nLab homotopy image

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 $Id \times Id : c \to c \times^h_d c$ 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 $c \to f(c)$ followed by a homotopy monomorphism $f(c) \to d$

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

In an $(\infty,1)$-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.

Related concepts

• image, coimage

References

A definition for model categories is def. 2.36 in

• Clark Barwick, On (enriched) left Bousfield localization of model categories (arXiv)

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