This page is about object classifier objects in (∞,1)-toposes. For the unnrelated notion of the classifying topos of the theory of objects see at classifying topos for the theory of objects.
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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.
As discussed also at stuff, structure, property, the classifying objects in higher topos theory classify more general morphisms.
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 $\kappa$ there is a classifying object for the class of all $\kappa$-compact morphisms in $X$.
This statement is originally due to Charles Rezk. It is reproduced as (Lurie HTT, theorem 6.1.6.8).
In terms of homotopy type theory these object classifiers are types of types. See there for more details and see at relation between category theory and type theory.
An object classifier is a (small) self-reflections of the $\infty$-topos inside itself (type of types, internal universe). See also (WdL, book 2, section 1).
Let $\mathcal{C}$ be a (∞,1)-category, and let $S$ be a class of 1-morphisms of $\mathcal{C}$ which is stable under (∞,1)-pullback. Then an $S$-classifier is a terminal object in the sub-category of arrows $\mathcal{C}^{\Delta^1}$ of $S$ whose morphisms are (∞,1)-pullback squares in $\mathcal{C}$.
Explicitly, an $S$-classifier consists of
a morphism $\widehat {S Type} \longrightarrow S Type$ in $S$
such that for each $X \stackrel{p}{\to} B$ in $S$, there exists an essentially unique (∞,1)-pullback square of the form
in $\mathcal{C}$
Here $'X'$ may be called the name, or the classifying morphism, or the modulating morphism or the internal reflection of $X$ over $B$,
For example,
When $S$ is the class of all monomorphisms in $C$, an $S$-classifier is called a subobject classifier. For instance, every topos has a subobject classifier.
When $S$ is the class of all morphisms in $C$, an $S$-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 $S$ is the class of all relatively $\kappa$-compact morphisms (for some regular cardinal $\kappa$–see below for the definition), an $S$-classifier is called a $\kappa$-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 $X$, a subobject classifier classifies the subobjects of $X$. For each object $X$, an object classifier classifies the objects over $X$.
So with that $\kappa$ fixed, we may write
for such a “universal bundle of $\kappa$-small objects”. Intuitively this is easy to describe: a point in $Type$ corresponds to a $\kappa$-small object, hence is the “name” $'X'$ or “code for” a $\kappa$-small object, and the fiber in $\widehat Type$ over that point is the very object $X$ itself.
If one gives the projection of the universal object bundle $\widehat Type \to Type$ a name, such as $El$, and writes $El^{-1}(-)$ for its preimages then $X \simeq El^{-1}('X')$. This is, with the ${(-)}^{-1}$-suppressed, the notation used at Type universes a la Tarski.
Let $C$ be an (∞,1)-category and $S \in C_1$ a class of morphisms that is stable under (∞,1)-pullback in $C$.
Let $Cod_C$ be the codomain fibration of $X$, i.e. the (∞,1)-category of (∞,1)-functors
equipped with the Cartesian fibration $Cod_C \to C$ induced from the endpoint inclusion $\Delta[0] \to \Delta[1]$.
Write
$Cod_C^S$ for the full sub-(∞,1)-category of $Cod_C$ on the object in $S$;
$Cod_C^{(S)}$ the non-full subcategory whose objects are the elements of $S$, and whose morphisms are squares that are pullback diagrams.
Then evaluation at $\Delta[0] \to \Delta[1]$ yields
a Cartesian fibration $Cod_C^S \to C$;
a right fibration $Cod_C^{(S)} \to C$.
We say a morphism $f :x \to y$ in $C$ classifies $S$ – or simply that $y$ classifies $S$ – if it is the terminal object in $Cod_C^{(S)}$.
This is HTT, notation 6.1.3.4 and HTT, def. 6.1.6.1.
A subobject classifier for $C$ is an object that classifies the class $S$ of monomorphisms/(-1)-truncated morphisms in $C$.
This is (HTT, def. 6.1.6.1).
The $(\infty,1)$-category ∞Grpd has a a subobject classifier: the 0-groupoid/set $\{\emptyset,*\}$ with two elements (the two (-1)-truncated $\infty$-groupoids).
Every (∞,1)-topos has a subobject classifier.
This appears as (HTT, prop. 6.1.6.3) 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 $(\infty,1)$-topos we thus expect an object that classifies all morphisms, in that the assignment
of an object $c\in C$ to the core of its over (∞,1)-category yields a (∞,1)-functor $C^{op} \to \infty Grpd$ that is representable.
Indeed, this is essentially the case – up to size issues, that the following definitions take care of.
For $\kappa$ some cardinal, say a morphism $f : x \to y$ in $C$ is relatively k-compact if for all (∞,1)-pullbacks along $h : y' \to y$ to $\kappa$-compact objects, $y'$, the pulled back object $h^* x'$ is itself a $\kappa$-compact object.
A presentable (∞,1)-category $C$ is an (∞,1)-topos precisely if
it has universal colimits;
for sufficiently large regular cardinals $\kappa$, $C$ has a classifying object for relatively $\kappa$-compact morphisms.
This is due to Charles Rezk. The statement appears as HTT, theorem 6.1.6.8.
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 $C^{op} \to \infty Gpd$ defined by $x\mapsto Core(C/x)$. 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 $\kappa$-small object classifier in the $(\infty,1)$-topos ∞Grpd of ∞-groupoids is itself the core of the (∞,1)-category $\infty Grpd_\kappa$ of $\kappa$-small $\infty$-groupoids. Observing that the connected components of this are the delooping $B Aut(F)$ of the automorphism ∞-group of a given homotopy type $[F]$, and using that ∞Grpd is presented by Top $\simeq$ 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 $\kappa$-compact object classifier in ∞Grpd is
the core of the full sub-(∞,1)-category of ∞Grpd on the $\kappa$-small ∞-groupoids.
The corresponding universal bundle is presented by the map of simplicial sets
which is the pullback of simplicial sets
of the universal right fibration along the defining inclusion of (the Kan complex presenting) $Type_\kappa$.
In ∞Grpd the relatively $\kappa$-compact morphisms, $X \to Y$, def. 3, are precisely those all whose homotopy fibers
over all objects $y \in Y$ are $\kappa$-small infinity-groupoids.
We may write $Y$ as an (∞,1)-colimit over itself (see there)
and then use the fact that ∞Grpd – being an (∞,1)-topos – has universal colimits, to obtain the (∞,1)-pullback diagram
exhibiting $X$ as an $(\infty,1)$-colimit of $\kappa$-small objects over $Y$. By stability of $\kappa$-compact objects under $\kappa$-small colimits (see here) it follows that $X$ is $\kappa$-compact if $Y$ 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 $\infty Grpd_\kappa$ factors through the core $Type_\kappa$ it follows that $Type_\kappa$ classifies all morphisms $X \to Y$ in ∞Grpd whose homotopy fibers
are $\kappa$-compact.
The claim then follows with lemma 1.
Let $C$ be an (∞,1)-category and $\mathbf{H} = PSh_{\infty}(C)$ the (∞,1)-category of (∞,1)-presheaves over $C$.
By the (∞,1)-Yoneda lemma, the $\kappa$-compact object classifier here should be the presheaf which assigns to $U \in C$ the $\infty$-groupoid of relatively $\kappa$-compact morphisms $X \to U$ in $PSh_\infty(C)$.