Category of elements on arvau

Wikipedia / 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. It and its generalization are also known as the Grothendieck construction (named after Alexander Grothendieck) especially in the theory of descent, in the theory of stacks, and in fibred category theory.

The Grothendieck construction is an instance of straightening (or rather unstraightening).

Significance

In categorical logic, the construction is used to model the relationship between a type theory and a logic over that type theory, and allows for the translation of concepts from indexed category theory into fibred category theory, such as Lawvere's concept of hyperdoctrine.

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. See also Density theorem (category theory) for an example usage.

Motivation

is a family of sets indexed by another set, one can form the disjoint union or coproduct

. The disjoint union set is naturally equipped with a "projection" map

. Jacobs has referred to these two perspectives as "display indexing" and "pointwise indexing".

Construction

be a category be a set-valued functor. The category of elements of (also denoted ) is the category whose:

Objects.
Morphisms.

is the comma category, where is a singleton (a set with one element).

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

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

For some applications, it is important to generalize the construction to even a contravariant pseudofunctor, where

an object,
a morphism,
; i.e.,

are coherent.

Examples

Group

is a group with one object and all morphisms invertible be a functor is then equivalent to specifying a group homomorphism denotes the group of automorphisms results in a category with one object, which can again be viewed as a group, and in this case, the resulting group is (isomorphic to) the semidirect product

Representable functor

Given a category C and a fixed object *, the contravariant functor represented by * associated to it by the Grothendieck construction is exactly the comma category. is a map over *.

Twisted arrows

Given a category C to be the hom-functor

denotes a product of categories is known as the category of twisted arrows in C. The opposite of it is known as the twisted diagonal of C.

Homotopy colimit

. The nerve is a simplicial set that is isomorphic to the homotopy colimit by Thomason's homotopy colimit theorem:

Sometimes, this is taken as a definition of a homotopy colimit.

is a simplicial diagram, one also gets the homotopy colimit of X as well.

As a cartesian fibration

be the forgetful functor is a cartesian fibration is a category fibered. The required lifting property then holds trivially.

induces a functor

has the required lifting property to be a cartesian morphism, completing the proof of the claim.

Formulation in ∞-categories

Using the language of ∞-categories, the Grothendieck construction can be stated in the following succinct way. Namely, it says there is an equivalence of ∞-categories:

between the functor category and the (2, 1)-category of cartesian fibrations..

as follows.. By the axiom of choice. We now make

will be coherent.

Notes

References

Peter Johnstone, Sketches of an Elephant (2002)