Category of elements

In category theory, if C is a category and F:C→Set is a set-valued functor, the category el(F) of elements of F (also denoted ∫^CF) is the following category:

• Objects are pairs ${\displaystyle (A,a)}$ where ${\displaystyle A\in \mathop {\rm {Ob}} (C)}$ and ${\displaystyle a\in FA}$.
• Morphisms ${\displaystyle (A,a)\to (B,b)}$ are arrows ${\displaystyle f:A\to B}$ of ${\displaystyle C}$ 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 ∗↓F, where ∗ is a one-point set. The category of elements of F comes with a natural projection el(F)→C that sends an object (A, a) to A, and an arrow (A,a)→(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∈Ĉ:=Set^Cop is a presheaf, the category of elements of P (again denoted by el(P), or, to make the distinction to the above definition clear, ∫_C P = ∫^CopP) is the following category:

• Objects are pairs ${\displaystyle (A,a)}$ where ${\displaystyle A\in \mathop {\rm {Ob}} (C)}$ and ${\displaystyle a\in P(A)}$.
• Morphisms ${\displaystyle (A,a)\to (B,b)}$ are arrows ${\displaystyle f:A\to B}$ of ${\displaystyle C}$ 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 (∗↓P)^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 small C, this construction can be extended into a functor ∫_C from Ĉ to Cat, the category of small categories. In fact, using the Yoneda lemma one can show that ∫_C P≅y↓P, where y:C→Ĉ is the Yoneda embedding. This isomorphism is natural in P and thus the functor ∫_C is naturally isomorphic to y↓–:Ĉ→Cat.

The category of elements of an operad algebra[]

Given a (colored) operad O and a functor, also called an algebra, A:O→Set, one obtains a new operad, called the category of elements and denoted ∫^OA, 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 ${\displaystyle A(f)(x_{1})=x_{2}.}$

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 in nLab