nLab
n-truncated object of an (infinity,1)-category

Context

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

Homotopy theory

Modalities, Closure and Reflection

Contents

Idea

An nn-truncated ∞-groupoid is an n-groupoid.

An nn-truncated topological space is a homotopy n-type: all homotopy groups above degree nn are trivial.

An nn-truncated object in a general (∞,1)-category is an object such that all hom-∞-groupoids into it are nn-truncated.

If an object in an (∞,1)-topos_ is kk-truncated for any (possibly large) kk, then it is nn-truncated precisely if all its categorical homotopy groups above degree nn are trivial.

The complementary notion of nn-truncated object is that of an n-connected object of an (∞,1)-category.

Definition

In terms of truncations

Definition

(nn-truncated \infty-groupoid)

An ∞-groupoid AGrpdA \in \infty Grpd is nn-truncated for nn \in \mathbb{N} if it is an n-groupoid:

Precisely: in the model of ∞-groupoids given by Kan complexes AA is nn-truncated if the simplicial homotopy groups π k(A,x)\pi_k(A,x) are trivial for all xx and all k>nk \gt n.

It makes sense for the following to adopt the convention that AA is called

  • (1)(-1)-truncated if it is empty or contractible – this is a (-1)-groupoid.

  • (2)(-2)-truncated if it is non-empty and contractible – this is a (-2)-groupoid.

(following HTT, p. 6).

To generalize this, let now CC be an arbitrary (∞,1)-category. For X,AX,A objects in CC write C(X,A)C(X,A) \in ∞ Grpd for the (∞,1)-categorical hom-space (if CC is given as a simplicially enriched category then this is just the SSet-hom-object which is guaranteed to be a Kan complex).

Using this, it shall be useful to slightly reformulate the above as follows:

Observation

An ∞-groupoid AA is nn-truncated precisely for all other ∞-groupoids XX the hom-\infty-groupoid Grpd(X,A)\infty Grpd(X,A) is nn-truncated.

In categorical terms this just says that (∞,k)-transformation between XX and AA whose components a k-morphisms in AA cannot be nontrivial for k>nk \gt n if there are no nontrivial k-morphisms with k>nk \gt n in AA.

Using this fact we can transport the notion of nn-truncation to any (∞,1)-category by testing it on hom-∞-groupoids:

Definition

(nn-truncated object in an (,1)(\infty,1)-category)

An object ACA \in C of an (∞,1)-category CC is nn-truncated, for nn \in \mathbb{N}, if for all XCX \in C the hom-∞-groupoid C(X,A)C(X,A) is nn-truncated.

This is HTT, def. 5.5.6.1.

Some terminology:

  • A 0-truncated object is also called discrete . Notice that this is categorically discrete as in discrete category, not discrete in the sense of topological spaces. An object in an (∞,1)-topos is discrete in this sense if, regarded as an ∞-groupoid with extra structure it has only trivial morphisms.

  • By the above convention on (-2)-truncated \infty-groupoids, it is only the terminal objects of CC that are (-2)-truncated.

  • Similarly, the (-1)-truncated objects are the subterminal objects.

Definition

(nn-truncated morphism in an (,1)(\infty,1)-category)

A morphism f:XYf : X \to Y of ∞-groupoids is nn-truncated if all of its homotopy fibers are nn-truncated by def. 2.

A morphism f:XYf : X \to Y in an (∞,1)-category CC is nn-truncated if for all WCW \in C the postcomposition morphism

C(W,f):C(W,X)C(W,Y) C(W,f) : C(W,X) \to C(W,Y)

is nn-truncated in ∞Grpd.

By the characterization of homotopy fiber of functor categories this is equivalent to saying that ff is kk-truncated when it is so regarded as an object of the over (∞,1)-category C /YC_{/Y}.

