nLab (sub)object classifier in an (infinity,1)-topos

This page is about object classifier objects in (∞,1)-toposes. For the unrelated notion of the classifying topos of the theory of objects see at classifying topos for the theory of objects.

Idea

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

• a presentable (∞,1)-category

• 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.

Remark

An object classifier is a (small) self-reflection of the $(\infty, 1)$-topos inside itself (type of types, internal universe). It possesses an internal (∞,1)-topos structure. See also (WdL, book 2, section 1).

Definition

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

  $$\array{ X &\longrightarrow& \widehat {S Type} \\ \downarrow^{\mathrlap{p}} && \downarrow \\ B &\stackrel{'X'}{\longrightarrow}& S Type } \,.$$

  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 faithful morphisms in $C$, an $S$-classifier is called a discrete object classifier. For instance, every (2,1)-topos has a discrete object 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

$$\array{ \widehat Type \\ \downarrow \\ Type }$$

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.

Details

n-truncated object classifier

Definition

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

$$Cod_C := Func(\Delta[1], C)$$

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.

Subobject classifier

Definition

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).

Example

The $(\infty,1)$-category ∞Grpd has a a subobject classifier: the 0-groupoid/set $\{\emptyset,*\}$ with two elements (the two (-1)-truncated $\infty$-groupoids).

Proposition

Every (∞,1)-topos has a subobject classifier.

This appears as (HTT, prop. 6.1.6.3) and the remark below that.

Discrete object classifier

Definition

A discrete object classifier for $C$ is an object that classifies the class $S$ of faithful morphism?/(0)-truncated? morphisms in $C$.

Example

The $(\infty,1)$-category ∞Grpd has a a discrete object classifier: the 1-groupoid/groupoid Set whose elements are sets (0-truncated $\infty$-groupoids).

Proposition

Every (∞,1)-topos has a discrete object classifier.

Object classifier

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

$$c \mapsto Core(C_{/c})$$

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.

Definition

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.

Theorem

A presentable (∞,1)-category $C$ is an (∞,1)-topos precisely if

1. it has universal colimits;

2. 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 (specifically, the representable 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.

Examples

Object classifier in $\infty Grpd$

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 by a universal Kan fibration, as listed in the References at associated ∞-bundle.

Proposition

The $\kappa$-compact object classifier in ∞Grpd is

$$Type_\kappa := Core(\infty Grpd_\kappa) \,,$$

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

$$\widehat Type_\kappa \to Type_\kappa$$

which is the pullback of simplicial sets

$$\array{ \widehat Type_\kappa &\to& Z_{\infty Grpd} \\ \downarrow && \downarrow \\ Type_\kappa &\to& \infty Grpd }$$

of the universal right fibration along the defining inclusion of (the Kan complex presenting) $Type_\kappa$.

Lemma

In ∞Grpd the relatively $\kappa$-compact morphisms, $X \to Y$, def. , are precisely those all whose homotopy fibers

$$X_{y} := X \times_{Y} \{y\}$$

over all objects $y \in Y$ are $\kappa$-small infinity-groupoids.

Proof

We may write $Y$ as an (∞,1)-colimit over itself (see there)

$$Y \simeq {\lim_{\to}}_{y \in Y} \{y\}$$

and then use the fact that ∞Grpd – being an (∞,1)-topos – has universal colimits, to obtain the (∞,1)-pullback diagram

$$\array{ {\lim_{\to}}_{y \in Y} X_y &\stackrel{\simeq}{\to} & X \\ \downarrow && \downarrow \\ {\lim_{\to}}_{y \in Y} \{y\} &\stackrel{\simeq}{\to}& Y }$$

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.

Proof of the proposition

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

$$X_y \simeq X \times_Y \{y\}$$

are $\kappa$-compact.

The claim then follows with lemma .

Object classifier in presheaf $(\infty,1)$-toposes

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 $PSh_\infty(C)$.

Related concepts

• univalent foundations for mathematics

• type of propositions, subobject classifier, partial map classifier

• type of sets?, category of sets, discrete object classifier

• type of types, univalence

• classifying morphism

• Awodey's conjecture

References

The general notion is due to:

• Jacob Lurie, section 6.1.6 of Higher Topos Theory (2009)

The object classifier in the archetypical special case of the $\infty$-topos ∞Grpd of $\infty$-groupoids, seen in the classical model structure on simplicial sets, is discussed in

• Chris Kapulkin, Peter LeFanu Lumsdaine, Vladimir Voevodsky, Univalence in simplicial sets [arXiv:1203.2553]

• Chris Kapulkin, Peter LeFanu Lumsdaine, The Simplicial Model of Univalent Foundations (after Voevodsky), Journal of the European Mathematical Society 23 (2021) 2071–2126 $[$arXiv:1211.2851, doi:10.4171/jems/1050$]$

as categorical semantics for univalent type universes in homotopy type theory.