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

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 $\mathrm{Id}\times \mathrm{Id}:c\to c{\times }_d^{h}c$ 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 $c\to f(c)$ 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.

This is definition 2.36 in

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