Category of elements

Template:Multiple issues 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 $F:C^{\mathrm{o}\mathrm{p}}\to 𝐒𝐞𝐭𝐬$ be a set-valued functor. The category Template:Math of elements of Template:Mvar (also denoted Template:Math) is the category whose:

• Objects are pairs $(A,a)$ where $A\in \mathrm{O}\mathrm{b}(C)$ 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 Template:Math, where Template:Math is a singleton (a set with one element).

The category of elements of Template:Mvar is naturally equipped with a projection functor Template:Math that sends an object Template:Math to Template:Mvar, and an arrow Template:Math to its underlying arrow in Template:Mvar.

As a functor from presheaves to small categories

For small Template:Mvar, this construction can be extended into a functor Template:Math from Template:Mvar to Template:Math, the category of small categories. Using the Yoneda lemma one can show that Template:Math, where Template:Math is the Yoneda embedding. This isomorphism is natural in Template:Mvar and thus the functor Template:Math is naturally isomorphic to Template:Math.

See also

• Grothendieck construction

References

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External links

• Template:Nlab