In terms of categorical homotopy groups

At least if the ambient (∞,1)-category is even an ∞-stack (∞,1)-topos there is an alternative, more intrinsic, characterization of nn-truncation in terms of categorical homotopy groups in an (∞,1)-topos:

Proposition

Suppose that an object XX in an ∞-stack (∞,1)-topos is kk-truncated for some kk \in \mathbb{N} (possibly very large).

Then for any nn \in \mathbb{N} this XX is nn-truncated precisely if all the categorical homotopy groups above degree nn are trivial.

Proof

This is HTT, prop 6.5.1.7.

Remark

Notice that this expected statement does require the assumption that XX is kk-truncated for some kk. Without any a priori truncation assumption on XX, there is no comparable statement about the relaton to categorical homotopy groups. See HTT, remark 6.5.1.8.

Properties

Recursive definition

Proposition

In an (,1)(\infty,1)-category CC with finite limits, a morphism f:XYf : X \to Y is kk-truncated (for k1k \geq -1) precisely if the diagonal morphism XX× YXX \to X \times_Y X is (k1)(k-1)-truncated.

This is HTT, lemma 5.5.6.15.

Proof

By definition ff is kk-truncated if for each object dCd \in C we have that C(d,f)C(d,f) is kk-truncated in ∞Grpd. Since the hom-functors C(d,)C(d,-) preserve (∞,1)-limits, we have in particular that XX× YXX \to X \times_Y X in CC is kk-truncated if C(d,X)C(d,X)× C(d,Y)C(d,X)C(d,X) \to C(d,X) \times_{C(d,Y)} C(d,X) is kk-truncated for all dd in ∞Grpd. Therefore it is sufficient to prove the statement for morphisms in C=C = ∞Grpd.

So let now f:XYf : X \to Y be a morphism of ∞-groupoids. We may find a fibration ϕ¯:X¯Y¯\bar \phi : \bar X \to \bar Y between Kan complexes in sSet that models f:XYf : X \to Y in the standard model structure on simplicial sets, and by the standard rules for homotopy pullbacks it follows that the object X× YXX \times_Y X in \infty-Grpd is then modeled by the ordinary pullback X¯× Y¯X¯\bar X \times_{\bar Y} \bar X in sSet. And the homotopy fibers of ff over yYy \in Y are then given by the ordinary fibers X¯ y\bar X_y of f¯\bar f in sSetsSet.

This way the statement is reduced to the following fact: a Kan complex X¯ y\bar X_y is kk-truncated precisely if the homotopy fibers of X¯ yX¯ y×X¯ y\bar X_y \to \bar X_y \times \bar X_y are (k1)(k-1)-truncated.

We write now XX for X¯ y\bar X_y, for simplicity. To see the last statement, let (a,b):*X×X(a,b) : * \to X \times X and compute the homotopy pullback

Q * (a,b) X X×X \array{ Q &\to& * \\ \downarrow && \downarrow^{\mathrlap{(a,b)}} \\ X &\to& X \times X }

as usual by replacing the right vertical morphism by the fibration (X×X) I× X×X(a,b)X×X(X \times X)^I \times_{X \times X} (a,b) \to X \times X and then forming the ordinary pullback. This shows that QQ is equivalent to the space of paths P a,bXP_{a,b}X in XX from aa to bb. (Use that gluing of path space objects at endpoints of paths produces a new path space, see for instance section 4 of BrownAHT).

If XX is connected, then choosing any path aba \to b gives an isomorphism from the the homotopy groups of P a,bXP_{a,b} X to those of the loop space Ω aX\Omega_a X. These latter are indeed those of XX, shifted down in degree by one (as described for instance at fiber sequence).

If XX is not connected, we can easily reduce to the case that it is.

General

Proposition

For CC an (,1)(\infty,1)-category and k2k \geq -2, the full sub-(∞,1)-category τ kC\tau_{\leq k} C is stable under all limits in CC.

This is HTT, prop. 5.5.6.5.

Proposition

Let H\mathbf{H} be an (∞,1)-topos. For all (2)n(-2) \leq n \leq \infty the class of nn-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 5.2.8.16. A construction of the factorization in terms of a model category presentation is in (Rezk, prop. 8.5). See also n-connected/n-truncated factorization system.

