An -truncated object of an ∞-stack (∞,1)-topos is the analog of an n-groupoid or homotopy n-type in the archetypical (∞,1)-topos ∞-Grpd/Top:
it is an object whose categorical homotopy groups are trivial for .
(-truncated -groupoid)
An ∞-groupoid is -truncated for if it is an n-groupoid:
Precisely: in the model of ∞-groupoid-groupoids given by Kan complexes an ∞-groupoid is -truncated if, the simplicial homotopy groups are trivial for all and all .
It makes sense for the following to adopt the convention that is called
-truncated if it is empty or contractible
-truncated if it is non-empty and contractible.
(following HTT, p. 6).
To generalize this, let now be an arbitrary (∞,1)-category. For objects in write ∞ Grpd for the (∞,1)-categorical hom-space (if 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:
An ∞-groupoid is -truncated precisely for all other ∞-groupoids the hom--groupoid is -truncated.
In categorical terms this just says that (∞,k)-transformation between and whose components a k-morphisms in cannot be nontrivial for if there are no nontrivial k-morphisms with in .
Using this fact we can transport the notion of -truncation to any (∞,1)-category by testing it on hom-∞-groupoids:
(-truncated object in an -category)
An object of an (∞,1)-category is -truncated, for , if for all the hom-∞-groupoid is -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 -groupoid, it is the terminal objects of that are (-2)-truncated.
At least if the ambient (∞,1)-category is even an ∞-stack (∞,1)-topos there is an alternative, more intrinsic, characterization of -truncation in terms of homotopy groups:
An object in an ∞-stack (∞,1)-topos is -truncated precisely when its categorical homotopy groups are trivial for .
This is HTT prop 6.5.1.7.
Under mild conditions there is for each a universal way to send an arbitrary object to its -truncation . This is a general version of decategorification where n-morphisms are identified if they are connected by an invertible -morphism.
For an (∞,1)-category and in write for the full subcategory of on its -truncated objects.
So for instance for we have .
If is an (∞,1)-category that is presentable then the canonical inclusion (∞,1)-functor
has an accessible? left adjoint
This is HTT 5.5.6.18.
Indeed, as the notation suggests, is the essential image of under . The image of an object under this operation is the -truncation of .
By the fact that the truncation functor is a left adjoint one obtains canonical morphisms
as the adjunct of the identity on , and then by iteration also canonical morphisms
For any the sequence
is the Postnikov tower in an (infinity,1)-category of . See there for more details.
The discussion of truncated objects is in section 5.5.6 of
The discussion of categorical homotopy groups in an (∞,1)-topos is in section 6.5.1.