Category of elements

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

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

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

The category of elements of a presheaf

In some texts (e.g. Mac Lane, Moerdijk) the category of elements is used for presheaves. We state it explicitly for completeness. If $P\in {\hat {C}}:=\mathbf {Set} ^{C^{op}}$ is a presheaf, the category of elements of P (again denoted by $\mathop {\rm {el}} (P)$, or, to make the distinction to the above definition clear, ∫_C P=∫^Cop P) is the category defined as follows:

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

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 $(\ast \downarrow P)^{\rm {op}}$. 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 ${\hat {C}}$ to $\mathbf {Cat}$, the category of small categories. In fact, using the Yoneda lemma one can show that ∫_CP $\cong \mathop {\textbf {y}} \downarrow P$, where $\mathop {\textbf {y}} :C\to {\hat {C}}$ is the Yoneda embedding. This isomorphism is natural in P and thus the functor ∫_C is naturally isomorphic to $\mathop {\textbf {y}} \downarrow -:{\hat {C}}\to {\textbf {Cat}}$.

The category of elements of an operad algebra

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

• Objects are pairs $(o,x)$ where $o\in \mathrm {Ob} (O)$ and $x\in A(o)$.

An arrow $(o_{1},x_{1})\to (o_{2},x_{2})$ is an arrow $f\colon o_{1}\to o_{2}$ in $O$ such that $A(f)(x_{1})=x_{2}.$

See also

• Grothendieck construction

References

• 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.

External links

• Category of elements at the nLab