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

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.

Context

Universes

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

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

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

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.

Definition

For simplicity, the following is phrased in classical categorical terms. Of course, it can be translated into “\infty”-language by prefixing every word by “\infty-”. We leave this translation to the reader.

Let CC be a category, and let SS be a class of morphisms of CC which is stable under pullback. Then an SS-classifier is a terminal object in the category of arrows of SS are objects and pullback squares in CC are morphisms.

Explicitly, an SS-classifier consists of

  • An arrow Σ:Σ 0Σ 1\Sigma: \Sigma_0 \to \Sigma_1 in SS such that

  • For each f:XYf: X \to Y in SS, there exists a unique pullback square

    X f Y Σ 0 Σ Σ 1 \begin{matrix} X & \overset{f}{\to} & Y \\ \downarrow& &\, \downarrow \\ \Sigma_0 &\overset{\Sigma}{\to} & \Sigma_1 \end{matrix}

    in CC.

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 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.

Details

Subobject classifier

Definition

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].

Write

  • 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 6.1.3.4 and HTT, def. 6.1.6.1.

Definition

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

This is (HTT, def. 6.1.6.1).

Example

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).

Proposition

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

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

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.

Definition

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.

Theorem

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 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 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.

Examples

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, as listed in the References at associated ∞-bundle.

Proposition

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.

Lemma

In ∞Grpd the relatively κ\kappa-compact morphisms, XYX \to Y, def. 3, 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.

Proof

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 1.

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).

References

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Revised on November 3, 2014 14:14:06 by Urs Schreiber (185.26.182.33)