Model category presentations

There are model structures for homotopy n-types that presentable (∞,1)-category present the full sub-(∞,1)-categories of nn-truncated objects in some ambient (,1)(\infty,1)-category. See there for more details.

Truncation

Definition

Under mild conditions there is for each nn a universal way to send an arbitrary object AA to its nn-truncation τ nA\tau_{\leq n} A. This is a general version of decategorification where n-morphisms are identified if they are connected by an invertible (n+1)(n+1)-morphism.

For CC an (∞,1)-category and n2n \geq -2 in \mathbb{Z} write τ nC\tau_{\leq n} C for the full subcategory of CC on its nn-truncated objects.

So for instance for C=C = ∞Grpd we have τ nGrpd=nGrpd\tau_{\leq n} \infty Grpd = n Grpd.

Proposition

If CC is an (∞,1)-category that is presentable then the canonical inclusion (∞,1)-functor

τ nCC \tau_{\leq n} C \hookrightarrow C

has an accessible left adjoint

τ n:CC n. \tau_{\leq n} : C \to C_{\leq n} \,.

This is HTT 5.5.6.18.

Indeed, as the notation suggests, C nC_{\leq n} is the essential image of CC under τ n\tau_{\leq n}. The image τ nA\tau_{\leq n} A of an object AA under this operation is the nn-truncation of AA.

So nn-truncated objects form a reflective sub-(∞,1)-category

τ nCτ nC. \tau_{\leq n} C \stackrel{\overset{\tau_{\leq n}}{\leftarrow}}{\hookrightarrow} C \,.

Properties

General

Observation/Corollary

A left exact functor F:CDF : C \to D between (,1)(\infty,1)-categories with finite limits sends kk-truncated objects/morphisms to kk-truncated objects/morphisms.

This is HTT, prop. 5.5.6.1.6.

Proof

Follows from the above recursive characterization of kk-truncated morphisms by the (k1)(k-1)-truncation of their diagonal, which is preserved by the finite limit preserving FF.

Proposition

A presentable (,1)(\infty,1)-functor F:CDF : C \to D between locally presentable (∞,1)-categories CC and DD commutes with truncation:

Fτ k Cτ k DF. F \circ \tau^C_{\leq k} \simeq \tau^D_{\leq k} \circ F \,.

This is HTT, prop. 5.5.6.28.

Proof

By the above lemma, FF restricts to a functor on the truncations. So we need to show that the diagram

C F D τ k (?) τ k τ kC F τ kD \array{ C &\stackrel{F}{\to}& D \\ {}^{\mathllap{\tau_{\leq k}}}\downarrow & (?) & \downarrow^{\mathrlap{\tau_{\leq k}}} \\ \tau_{\leq k } C &\stackrel{F}{\to}& \tau_{\leq k} D }

in (∞,1)Cat can be filled by a 2-cell. To see this, notice that the adjoint (∞,1)-functor of both composite morphisms exists (because that of FF exists by the adjoint (∞,1)-functor theorem and bcause adjoints of composites are composites of adjoints) and since the bottom morphism is just the restriction of the top morphism and the right adjoints of the vertical morphisms are full inclusions this adjoint diagram

C G D τ kC G τ kD. \array{ C &\stackrel{G}{\leftarrow}& D \\ \uparrow & & \uparrow \\ \tau_{\leq k } C &\stackrel{G}{\leftarrow}& \tau_{\leq k} D } \,.

evidently commutes, since it just expresses this restriction.

Proposition

If CC is an (∞,1)-topos, then truncation τ n:CC\tau_{\leq n} : C \to C preserves finite products.

This appears as HTT, lemma 6.5.1.2.

