homotopy image


Model category theory

model category



Universal constructions


Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras



for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-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 f:cdf : c \to d is in terms of universal subobjects – i.e. universal monomorphisms – through which ff factors.

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

Definition (homotopy image)

Let CC be an SS enriched model category satisfying some assumptions… .

  • A morphism f:cdf : c \to d in CC is called a homotopy monomorphism if the universal morphism Id×Id:cc× d hcId \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 ff is a factorization of ff into a cofibration cf(c)c \to f(c) followed by a homotopy monomorphism f(c)df(c) \to d

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

In an (,1)(\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 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.


A definition for model categories is def. 2.36 in

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

Last revised on July 2, 2012 at 22:58:18. See the history of this page for a list of all contributions to it.