Truncation | Higher Topos Theory notes

3.1. Truncation

Truncation organizes objects and morphisms according to the amount of homotopical information they contain. The definition is recursive: a morphism is $$n$$ -truncated when its diagonal is $$(n-1)$$ -truncated. We first recall this hierarchy and construct the truncation functors in any presentable category. We then develop the relative truncation calculus needed later, with particular attention to its interaction with base change, monomorphisms, and effective epimorphisms.

Definition 3.1.

Let $$C$$ be a category with pullbacks. A morphism $f\colon X \to Y$ in $$C$$ is called $$(-2)$$ -truncated if it is an isomorphism. For $n \geq -1$ , we say $$f$$ is $$n$$ -truncated if $\Delta_f\colon X \to X \times_Y X$ is $$(n-1)$$ -truncated. We denote by $C_{\leq n} \subseteq C$ the full subcategory of $$n$$ -truncated objects.

If the inclusion $C_{\leq n} \hookrightarrow C$ admits a left adjoint, we denote this adjoint by

\tau_n\colon C \to C_{\leq n},

and call it the $$n$$ -truncation functor.

Remark 3.2.

Being $$(-1)$$ -truncated is the same as being a monomorphism.

Remark 3.3.

The truncation functor $\tau_n\colon C \to C_{\leq n}$ always exists when $$C$$ is presentable, hence in particular when $$T$$ is a topos.

To see this, recall that $$C$$ admits a tensoring and a cotensoring by the category $\An$ . There are unique functors

- \otimes -\colon \An \times C \to C \qquadtext{ and } (-)^{(-)}\colon \An\catop \times C \to C

satisfying $* \otimes X \cong X \cong X^*$ for all $X \in C$ and preserving colimits (resp. limits) in the first variable. Here and below we use the convention $S^{-1} := \emptyset$ . Thus, for $$n=-2$$ , the map below is the terminal map $X \to X^{\emptyset} \cong *$ . By an easy induction argument, an object $X \in C$ is $$n$$ -truncated if and only if the map $X \to X^{S^{n+1}}$ induced by $S^{n+1} \to *$ is an isomorphism.

Since $$C$$ is presentable, there is a small collection of objects $\{Y\}$ that detect equivalences. Since $\Hom_C(Y, X^{S^{n+1}}) \simeq \Hom_C(Y \otimes S^{n+1}, X)$ , the $$n$$ -truncated objects are precisely the local objects with respect to the small collection of morphisms $\{S^{n+1} \otimes Y \to * \otimes Y = Y\}$ . In particular, the $$n$$ -truncated objects form a reflective subcategory of $$C$$ , admitting a reflector $\tau_n\colon C \to C_{\leq n}$ .

Exercise 3.4.

Show that a stable category does not admit any non-zero $$n$$ -truncated objects.

We now prove some basic properties of truncated objects.

Lemma 3.5.

The $$n$$ -truncated morphisms in $$C$$ are closed under base change.

Proof

Immediate from the definitions, details left to the reader.

Lemma 3.6.

A morphism $f\colon X \to Y$ is $$n$$ -truncated if and only if the induced map of animae $f_*\colon \Hom_C(Z,X) \to \Hom_C(Z,Y)$ is $$n$$ -truncated for all $Z \in C$ .

Proof

Immediate from Yoneda lemma and the fact that

\Hom_C(Z,-)

preserves pullbacks.

Lemma 3.7.

Consider morphisms $f\colon X \to Y$ and $g\colon Y \to Z$ .

If $$g$$ is $$n$$ -truncated, then $$f$$ is $$n$$ -truncated if and only if $$gf$$ is $$n$$ -truncated.
If $$gf$$ is $$n$$ -truncated and $$g$$ is $$(n+1)$$ -truncated, then $$f$$ is $$n$$ -truncated.

