(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
The -connected/-truncated factorization system is an orthogonal factorization system in an (∞,1)-category, specifically in an (∞,1)-topos, that generalizes the relative Postnikov systems of ∞Grpd: it factors any morphism through its (n+2)-image by an (n+2)-epimorphism followed by an (n+2)-monomorphism.
As ranges through these factorization systems form an ∞-ary factorization system.
Let be an (∞,1)-topos. For all the class of n-truncated morphisms in forms the right class in a orthogonal factorization system in an (∞,1)-category. The left class is that of n-connected morphisms in .
This appears as a remark in HTT, Example 5.2.8.16. A construction of the factorization in terms of a model category presentation is in (Rezk, prop. 8.5).
For this says that effective epimorphisms in an (∞,1)-category have the left lifting property against monomorphisms in an (∞,1)-category. Therefore one may say that the effective epimorphisms in an -topos are the strong epimorphisms.
For all , the -connected/-truncated factorization system is stable: the class of n-connected morphisms is preserved under (∞,1)-pullback.
This appears as (Lurie, prop. 6.5.1.16(6)).
It follows that:
For all , n-images are preserved by (∞,1)-pullback
Let with -image . By the pasting law its -pullback along any may be decomposed as two consecutive -pullbacks:
Here the pullback of the -truncated map in again -truncated since the right class of any orthogonal factorization system is stable under pullback. The analogous statement holds also for the -connected map by Prop. . Therefore the pullback of along is indeed , as shown.
A (-2)-truncated morphism is precisely an equivalence in an (∞,1)-category (see there or HTT, example 5.5.6.13).
Moreover, every morphism is (-2)-connected.
Therefore for the -connected/-truncated factorization system says (only) that equivalences have inverses, unique up to coherent homotopy.
A (-1)-truncated morphism is precisely a full and faithful morphism.
A (-1)-connected morphism is one whose homotopy fibers are inhabited.
In ∞Grpd a morphism between 0-truncated objects (sets)
is full and faithful precisely if it is an injection;
has non-empty fibers precisely if it is an epimorphism.
Therefore between 0-truncated objects the (-1)-connected/(-1)-truncated factorization system is the epi/mono factorization system and hence image factorization.
Generally, the (-1)-connected/(-1)-truncated factorization is through the -categorical 1-image, the homotopy image (see there for more details).
Let be two groupoids (homotopy 1-types) in ∞Grpd.
A morphism is 0-truncated precisely if it is a faithful functor.
A morphism is 0-connected precisely if it is a full functor and an essentially surjective functor: a essentially surjective and full functor
Therefore on homotopy 1-types the 0-connected/0-truncated factorization system is the (eso+full, faithful) factorization system.
The general abstract statement is in
A model category-theoretic discussion is in section 8 of
Discussion in homotopy type theory is in
Last revised on April 17, 2024 at 17:10:31. See the history of this page for a list of all contributions to it.