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derived hom-functor

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Idea

The (right) derived functor of an enriched hom-functor.

Definition

For enriched model categories

We discuss the definition of the derived hom-functor (Def. below) for enriched model categories (recalled as Def. below). The key point is to observe (Lemma below) that the axioms on enriched model categories imply that the ordinary enriched hom-functor is a homotopical functor when its first argument is cofibrant and its second argument is fibrant, so that its right derived functor exists.

\,

Definition

(enriched model category)

Let 𝒱\mathcal{V} be a monoidal model category. Then a 𝒱\mathcal{V}-enriched model category is a category 𝒞\mathcal{C} equipped with

  1. the structure of 𝒱\mathcal{V}-enriched category, which is also tensored and cotensored over 𝒱\mathcal{V};

  2. the structure of a model category,

such that these two structures are compatible in the following way:

  • for every cofibration XfYX \overset{f}{\to} Y and every fibration AgBA \overset{g}{\to} B in 𝒞\mathcal{C}, the induced pullback powering-morphism of hom-objects in 𝒱\mathcal{V}

    (1)𝒞(Y,A)𝒞(X,A)×𝒞(X,B)𝒞(Y,B) \mathcal{C}(Y,A) \longrightarrow \mathcal{C}(X,A) \underset{\mathcal{C}(X,B)}{\times} \mathcal{C}(Y,B)

    is a fibration, and is a weak equivalence as soon as one of the two morphisms is a weak equivalence in 𝒞\mathcal{C}.

Lemma

(in enriched model category the enriched hom-functor out of cofibrant into fibrant is homotopical functor)

Let 𝒞\mathcal{C} be an enriched model category (Def. ).

If Y𝒞Y \in \mathcal{C} is a cofibrant object, then the enriched hom-functor out of YY

𝒞(Y,):𝒞𝒱 \mathcal{C}(Y,-) \;\colon\; \mathcal{C} \longrightarrow \mathcal{V}

preserves fibrations and acyclic fibrations.

If A𝒞A \in \mathcal{C} is a fibrant object, then the enriched hom-functor into AA

𝒞(,A):𝒞 op𝒱 \mathcal{C}(-,A) \;\colon\; \mathcal{C}^{op} \longrightarrow \mathcal{V}

sends cofibrations and acyclic cofibrations in 𝒞\mathcal{C} to fibrations and acyclic fibrations, respectively, in 𝒱\mathcal{V}.

Proof

In the first case, consider the comparison morphism (1) for X=X =\emptyset the initial object, in the second case consider it for B=*B = \ast the terminal object.

Since 𝒞\mathcal{C} is tensored and cotensored over 𝒱\mathcal{V}, it follows (by this Prop that

𝒞(,)*AAandAA𝒞(,*)*𝒱. \mathcal{C}(\emptyset, -) \;\simeq\; \ast \phantom{AA} \text{and} \phantom{AA} \mathcal{C}(-,\ast) \;\simeq\; \ast\; \;\;\; \in \mathcal{V} \,.

This means that in the first case the comparison morphism

𝒞(Y,A)𝒞(X,A)×𝒞(X,B)𝒞(Y,B) \mathcal{C}(Y,A) \longrightarrow \mathcal{C}(X,A) \underset{\mathcal{C}(X,B)}{\times} \mathcal{C}(Y,B)

(1) becomes equal to the top morphism in the following diagram

𝒞(Y,A) 𝒞(Y,g) 𝒞(Y,B) * AAA * \array{ \mathcal{C}(Y,A) &\overset{\mathcal{C}(Y,g)}{\longrightarrow}& \mathcal{C}(Y,B) \\ \Big\downarrow && \Big\downarrow \\ \ast &\underset{\phantom{AAA}}{\longrightarrow}& \ast }

while in the second case it becomes equal to the left morphism in

𝒞(Y,A) 𝒞(Y,g) * 𝒞(f,A) 𝒞(X,A) AAA * \array{ \mathcal{C}(Y,A) &\overset{\phantom{\mathcal{C}(Y,g)}}{\longrightarrow}& \ast \\ {}^{\mathllap{ \mathcal{C}(f,A) }}\Big\downarrow && \Big\downarrow \\ \mathcal{C}(X,A) &\underset{\phantom{AAA}}{\longrightarrow}& \ast }

Hence the claim follows by the defining condition on the comparison morphism (1) in an enriched model category.

Definition

(derived hom-functor of an enriched model category)

Let 𝒞\mathcal{C} be an enriched model category (Def. ).

By Lemma and by Ken Brown's lemma, the enriched hom-functor has a right derived functor (this Def.) when its first argument is cofibrant and its second argument is fibrant. This is called the derived hom-functor

hom:Ho(𝒞) op×Ho(𝒞)Ho(𝒱) \mathbb{R}hom \;\colon\; Ho(\mathcal{C})^{op} \times Ho(\mathcal{C}) \longrightarrow Ho(\mathcal{V})

In the presence of functorial cofibrant resolution QQ and fibrant resolution PP this is given by the ordinary enriched hom-functor 𝒞(,)\mathcal{C}(-,-) as

hom(X,Y)𝒞(QX,PY). \mathbb{R}hom(X,Y) \;\simeq\; \mathcal{C}(Q X, P Y) \,.

Examples

Last revised on May 14, 2019 at 08:34:09. See the history of this page for a list of all contributions to it.