Proof

First notice that the statement is true for C=C = ∞Grpd. For instance we can use the example In ∞Grpd and Top, model ∞-groupoids by Kan complexes and notice that then τ n\tau_{\leq n} is given by the truncation functor tr n+1:sSet[Δ n+1 op,sSet]tr_{n+1} : sSet \to [\Delta^{op}_{\leq n+1}, sSet]. This is also a right adjoint and as such preserves in particular product in sSetsSet, which are (,1)(\infty,1)-products in Grpd\infty Grpd.

From that we deduce that the statement is true for CC any (∞,1)-category of (∞,1)-presheaves C=PSh (,1)(K)=Func (,1)(K op,Grpd)C = PSh_{(\infty,1)}(K) = Func_{(\infty,1)}(K^{op}, \infty Grpd) because all relevant operations there are objectwise those in Grpd\infty Grpd.

So far this shows even that on presheaf (,1)(\infty,1)-toposes all products (not necessarily finite) are preserved by truncation.

A general (∞,1)-topos CC is (by definition) a left exact reflective sub-(∞,1)-category of a presheaf (,1)(\infty,1)-topos,

CiLPSh (,1)(K). C \stackrel{\overset{L}{\leftarrow}}{\underset{i}{\hookrightarrow}} PSh_{(\infty,1)}(K) \,.

Let ji(X j)\prod_{j} i(X_j) be the product of the objects in question taken in PSh(K)PSh(K). By the above there we have an equivalence

τ k ji(X j) jτ i(X j). \tau_{\leq k} \prod_j i(X_j) \stackrel{\simeq}{\to} \prod_j \tau_{\leq} i(X_j) \,.

Now applying LL to this equivalence and using now that LL preserves the finite product, this gives an equivalence

Lτ k ji(X j)L jτ i(X j) jLτ i(X j) L \tau_{\leq k} \prod_j i(X_j) \stackrel{\simeq}{\to} L \prod_j \tau_{\leq} i(X_j) \simeq \prod_j L \tau_{\leq} i(X_j)

in CC. The claim follows now with the above result that Lτ nτ nLL \circ \tau_{\leq n} \simeq \tau_{\leq n} \circ L.

Postnikov tower

By the fact that the truncation functor τ n\tau_{\leq n} is a left adjoint one obtains canonical morphisms

τ nAA \tau_{\leq n}A \to A

as the adjunct of the identity? on AA, and then by iteration also canonical morphisms

τ (n+1)Aτ nA. \tau_{\leq (n+1)} A \to \tau_{\leq n} A \,.

For any ACA \in C the sequence

τ 2Aτ 1Aτ 0A \cdots \to \tau_{\leq 2}A \to \tau_{\leq 1} A \to \tau_{\leq 0} A

is the Postnikov tower in an (∞,1)-category of AA. See there for more details.

Homotopy type theory syntax

Discussion of nn-truncation of types in homotopy type theory via higher inductive types is in (Brunerie). This sends a type to an h-level (n+2)(n+2)-type.

The (1)(-1)-truncation in the context is forming the bracket type hProp.

Relation to homotopy groups

In an (,1)(\infty,1)-topos CC there is a notion of categorical homotopy groups in an (∞,1)-topos. For the (,1)(\infty,1)-topos ∞Grpd given by the model of Kan complexes this coincides with the notion of simplicial homotopy groups:

Observation

An object AA in the (∞,1)-topos ∞Grpd is nn-truncated precisely if its categorical homotopy groups π k(A)\pi_k(A) vanish for all k>nk \gt n.

This simple relation between nn-truncation and categorical homotopy groups is almost, but not exactly true in an arbitrary (∞,1)-topos.

Proposition

Let H\mathbf{H} be an (∞,1)-topos and AHA \in \mathbf{H} an nn-truncated object.

Then

  1. for k>nk \gt n we have for the categorical homotopy groups π k(A)=*\pi_k(A) = *;

  2. if (for n0n \geq 0) π n(A)=*\pi_n(A) = *, then XX is in fact (n1)(n-1)-truncated.

