$ ,
its morphisms $(A,a)to (B,b)$ are the morphisms $f:Ato B$ of $C$ such that $a = P (f)(b)$ .

This construction can be extended into a functor $mathbf{El}$ from $hat C$ to $mathbf{Cat}$ , the category of small categories. In mathematics, specifically in category theory, the 2-category of small categories is the 2-category whose objects are small categories, whose arrows are functors and whose 2-arrows are natural transformations. ...

Categories: Category theory stubs | Category theory

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