nLab
Brown functor

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In the context of Brown representability

Definition

Let 𝒞\mathcal{C} be a locally presentable (∞,1)-category. A functor

F:Ho(𝒞) opSet F \;\colon\; Ho(\mathcal{C})^{op} \longrightarrow Set

(from the opposite of the homotopy category of 𝒞\mathcal{C} to Set)

is called a Brown functor if

  1. it sends small coproducts to products;

  2. it sends (∞,1)-pushouts in 𝒞Ho(𝒞)\mathcal{C}\to Ho(\mathcal{C}) to weak pullbacks in Set.

Remark

A weak pullback is a diagram that satisfies the existence clause of a pullback, but not necessarily the uniqueness condition. Hence the second clause in def. 1 says that for a (∞,1)-pushout square

Z X Y XZY \array{ Z &\longrightarrow& X \\ \downarrow &\swArrow& \downarrow \\ Y &\longrightarrow& X \underset{Z}{\sqcup}Y }

in 𝒞\mathcal{C}, then the induced universal morphism

F(XZY)epiF(X)×F(Z)F(Y) F\left(X \underset{Z}{\sqcup}Y\right) \stackrel{epi}{\longrightarrow} F(X) \underset{F(Z)}{\times} F(Y)

into the actual pullback is an epimorphism.

Proposition

(Brown representability theorem)

Let 𝒞\mathcal{C} be a locally presentable (∞,1)-category.

If

  1. 𝒞\mathcal{C} is generated by a set

    {S i𝒞} iI \{S_i \in \mathcal{C}\}_{i \in I}

    of compact objects (i.e. every object of 𝒞\mathcal{C} is an (∞,1)-colimit of the objects S iS_i.)

  2. each S iS_i admits the structure of a cogroup object in the homotopy category Ho(𝒞)Ho(\mathcal{C}),

then a functor

F:Ho(𝒞) opSet F \;\colon\; Ho(\mathcal{C})^{op} \longrightarrow Set

(from the opposite of the homotopy category of 𝒞\mathcal{C} to Set)

is representable precisely if it is a Brown functor, def. 1.

In the context of categories of fibrant objects

A right Brown functor between categories of fibrant objects is a functor that preserves weak equivalences between fibrant objects.

Dually for a left Brown functor (for instance between Waldhausen categories).

Revised on March 11, 2016 10:04:45 by Urs Schreiber (86.187.101.31)