model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-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.
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.
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 $f(c) \to d$
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 $(\infty,1)$-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.