Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
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.
In a model category
One of the definitions of the image of a morphism is in terms of universal subobjects – i.e. universal monomorphisms – through which factors.
This definition can be generalized to the context of (infinity,1)-categories presented by a model category.
Definition (homotopy image)
Let be an enriched model category satisfying some assumptions… .
A morphism in is called a homotopy monomorphism if the universal morphism into its homotopy pullback along itself is an isomorphism in the homotopy category.
The homotopy image of is a factorization of into a cofibration followed by a homotopy monomorphism
- such that for any other such factorization there exists a unique morphism in the homotopy category making the obvious triangles commute.
In an -topos
The above definition of homotopy monomorphism presents precisely the notion of monomorphism in an (∞,1)-category : a (-1)-truncated morphism. Because (HTT, lemma 22.214.171.124) 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.
A definition for model categories is def. 2.36 in
For the definition in -topos theory see the references at n-connected/n-truncated factorization system.
Revised on July 2, 2012 22:58:18
by Urs Schreiber