model category

for ∞-groupoids

# Contents

## Idea

The notion of homotopy image generalizes the notion of image of a morphism in a category to that of a morphism in a presentable (infinity,1)-category or model category.

## Definition

### In a model category

One of the definitions of the image of a morphism $f:c\to d$ is in terms of universal subobjects – i.e. universal monomorphisms – through which $f$ factors.

This definition can be generalized to the context of (infinity,1)-categories presented by a model category.

###### Definition (homotopy image)

Let $C$ be an $S$ enriched model category satisfying some assumptions… .

• A morphism $f:c\to d$ in $C$ is called a homotopy monomorphism if the universal morphism $\mathrm{Id}×\mathrm{Id}:c\to c{×}_{d}^{h}c$ into its homotopy pullback along itself is an isomorphism in the homotopy category.

• The homotopy image of $f$ is a factorization of $f$ into a cofibration $c\to f\left(c\right)$ followed by a homotopy monomorphism $f\left(c\right)\to d$

• such that for any other such factorization $c\to e\to d$ there exists a unique morphism $f\left(c\right)\to e$ in the homotopy category making the obvious triangles commute.

### In an $\left(\infty ,1\right)$-topos

The above definition of homotopy monomorphism presents precisely the notion of monomorphism in an (∞,1)-category : a (-1)-truncated morphism. Because (HTT, lemma 5.5.6.15) a morphism is (-1)-truncated precisely if its diagonal is (-2)-truncated, hence is an equivalence.

Therefore in an (∞,1)-topos the homtopy image of a morphism is a presentation for the (-1)-connected/(-1)-truncated factorization of the morphism.

See ∞-image.

## References

A definition for model categories is def. 2.36 in

For the definition in $\left(\infty ,1\right)$-topos theory see the references at n-connected/n-truncated factorization system.

Revised on July 2, 2012 22:58:18 by Urs Schreiber (89.204.130.180)