Proof

(1) We prove the claim by induction on

\(n\)

. The case

\(n = -2\)

is clear by 2-out-of-3.For

n \geq -1

, consider the following commutative triangle:

The right diagonal map is a base change of

\Delta_g\colon Y \to Y \times_Z Y

, hence is

\((n-1)\)

-truncated. By the induction hypothesis,

\Delta_f

\((n-1)\)

-truncated if and only if

\Delta_{gf}

\((n-1)\)

-truncated. This completes the proof.(2) We may factor

\(f\)

as the composite

X \xrightarrow{(\id_X,f)} X \times_Z Y \xrightarrow{\pr_Y} Y.

The first map is a base change of the

\(n\)

-truncated map

\Delta_g\colon Y \to Y \times_Z Y

. The second map is a base change of the

\(n\)

-truncated map

gf\colon X \to Z

. By (1) and Lemma 3.5,

\(f\)

is also

\(n\)

-truncated.

Lemma 3.8.

Consider a functor $F\colon C \to D$ .

If $$F$$ preserves pullbacks, then it preserves $$n$$ -truncated objects for all $n \geq -2$ .
If $$F$$ furthermore admits a right adjoint, and $$C$$ and $$D$$ admit $$n$$ -truncation functors, then $$F$$ commutes with $$n$$ -truncation: for any $X \in C$ the canonical map
$\tau_n(F(X)) \to F(\tau_n X)$
is an isomorphism.

Proof

Part (1) is clear, since

\(n\)

-truncatedness is formulated in terms of pullbacks.For part (2), the claim is equivalent to the claim that the right adjoint

G\colon D \to C

\(F\)

preserves

\(n\)

-truncated objects. This is an instance of (1).

Lemma 3.9.

For every $n\geq -2$ , the truncation functor $\tau_n\colon T\to T_{\leq n}$ of a topos preserves finite products.

Proof

Choose a left exact localization

L\colon \PSh(C)\to T

with fully faithful right adjoint

\(R\)

. In a presheaf topos, truncation is computed pointwise, and truncation of animae preserves finite products. It follows that

\tau_n^{\mathrm{pre}}

preserves finite products in

\PSh(C)

. By Lemma 3.8, the truncation functor on

\(T\)

is naturally isomorphic to the composite

L\tau_n^{\mathrm{pre}}R

. The functors

\(L\)

and

\(R\)

preserve finite limits, so this composite preserves finite products.

Corollary 3.10.

For a morphism $f\colon X \to Y$ in a topos $$T$$ , the pullback functor $f^*\colon T_{/Y} \to T_{/X}$ preserves $$n$$ -truncated objects and commutes with $$n$$ -truncation: for a map $Z \to Y$ we have $\tau_n(f^*Z/X) \cong f^*\tau_{n}(Z/Y)$ .

Proof

Lemma 3.8 applies: the functor

\(f^*\)

preserves limits (as it has a left adjoint) and preserves colimits (by universality of colimits), so that it admits a right adjoint

f_*\colon T_{/X} \to T_{/Y}

by the adjoint functor theorem.

Proposition 3.11. ([Anel et al. 2020, Proposition 2.2.6])

Consider a pushout square

in which the map $A \hookrightarrow B$ is a monomorphism. Then also $C \to D$ is a monomorphism and the square is a pullback square.

Proof

Consider the cube

The top and bottom faces are pushouts, and the left and back faces are pullbacks. By descent (Mather's first cube lemma), the front and right faces are also pullback squares. This says precisely that the starting square is a pullback square and that the map

C \to D

is a monomorphism.

Lemma 3.12.

Let $U \hookrightarrow X$ be a monomorphism. Then also $\tau_0 U \hookrightarrow \tau_0 X$ is a monomorphism, and the square

is a pullback square.

