nLab Brown functor

In the context of Brown representability

Definition

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

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

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

is called a Brown functor if

1. it sends small coproducts to products;

2. it sends (∞,1)-pushouts in $\mathcal{C}\to Ho(\mathcal{C})$ 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. says that for a (∞,1)-pushout square

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

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

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

into the actual pullback is an epimorphism.

Proposition

(Brown representability theorem)

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

If

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

   $$\{S_i \in \mathcal{C}\}_{i \in I}$$

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

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

then a functor

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

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

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

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).