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



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

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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.


In terms of truncations


(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 is useful to reformulate the above as follows slightly:


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

In other category-theoretic terms this says that (∞,k)-transformation between XX and AA, whose components are 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- \infty -groupoids:


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

Given an (∞,1)-category 𝒞\mathcal{C}, an object A𝒞A \in \mathcal{C} is called nn-truncated, for nn \in \mathbb{N}, if for all objects X𝒞X \in \mathcal{C} the hom-∞-groupoid 𝒞(X,A)\mathcal{C}(X,A) is nn-truncated according to Def. .

This is HTT, def.

Some terminology:


(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. .

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.

This is HTT, def.

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}. (See also HTT, rem.

Unwinding the definitions and applying the long exact sequence of homotopy groups, we have:


For n1n \geq -1, a morphism f:XYf : X \to Y of ∞-groupoids is nn-truncated iff, for every point xXx \in X, we have π n+1(X,x)π n+1(Y,f(x))\pi_{n+1}(X, x) \to \pi_{n+1}(Y, f(x)) is a monomorphism and for every kn+2k \geq n+2, π k(X,x)π k(Y,f(x))\pi_k(X, x) \to \pi_k(Y, f(x)) is an isomorphism.

ff is (2)(-2)-trunacated iff it is a weak homotopy equivalence.

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:


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.


This is HTT, prop


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 relation to categorical homotopy groups. See HTT, remark


Recursive definition


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


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 now write 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 with 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 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.



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 \infty -limits in CC.

This is HTT, prop.


Let H\mathbf{H} be an (∞,1)-topos. For all (2)n<(-2) \leq n \lt \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 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.



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.


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:Cτ nC. \tau_{\leq n} : C \to \tau_{\leq n} C \,.

This is HTT

Indeed, as the notation suggests, τ nC\tau_{\leq n} C 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 \,.




For any small \infty-category CC, τ nPSh(C,Grpd)PSh(C,nGrpd)\tau_{\leq n} PSh(C, \infty Grpd) \simeq PSh(C, n Grpd) , and truncation acts pointwise.


If PP is an nn-truncated ∞-presheaf, then P(c)PSh(C)(C(,c),P)P(c) \simeq PSh(C)(C(-, c), P) is nn-truncated; thus PP takes values in nGrpdn Grpd.

Conversely, if PP takes values in nGrpdn Grpd, then the fact every presheaf is a colimit of representables implies hom(Q,P)hom(Q, P) is a limit of nn-truncated spaces and is thus nn-truncated.

Given this identification of the subcategory of nn-truncated objects, we can see that the truncation-inclusion adjunction between nGrpdn Grpd and Grpd\infty Grpd induces an adjunction whose right adjoint is the inclusion PSh(C,nGrpd)PSh(C,Grpd)PSh(C, n Grpd) \to PSh(C, \infty Grpd)


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

(HTT, prop.

This 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.


A left exact 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.


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.


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


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,Set]tr_{n+1} : sSet \to [\Delta^{op}_{\leq n+1}, Set]. This is also a right adjoint and as such preserves in particular products 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 ki(X j) jLτ ki(X j) L \tau_{\leq k} \prod_j i(X_j) \stackrel{\simeq}{\to} L \prod_j \tau_{\leq k} i(X_j) \simeq \prod_j L \tau_{\leq k} 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

Aτ nA A \to \tau_{\leq n}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.

See at n-truncation modality.

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:


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.


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


  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


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.


Truncated morphisms


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.


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


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. 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).


Let f:XYf : X \to Y be a morphism of stacks that 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).


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. .

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 \,.

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}.


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.



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. 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).


Let XX be an object that is nn-truncated. This means that X*X \to * is an nn-truncated morphism. So by prop. 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 diagonal 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)-truncatedcontractible-if-inhabited(-1)-groupoid/​truth value(0,1)-sheaf/​idealmere proposition/​h-proposition
h-level 20-truncatedhomotopy 0-type0-groupoid/​setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/​groupoid(2,1)-sheaf/​stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoid(3,1)-sheaf/​2-stackh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoid(4,1)-sheaf/​3-stackh-3-groupoid
h-level n+2n+2nn-truncatedhomotopy n-typen-groupoid(n+1,1)-sheaf/​n-stackh-nn-groupoid
h-level \inftyuntruncatedhomotopy type∞-groupoid(∞,1)-sheaf/​∞-stackh-\infty-groupoid


In \infty-category theory

Discussion in ( , 1 ) (\infty,1) -category:

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

Discussion in terms of model category presentations:

A classical article that amplifies the expression of Postnikov towers in sSet (hence in Grpd Grpd_{\infty} ) in terms of coskeleta:

Discussion in the context of modal homotopy type theory:

More discussion of the internal perspective:

In homotopy type theory

Discussion of homotopy n n -types/ n n -truncated objects in homotopy type theory:

Last revised on June 25, 2024 at 07:42:40. See the history of this page for a list of all contributions to it.