Proof

Choose a left exact localization

L\colon \PSh(C) \to T

with fully faithful right adjoint

R\colon T \hookrightarrow \PSh(C)

, as in Theorem 2.42. Since

\(R\)

preserves limits, the map

R(U) \to R(X)

is again a monomorphism. Truncation and limits in the presheaf topos are computed pointwise. For every

c \in C

, the monomorphism of animae

R(U)(c) \hookrightarrow R(X)(c)

is the inclusion of a union of connected components. Consequently,

\tau_0R(U)(c) \to \tau_0R(X)(c)

is a monomorphism and the corresponding naturality square is a pullback. It follows pointwise that the square

is a pullback in

\PSh(C)

. Applying the left exact functor

\(L\)

preserves this pullback square and the monomorphism on the right. Moreover, Lemma 3.8 identifies

L\tau_0R(U)

and

L\tau_0R(X)

with

\tau_0U

and

\tau_0X

, respectively. This gives the asserted square in

\(T\)

We now prove a crucial fact: effective epimorphisms in a topos can be tested on 0-truncations.

Lemma 3.13. (Key lemma)

Let $$T$$ be a topos.

The map $X \to \tau_0 X$ is an effective epimorphism for every $X \in T$ .
A morphism $f\colon Y \to X$ is an effective epimorphism if and only if the map $\tau_0 Y \to \tau_0 X$ is an effective epimorphism.

Remark 3.14.

The corresponding argument in Lurie (2009) contains a circular dependency. Proposition~7.2.1.14 is the analogue of the lemma above. Its proof uses Lemma~7.2.1.13, which invokes Proposition~6.5.1.20; the proof of Proposition~6.5.1.20 in turn appeals to Proposition~7.2.1.14. Proposition~6.5.1.12 also uses Proposition~7.2.1.14.

There is a related gap in the proof of [Lurie 2009, Proposition 6.5.1.18]: one still has to show that the diagonal of a $$0$$ -connected morphism is an effective epimorphism. The proof below establishes the effective-epimorphism criterion independently, and we will use it to supply this step in Theorem 3.22.

Proof

(1) Factor the map

X \to \tau_0 X

into an effective epimorphism followed by a monomorphism:

X \twoheadrightarrow U \hookrightarrow \tau_0 X.

We must show that the second map is an isomorphism. Since it is a monomorphism (

\((-1)\)

-truncated) and

\tau_0 X

\(0\)

-truncated, Lemma 3.7 implies that

\(U\)

is also

\(0\)

-truncated. By definition,

\tau_0 X

is the initial

\(0\)

-truncated object equipped with a map from

\(X\)

. Therefore the map

U \hookrightarrow \tau_0 X

admits a section

\tau_0 X \to U

. In particular, it is an isomorphism by Corollary 2.39.(2) The “only if” direction is clear from the commutative diagram:

Indeed, by part (1) the map

X \to \tau_0 X

is an effective epimorphism. If

\(f\)

is also an effective epimorphism, then so is the composite

Y \to \tau_0 X

by Lemma 2.37. Then

\tau_0 Y \to \tau_0 X

is also an effective epimorphism by Lemma 2.40. Conversely, assume that

\tau_0 f

is an effective epimorphism. Consider the epi–mono factorization

Y \twoheadrightarrow U \hookrightarrow X

\(f\)

and the induced diagram

By Lemma 3.12, the map

\tau_0 U \to \tau_0 X

is again a monomorphism, and the right-hand square is a pullback square. Since the composite

\tau_0 Y \to \tau_0 U \to \tau_0 X

is an effective epimorphism by assumption, Lemma 2.40 implies that

\tau_0 U \to \tau_0 X

is an effective epimorphism. Since it is also a monomorphism, it is an isomorphism by Corollary 2.39. Since the right-hand square is a pullback square,

U \to X

is also an isomorphism, as desired.

Corollary 3.15.

A map $*\to X$ in a topos $$T$$ is an effective epimorphism if and only if $\tau_0 X$ is the terminal object of $$T$$ .

References

Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal. A generalized Blakers-Massey theorem. J. Topol., 13 (4), 1521–1553. 2020.
Jacob Lurie. Higher topos theory. Ann. Math. Stud. 170, Princeton, NJ: Princeton University Press. 2009.