Category of elements

In some texts (e.g. Mac Lane, Moerdijk) the category of elements is used for presheaves. We state it explicitly for completeness. 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=∫^Cop 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 ${\displaystyle \mathop {\textbf {y}} \downarrow -:{\hat {C}}\to {\textbf {Cat}}}$.

Given a (colored) operad ${\displaystyle O}$ and a functor, also called an algebra, ${\displaystyle A\colon O\to \mathbf {Set} }$, one obtains a new operad, called the category of elements and denoted ${\displaystyle \textstyle {\int }^{O}A}$, generalizing the above story for categories. It has the following description:

• Objects are pairs ${\displaystyle (o,x)}$ where ${\displaystyle o\in \mathrm {Ob} (O)}$ and ${\displaystyle x\in A(o)}$.
• An arrow ${\displaystyle (o_{1},x_{1})\to (o_{2},x_{2})}$ is an arrow ${\displaystyle f\colon o_{1}\to o_{2}}$ in ${\displaystyle O}$ such that ${\displaystyle A(f)(x_{1})=x_{2}.}$

• Grothendieck construction

• Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5 (2nd ed.). Springer-Verlag. ISBN 0-387-98403-8.
• Mac Lane, Saunders; Moerdijk, Ieke (1992). Sheaves in Geometry and Logic. Universitext (corrected ed.). Springer-Verlag. ISBN 0-387-97710-4.

• Category of elements in nLab