nLab
category of presheaves

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Definition

For C a small category, its category of presheaves is the functor category

PSh(C):=[C op,Set]PSh(C) := [C^{op}, Set]

from the opposite category of C to Set.

An object in this category is a presheaf. See there for more details.

Properties

General

Characterization

Theorem

A category E is equivalent to a presheaf topos if and only if it is cocomplete, atomic, and regular.

This is due to Marta Bunge.

Cartesian closed monoidal structure

As every topos, a category of presheaves is a cartesian closed monoidal category.

For details on the closed structure see

Presheaves on over-categories and over-categories of presheaves

Let C be a category, c an object of C and let C/c be the over category of C over c. Write PSh(C/c)=[(C/c) op,Set] for the category of presheaves on C/c and write PSh(C)/Y(y) for the over category of presheaves on C over the presheaf Y(c), where Y:C→PSh(c) is the Yoneda embedding.

Proposition

There is an equivalence of categories

e:PSh(C/c)→≃PSh(C)/Y(c).e : PSh(C/c) \stackrel{\simeq}{\to} PSh(C)/Y(c) \,.
Proof

The functor e takes F∈PSh(C/c) to the presheaf F′:d↦⊔ f∈C(d,c)F(f) which is equipped with the natural transformation η:F′→Y(c) with component map η d⊔ f∈C(d,c)F(f)→C(d,c).

A weak inverse of e is given by the functor

e¯:PSh(C)/Y(c)→PSh(C/c)\bar e : PSh(C)/Y(c) \to PSh(C/c)

which sends η:F′→Y(C)) to F∈PSh(C/c) given by

F:(f:d→c)↦F′(d)∣ c,F : (f : d \to c) \mapsto F'(d)|_c \,,

where F′(d)∣ c is the pullback

F′(d)∣ c → F′(d) ↓ ↓ η d pt →f C(d,c).\array{ F'(d)|_c &\to& F'(d) \\ \downarrow && \downarrow^{\eta_d} \\ pt &\stackrel{f}{\to}& C(d,c) } \,.
Example

Suppose the presheaf F∈PSh(C/c) does not actually depend on the morphsims to C, i.e. suppose that it factors through the forgetful functor from the over category to C:

F:(C/c) op→C op→Set.F : (C/c)^{op} \to C^{op} \to Set \,.

Then F′(d)=⊔ f∈C(d,c)F(f)=⊔ f∈C(d,c)F(d)≃C(d,c)×F(d) and hence F′=Y(c)×F with respect to the closed monoidal structure on presheaves.

See also functors and comma categories.

For the analog statement in (∞,1)-category theory see

For (∞,1)-category theory see (∞,1)-category of (∞,1)-presheaves.

Locally presentable categories: Large categories whose objects arise from small generators under small relations.

(n,r)-categories…satisfying Giraud's axiomsinclusion of left exaxt localizationsgenerated under colimits from small objectslocalization of free cocompletiongenerated under filtered colimits from small objects
(0,1)-category theory(0,1)-toposes↪algebraic lattices≃ Porst’s theoremsubobject lattices in accessible reflective subcategories of presheaf categories
category theorytoposes↪locally presentable categories≃ Adámek-Rosický’s theoremaccessible reflective subcategories of presheaf categories↪accessible categories
model category theorymodel toposes↪combinatorial model categories≃ Dugger’s theoremleft Bousfield localization of global model structures on simplicial presheaves
(∞,1)-topos theory(∞,1)-toposes↪locally presentable (∞,1)-categories≃
Simpson’s theorem		accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories↪accessible (∞,1)-categories

Revised on October 15, 2012 17:59:16 by Urs Schreiber (82.113.99.246)
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