of all homotopy types
structures in a cohesive (∞,1)-topos
A crucial ingredient in a topos is a subobject classifier. From the point of view of homotopy theory, that this has to do with subobjects turns out to be a coincidence of low dimensions: subobjects are (-1)-truncated morphisms.
One way to characterize an (∞,1)-topos is as
with universal colimits
such that for all sufficiently large regular cardinals there is a classifying object for the class of all -compact morphisms in .
Let be a (∞,1)-category, and let be a class of 1-morphisms of which is stable under (∞,1)-pullback. Then an -classifier is a terminal object in the sub-category of arrows of whose morphisms are (∞,1)-pullback squares in .
Explicitly, an -classifier consists of
a morphism in
such that for each in , there exists an essentially unique (∞,1)-pullback square of the form
When is the class of all monomorphisms in , an -classifier is called a subobject classifier. For instance, every topos has a subobject classifier.
When is the class of all morphisms in , an -classifier is called an object classifier. However, due to size issues, interesting categories tend not to have such objects, which is one reason to be interested in the next example:
When is the class of all relatively -compact morphisms (for some regular cardinal –see below for the definition), an -classifier is called a -compact-object classifier.
Note on terminology: In all cases, the “things” classified by an “(adjectives) object classifier” are arrows – this is no different from the most famous case of subobject classifiers, which classify monos. For each object , a subobject classifier classifies the subobjects of . For each object , an object classifier classifies the objects over .
So with that fixed, we may write
for such a “universal bundle of -small objects”. Intuitively this is easy to describe: a point in corresponds to a -small object, hence is the “name” or “code for” a -small object, and the fiber in over that point is the very object itself.
If one gives the projection of the universal object bundle a name, such as , and writes for its preimages then . This is, with the -suppressed, the notation used at Type universes a la Tarski.
equipped with the Cartesian fibration induced from the endpoint inclusion .
for the full sub-(∞,1)-category of on the object in ;
the non-full subcategory whose objects are the elements of , and whose morphisms are squares that are pullback diagrams.
Then evaluation at yields
We say a morphism in classifies – or simply that classifies – if it is the terminal object in .
This is (HTT, def. 126.96.36.199).
This appears as (HTT, prop. 188.8.131.52) and the remark below that.
Remark/Warning. The point of having subobjects and hence monomorphisms classified by an object in an ordinary topos may be thought of as being solely due to the fact that in a 1-topos, any object necessarily classifies a poset i.e. a (0,1)-category of morphisms, and the point of subobjects/monomorphisms of a given object is that they do not have automorphisms.
In an -topos we thus expect an object that classifies all morphisms, in that the assignment
Indeed, this is essentially the case – up to size issues, that the following definitions take care of.
The proof essentially consists of showing that by the adjoint functor theorem, the existence of object classifiers is equivalent to continuity of the core self-indexing defined by . In the presence of universal colimits, this latter condition is equivalent to all colimits being van Kampen colimits, which in turn yields the connection to the Giraud-type exactness properties.
We discuss that the -small object classifier in the -topos ∞Grpd of ∞-groupoids is itself the core of the (∞,1)-category of -small -groupoids. Observing that the connected components of this are the delooping of the automorphism ∞-group of a given homotopy type , and using that ∞Grpd is presented by Top sSet (see also at homotopy hypothesis) this recovers classical theorems about the classification of fibrations in simplicial sets/topological spaces, as listed in the References at associated ∞-bundle.
The -compact object classifier in ∞Grpd is
which is the pullback of simplicial sets
We may write as an (∞,1)-colimit over itself (see there)
exhibiting as an -colimit of -small objects over . By stability of -compact objects under -small colimits (see here) it follows that is -compact if is.
Since right fibrations are stable under pullback (see here), this is still a right fibration. Since, up to equivalence, every morphism into a Kan complex is a right fibration (see here), and since every morphism out of a Kan complex into factors through the core it follows that classifies all morphisms in ∞Grpd whose homotopy fibers
The claim then follows with lemma 1.
By the (∞,1)-Yoneda lemma, the -compact object classifier here should be the presheaf which assigns to the -groupoid of relatively -compact morphisms in .