This implies

Corollary

If AHA \in \mathbf{H} is truncated at all (for any value), then it is nn-truncated precisely if all categorical homotopy groups vanish π k(A)=*\pi_k(A) = * for k>nk \gt n.

Notice. If AA on the other hand is not truncated at all, then all its homotopy groups may be trivial and AA may still not be equivalent to the terminal object. This means that Whitehead's theorem may fail in a general (∞,1)-topos for untruncated objects. It holds, however, in hypercomplete (∞,1)-toposes.

Examples

Truncated morphisms

General

A morphism f:X0f : X \to 0 is

Between groupoids

For morphisms between 1-groupoids, the notion of nn-truncation for low nn reproduces standard concepts from ordinary category theory.

Proposition

A functor f:XYf : X \to Y between groupoids, is nn-truncated precisely when regarded as a morphism in ∞Grpd it is

Proof

Notice that ff being faithful means precisely that it induces a monomorphism on the first homotopy groups.

For x:*Xx : * \to X any point and F f(x)F_{f(x)} the corresponding homotopy fiber of ff, the long exact sequence of homotopy groups gives that π 1(F)\pi_1(F) is the kernel of an injective map

π 1(F)π 1(X)π 1(Y,f(x)), \cdots \to \pi_1(F) \to \pi_1(X) \hookrightarrow \pi_1(Y,f(x)) \to \cdots \,,

hence π 1(F y)=*\pi_1(F_{y}) = * for all points yy in the essential image of ff. For yy not in the essential image we have F yF_y \simeq \emptyset. In either case it follows that FF is 0-truncated.

By def. 3 this is the defining condition for ff to be 0-truncated.

Between stacks

Let CC be a site and write Sh (2,1)(C)Sh (,1)(C)Sh_{(2,1)}(C) \hookrightarrow Sh_{(\infty,1)}(C) for the (2,1)-topos of stacks/(2,1)-sheaves inside the (∞,1)-sheaf (∞,1)-topos of all ∞-stacks/(∞,1)-sheaves.

Write L W[C op,Grpd]L_W [C^{op}, Grpd] for the simplicial localization of groupoid valued presheaves in CC and write [C op,sSet] proj,loc[C^{op}, sSet]_{proj,loc} for the local projective model structure on simplicial presheaves that presents Sh (,1)(C)Sh_{(\infty,1)}(C).

Proposition

Let f:XYf : X \to Y be a morphism of stacks which has a presentation by a degreewise faithful functor that, under the nerve, goes between fibrant simplicial presheaves.

Then ff is 0-truncated as a morphism in Sh (,1)(C)Sh_{(\infty,1)}(C).

Proof

We need to check that for any \infty-stack AA the morphism Sh (A,f)Sh_\infty(A,f) is 0-truncated in ∞Grpd. We may choose a cofibrant model for AA in [C op,sSet] proj,loc[C^{op}, sSet]_{proj,loc} and by assumption that XX and YY is fibrant we have that the ordinary hom of simplicial presheaves [C op,sSet](A,f)[C^{op}, sSet](A, f) is the correct derived hom space morphism. This is itself (the nerve of) a faithful functor, hence the statement follows with prop. 9.

In Grpd\infty Grpd and in TopTop

An object in ∞Grpd is nn-truncated precisely if it is an n-groupoid. To some extent this is so by definition. Equivalently, an object in Top is nn-truncated if it is (in the equivalence class of) a homotopy n-type.

