nLab (n-connected, n-truncated) factorization system



Factorization systems

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

structures in a cohesive (∞,1)-topos

(,1)(\infty,1)-Category theory



The nn-connected/nn-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 nn ranges through (1),0,1,2,3,(-1), 0, 1, 2, 3, \cdots these factorization systems form an ∞-ary factorization system.



Let H\mathbf{H} be an (∞,1)-topos. For all (2)n<(-2) \leq n \lt \infty the class of n-truncated morphisms in H\mathbf{H} forms the right class in a orthogonal factorization system in an (∞,1)-category. The left class is that of n-connected morphisms in H\mathbf{H}.

This appears as a remark in HTT, Example A construction of the factorization in terms of a model category presentation is in (Rezk, prop. 8.5).


For n=1n = -1 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 (,1)(\infty,1)-topos are the strong epimorphisms.




For all nn, the nn-connected/nn-truncated factorization system is stable: the class of n-connected morphisms is preserved under (∞,1)-pullback.

This appears as (Lurie, prop.

It follows that:


For all nn, n-images are preserved by (∞,1)-pullback


Let XfYX \xrightarrow{f} Y with nn-image im n(f)im_n(f). By the pasting law its \infty -pullback along any g:XXg \,\colon\, X' \xrightarrow{\;} X may be decomposed as two consecutive \infty-pullbacks:

Here the pullback of the nn-truncated map in again nn-truncated since the right class of any orthogonal factorization system is stable under pullback. The analogous statement holds also for the nn-connected map by Prop. . Therefore the pullback of im n(f)im_n(f) along gg is indeed im n(g *f)im_n(g^\ast f), as shown.


The case n=2n = -2

A (-2)-truncated morphism is precisely an equivalence in an (∞,1)-category (see there or HTT, example

Moreover, every morphism is (-2)-connected.

Therefore for n=2n = -2 the nn-connected/nn-truncated factorization system says (only) that equivalences have inverses, unique up to coherent homotopy.

The case n=1n = -1

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 \infty-categorical 1-image, the homotopy image (see there for more details).

The case n=0n = 0

Let X,YX,Y be two groupoids (homotopy 1-types) in ∞Grpd.

A morphism XYX \to Y is 0-truncated precisely if it is a faithful functor.

A morphism XYX \to Y 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.