on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
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.
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.
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
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.
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.