So we have for nn \in \mathbb{N} a reflective sub-(∞,1)-category

nGrpdτ nGrpd. n Grpd \stackrel{\overset{\tau_{\leq n}}{\leftarrow}}{\hookrightarrow} \infty Grpd \,.
Observation

If we model the (∞,1)-category ∞Grpd as the Kan complex-enriched category/fibrant simplicial category KanCplxKanCplx \subset sSet of Kan complexes, then the truncation adjunction

(τ ni):nGrpdτ nGrpd. (\tau_{\leq n } \dashv i) : n Grpd \stackrel{\overset{\tau_{\leq n}}{\leftarrow}}{\hookrightarrow} \infty Grpd \,.

is modeled by the simplicial coskeleton sSet-enriched adjunction

(tr n+1cosk n+1):KanCplx n+1cosk n+1tr n+1KanCplx, (tr_{n+1} \dashv cosk_{n+1}) : KanCplx_{n+1} \stackrel{\overset{tr_{n+1}}{\leftarrow}}{\underset{cosk_{n+1}}{\to}} KanCplx \,,

where KanCplx n+1KanCplx_{n+1} is the subcategory of [Δ n+1 op,Set][\Delta^{op}_{\leq n+1}, Set] on those truncated simplicial sets that are truncations of Kan complexes, regarded as a Kan-complex-enriched category by the embedding via cosk n+1cosk_{n+1}.

Proof

Notice that every Kan complex XX which is nn-truncated is homotopy equivalent to one in the image of cosk n+1cosk_{n+1}, namely to cosk n+1tr n+1Xcosk_{n+1} tr_{n+1} X, because by one of the properties of cosk n+1cosk_{n+1} we have that the unit

Xcosk n+1tr n+1X X \to cosk_{n+1} tr_{n+1} X

induces isomorphisms on homotopy groups π k\pi_k for knk \leq n.

This shows that KanCplx n+1KanCplx_{n+1} is indeed a full sub-(∞,1)-category of KanCplxKanCplx on nn-truncated objects

Moreover, by the fact discussed at Simplicial and derived adjunctions at adjoint (∞,1)-functor we have that the sSet-enriched adjunction (tr n+1cosk n+1)(tr_{n+1} \dashv cosk_{n+1}) on KanCplxKanCplx indeed presents a pair of adjoint (∞,1)-functors on ∞Grpd. So tr n+1:KanCplxKanCplxtr_{n+1} : KanCplx \to KanCplx indeed presents the left adjoint τ :GrpdnGrpd\tau_{\leq} : \infty Grpd \to n Grpd to the inclusion nGrpdGrpdn Grpd \hookrightarrow \infty Grpd.

Diagonals

Example

In ordinary category theory we have that a morphism is a monomorphism (as discussed there), precisely if its diagonal is an isomorphism. Embedded into (∞,1)-category this becomes the special case of prop. 2 for n=0n = 0: a morphism is (-1)-truncated (hence a monomorphism in an (∞,1)-category), precisely if its diagonal is (-2)-truncated (hence an equivalence in an (∞,1)-category).

Example

Let XX be an object that is nn-truncated. This means that X*X \to * is an nn-truncated morphism. So by prop. 2 the diagonal on that object

Δ:XX×X \Delta : X \to X \times X

is an (n1)(n-1)-truncated morphism, and precisely if it is (n1)(n-1)-truncated is XX nn-truncated.

In particular, the diaginal is a monomorphism in an (∞,1)-category, hence (-1)-truncated, precisely if XX is 00-truncated (an h-set).

homotopy leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/unit type/contractible type
h-level 1(-1)-truncated(-1)-groupoid/truth valuemere proposition, h-proposition
h-level 20-truncateddiscrete space0-groupoid/setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/groupoid(2,1)-sheaf/stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoidh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoidh-3-groupoid
h-level n+2n+2nn-truncatedhomotopy n-typen-groupoidh-nn-groupoid
h-level \inftyuntruncatedhomotopy type∞-groupoid(∞,1)-sheaf/∞-stackh-\infty-groupoid

References

The discussion of truncated objects in an (,1)(\infty,1)-category is in section 5.5.6 of

The discussion of categorical homotopy groups in an (∞,1)-topos is in section 6.5.1.

A discussion in terms of model category presentations is in section 7 of

A classical article that amplifies the expression of Postnikov towers in terms of coskeletons is

Discussion in the context of homotopy type theory is in

Revised on November 16, 2013 07:18:20 by Fosco Loregian (95.232.182.7)