Category of elements

In category theory, a branch of mathematics, the category of elements of a presheaf is a category associated to that presheaf whose objects are the elements of sets in the presheaf.

The category of elements of a simplicial set is fundamental in simplicial homotopy theory, a branch of algebraic topology. More generally, the category of elements plays a key role in the proof that every weighted colimit can be expressed as an ordinary colimit, which is in turn necessary for the basic results in theory of pointwise left Kan extensions, and the characterization of the presheaf category as the free cocompletion of a category.

Definition
Let $$C$$ be a category and let $$F: C^{\rm op}\to \mathbf{Sets}$$ be a set-valued functor. The category $el(F)$ of elements of $F$ (also denoted $∫_{C} F$) is the category whose:
 * Objects are pairs $$(A,a)$$ where $$A \in \mathop{\rm Ob}(C)$$ and $$a \in FA$$.
 * Morphisms $$(A,a) \to (B,b)$$ are arrows $$f: A \to B$$ of $$C$$ such that $$(Ff)b = a$$.

An equivalent definition is that the category of elements of $$F$$ is the comma category $&lowast;↓F$, where $&lowast;$ is a singleton (a set with one element).

The category of elements of $F$ is naturally equipped with a projection functor $&Pi;: ∫_{C} F→C$ that sends an object $(A, a)$ to $A$, and an arrow $(A,a)→(B,b)$ to its underlying arrow in $C$.

As a functor from presheaves to small categories
For small $C$, this construction can be extended into a functor $∫_{C}$ from $Ĉ$ to $Cat$, the category of small categories. Using the Yoneda lemma one can show that $∫_{C} P&cong;y↓P$, where $y:C→Ĉ$ is the Yoneda embedding. This isomorphism is natural in $P$ and thus the functor $∫_{C}$ is naturally isomorphic to $y↓–:Ĉ→Cat$.