(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.



(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory





Extra stuff, structure and property



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.

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

This statement is originally due to Charles Rezk. It is reproduced as (Lurie HTT, theorem

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-reflection of the (,1)(\infty, 1)-topos inside itself (type of types, internal universe). It possesses an internal (∞,1)-topos structure. See also (WdL, book 2, section 1).


Let 𝒞\mathcal{C} be a (∞,1)-category, and let SS be a class of 1-morphisms of 𝒞\mathcal{C} which is stable under (∞,1)-pullback. Then an SS-classifier is a terminal object in the sub-category of arrows 𝒞 Δ 1\mathcal{C}^{\Delta^1} of SS whose morphisms are (∞,1)-pullback squares in 𝒞\mathcal{C}.

Explicitly, an SS-classifier consists of

  • a morphism SType^SType\widehat {S Type} \longrightarrow S Type in SS

  • such that for each XpBX \stackrel{p}{\to} B in SS, there exists an essentially unique (∞,1)-pullback square of the form

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

    in 𝒞\mathcal{C}

Here X'X' may be called the name, or the classifying morphism, or the modulating morphism or the internal reflection of XX over BB,

For example,

  • When SS is the class of all monomorphisms in CC, an SS-classifier is called a subobject classifier. For instance, every topos has a subobject classifier.

  • When SS is the class of all faithful morphisms in CC, an SS-classifier is called a discrete object classifier. For instance, every (2,1)-topos has a discrete object classifier.

  • When SS is the class of all morphisms in CC, an SS-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 SS is the class of all relatively κ\kappa-compact morphisms (for some regular cardinal κ\kappa–see below for the definition), an SS-classifier is called a κ\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 XX, a subobject classifier classifies the subobjects of XX. For each object XX, an object classifier classifies the objects over XX.

So with that κ\kappa fixed, we may write

Type^ Type \array{ \widehat Type \\ \downarrow \\ Type }

for such a “universal bundle of κ\kappa-small objects”. Intuitively this is easy to describe: a point in TypeType corresponds to a κ\kappa-small object, hence is the “name” X'X' or “code for” a κ\kappa-small object, and the fiber in Type^\widehat Type over that point is the very object XX itself.

If one gives the projection of the universal object bundle Type^Type\widehat Type \to Type a name, such as ElEl, and writes El 1()El^{-1}(-) for its preimages then XEl 1(X)X \simeq El^{-1}('X'). This is, with the () 1{(-)}^{-1}-suppressed, the notation used at Type universes a la Tarski.


n-truncated object classifier


Let CC be an (∞,1)-category and SC 1S \in C_1 a class of morphisms that is stable under (∞,1)-pullback in CC.

Let Cod CCod_C be the codomain fibration of XX, i.e. the (∞,1)-category of (∞,1)-functors

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

equipped with the Cartesian fibration Cod CCCod_C \to C induced from the endpoint inclusion Δ[0]Δ[1]\Delta[0] \to \Delta[1].


  • Cod C SCod_C^S for the full sub-(∞,1)-category of Cod CCod_C on the object in SS;

  • Cod C (S)Cod_C^{(S)} the non-full subcategory whose objects are the elements of SS, and whose morphisms are squares that are pullback diagrams.

Then evaluation at Δ[0]Δ[1]\Delta[0] \to \Delta[1] yields

We say a morphism f:xyf :x \to y in CC classifies SS – or simply that yy classifies SS – if it is the terminal object in Cod C (S)Cod_C^{(S)}.

This is HTT, notation and HTT, def.

Subobject classifier


A subobject classifier for CC is an object that classifies the class SS of monomorphisms/(-1)-truncated morphisms in CC.

This is (HTT, def.


The (,1)(\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. and the remark below that.

Discrete object classifier


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


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


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 (,1)(\infty,1)-topos we thus expect an object that classifies all morphisms, in that the assignment

cCore(C /c) c \mapsto Core(C_{/c})

of an object cCc\in C to the core of its over (∞,1)-category yields a (∞,1)-functor C opGrpdC^{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 κ\kappa some cardinal, say a morphism f:xyf : x \to y in CC is relatively k-compact if for all (∞,1)-pullbacks along h:yyh : y' \to y to κ\kappa-compact objects, yy', the pulled back object h *xh^* x' is itself a κ\kappa-compact object.


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

  1. it has universal colimits;

  2. for sufficiently large regular cardinals κ\kappa, CC has a classifying object for relatively κ\kappa-compact morphisms.

This is due to Charles Rezk. The statement appears as HTT, theorem

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 opGpdC^{op} \to \infty Gpd defined by xCore(C/x)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.


Object classifier in Grpd\infty Grpd

We discuss that the κ\kappa-small object classifier in the (,1)(\infty,1)-topos ∞Grpd of ∞-groupoids is itself the core of the (∞,1)-category Grpd κ\infty Grpd_\kappa of κ\kappa-small \infty-groupoids. Observing that the connected components of this are the delooping BAut(F)B Aut(F) of the automorphism ∞-group of a given homotopy type [F][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.


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

Type κ:=Core(Grpd κ), Type_\kappa := Core(\infty Grpd_\kappa) \,,

the core of the full sub-(∞,1)-category of ∞Grpd on the κ\kappa-small ∞-groupoids.

The corresponding universal bundle is presented by the map of simplicial sets

Type^ κType κ \widehat Type_\kappa \to Type_\kappa

which is the pullback of simplicial sets

Type^ κ Z Grpd Type κ Grpd \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 κType_\kappa.


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

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

over all objects yYy \in Y are κ\kappa-small infinity-groupoids.


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

Ylim yY{y} 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

lim yYX y X lim yY{y} Y \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 XX as an (,1)(\infty,1)-colimit of κ\kappa-small objects over YY. By stability of κ\kappa-compact objects under κ\kappa-small colimits (see here) it follows that XX is κ\kappa-compact if YY 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 Grpd κ\infty Grpd_\kappa factors through the core Type κType_\kappa it follows that Type κType_\kappa classifies all morphisms XYX \to Y in ∞Grpd whose homotopy fibers

X yX× Y{y} X_y \simeq X \times_Y \{y\}

are κ\kappa-compact.

The claim then follows with lemma .

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

Let CC be an (∞,1)-category and H=PSh (C)\mathbf{H} = PSh_{\infty}(C) the (∞,1)-category of (∞,1)-presheaves over CC.

By the (∞,1)-Yoneda lemma, the κ\kappa-compact object classifier here should be the presheaf which assigns to UCU \in C the \infty-groupoid of relatively κ\kappa-compact morphisms XUX \to U in PSh (C)PSh_\infty(C).


Last revised on March 2, 2021 at 11:35:22. See the history of this page for a list of all contributions to it.