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

Idea

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

Definitions

Proposition

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

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).

Remark

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

Properties

Stability

Proposition

For all $n$, the $n$-connected/$n$-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:

Proposition

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

Proof

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

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

Examples

The case $n = -2$

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

The case $n = -1$

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 $n = 0$

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

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

A morphism $X \to Y$ 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.

References

The general abstract statement is in

• Jacob Lurie, Higher Topos Theory

A model category-theoretic discussion is in section 8 of

• Charles Rezk, Toposes and homotopy toposes (pdf)

Discussion in homotopy type theory is in

• Univalent Foundations Project, section 7.6 of Homotopy Type Theory – Univalent Foundations of Mathematics