of all homotopy types
structures in a cohesive (∞,1)-topos
When one passes all the way to -toposes, there should be objects that classify all morphisms, subject to some bound on size. This is made precise in the context of (∞,1)-topos theory.
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 .
This statement is originally due to Charles Rezk. It is reproduced as theorem 220.127.116.11 in
For simplicity, the following is phrased in classical categorical terms. Of course, it can be translated into ”“-language by prefixing every word by ”-“. We leave this translation to the reader.
Let be a category, and let be a class of morphisms of which is stable under pullback. Then an -classifier is a terminal object in the category of arrows of are objects and pullback squares in are morphisms.
Explicitly, an -classifier consists of
An arrow in such that
For each in , there exists a unique pullback square
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 .
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. 18.104.22.168).
This appears as (HTT, prop. 22.214.171.124) 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 .