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

Idea

An $n$-truncated object of an ∞-stack (∞,1)-topos $H$ is the analog of an n-groupoid or homotopy n-type in the archetypical (∞,1)-topos ∞-Grpd/Top:

it is an object $X\in H$ whose categorical homotopy groups ${\pi }_k(X)$ are trivial for $k>n$.

Definition

In terms of truncations

Definition

($n$-truncated $\mathrm{\infty }$-groupoid)

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

Precisely: in the model of ∞-groupoid-groupoids given by Kan complexes an ∞-groupoid is $n$-truncated if, the simplicial homotopy groups ${\pi }_k(A,x)$ are trivial for all $x$ and all $k>n$.

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

• $(-1)$-truncated if it is empty or contractible

• $(-2)$-truncated if it is non-empty and contractible.

(following HTT, p. 6).

To generalize this, let now $C$ be an arbitrary (∞,1)-category. For $X,A$ objects in $C$ write $C(X,A)\in$ ∞ Grpd for the (∞,1)-categorical hom-space (if $C$ 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 $A$ is $n$-truncated precisely for all other ∞-groupoids $X$ the hom-$\mathrm{\infty }$-groupoid $\mathrm{\infty }\mathrm{Grpd}(X,A)$ is $n$-truncated.

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

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

Definition

($n$-truncated object in an $(\mathrm{\infty }\mathrm{,1})$-category)

An object $A\in C$ of an (∞,1)-category $C$ is $n$-truncated, for $n\in \mathbb{N}$, if for all $X\in C$ the hom-∞-groupoid $C(X,A)$ is $n$-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 concention on (-2)-truncated $\mathrm{\infty }$-groupoid, it is the terminal objects of $C$ that are (-2)-truncated.

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 $n$-truncation in terms of homotopy groups:

Proposition

An object $X$ in an ∞-stack (∞,1)-topos is $n$-truncated precisely when its categorical homotopy groups ${\pi }_k(X)$ are trivial for $k>n$.

Proof

This is HTT prop 6.5.1.7.

Truncation

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

For $C$ an (∞,1)-category and $n\ge -2$ in $\mathbb{Z}$ write ${\tau }_{\le n}C$ for the full subcategory of $C$ on its $n$-truncated objects.

So for instance for $C=\mathrm{\infty }\mathrm{Grpd}$ we have ${\tau }_{\le n}\mathrm{\infty }\mathrm{Grpd}=n\mathrm{Grpd}$.

Proposition

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

$${\tau }_{\le n}C\hookrightarrow C$$\tau_{\leq n} C \hookrightarrow C

has an accessible? left adjoint

$${\tau }_{\le n}:C\to C_{\le n}.$$\tau_{\leq n} : C \to C_{\leq n} \,.

Proof

This is HTT 5.5.6.18.

Indeed, as the notation suggests, $C_{\le n}$ is the essential image of $C$ under ${\tau }_{\le n}$. The image ${\tau }_{\le n}A$ of an object $A$ under this operation is the $n$-truncation of $A$.

Postnikov tower

By the fact that the truncation functor ${\tau }_{\le n}$ is a left adjoint one obtains canonical morphisms

$${\tau }_{\le n}A\to A$$\tau_{\leq n}A \to A

as the adjunct of the identity on $A$, and then by iteration also canonical morphisms

$${\tau }_{\le (n+1)}A\to {\tau }_{\le n}A.$$\tau_{\leq (n+1)} A \to \tau_{\leq n} A \,.

For any $A\in C$ the sequence

$$\cdots \to {\tau }_{\le 2}\to {\tau }_{\le 1}A\to {\tau }_{\le 0}A$$\cdots \to \tau_{\leq 2} \to \tau_{\leq 1} A \to \tau_{\leq 0} A

is the Postnikov tower in an (infinity,1)-category of $A$. See there for more details.

References

The discussion of truncated objects is in section 5.5.6 of

• Jacob Lurie, Higher Topos Theory

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