Category of elements

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In category theory, if C is a category and ${\displaystyle F:C\to \mathbf {Set} }$ is a set-valued functor, the category of elements of F ${\displaystyle {\mathop {\rm {el}}}(F)}$ (also denoted by ∫CF) is the category defined as follows:

• Objects are pairs ${\displaystyle (A,a)}$ where ${\displaystyle A\in {\mathop {\rm {Ob}}}(C)}$ and ${\displaystyle a\in FA}$.
• An arrow ${\displaystyle (A,a)\to (B,b)}$ is an arrow ${\displaystyle f:A\to B}$ in C such that ${\displaystyle (Ff)a=b}$.

A more concise way to state this is that the category of elements of F is the comma category ${\displaystyle \ast \downarrow F}$, where ${\displaystyle \ast }$ is a one-point set. The category of elements of F comes with a natural projection ${\displaystyle {\mathop {\rm {el}}}(F)\to C}$ that sends an object (A,a) to A, and an arrow ${\displaystyle (A,a)\to (B,b)}$ to its underlying arrow in C.

The category of elements of a presheaf

Somewhat confusingly in some texts (e.g. Mac Lane, Moerdijk), the category of elements for a presheaf is defined differently. If ${\displaystyle P\in {\hat {C}}:=\mathbf {Set} ^{C^{op}}}$ is a presheaf, the category of elements of P (again denoted by ${\displaystyle {\mathop {\rm {el}}}(P)}$, or, to make the distinction to the above definition clear, ∫C P) is the category defined as follows:

• Objects are pairs ${\displaystyle (A,a)}$ where ${\displaystyle A\in {\mathop {\rm {Ob}}}(C)}$ and ${\displaystyle a\in P(A)}$.
• An arrow ${\displaystyle (A,a)\to (B,b)}$ is an arrow ${\displaystyle f:A\to B}$ in C such that ${\displaystyle (Pf)b=a}$.

As one sees, the direction of the arrows is reversed. One can, once again, state this definition in a more concise manner: the category just defined is nothing but ${\displaystyle (\ast \downarrow P)^{\rm {op}}}$. Consequentially, in the spirit of adding a "co" in front of the name for a construction to denote its opposite, one should rather call this category the category of coelements of P.

For C small, this construction can be extended into a functor ∫C from ${\displaystyle {\hat {C}}}$ to ${\displaystyle \mathbf {Cat} }$, the category of small categories. In fact, using the Yoneda lemma one can show that ∫CP ${\displaystyle \cong {\mathop {\textbf {y}}}\downarrow P}$, where ${\displaystyle {\mathop {\textbf {y}}}:C\to {\hat {C}}}$ is the Yoneda embedding. This isomorphism is natural in P and thus the functor ∫C is naturally isomorphic to ${\displaystyle {\mathop {\textbf {y}}}\downarrow -:{\hat {C}}\to {\textbf {Cat}}}$.