nLab
cohesive (infinity,1)-topos

This entry is about a refinement of the concept of cohesive topos. The definition here expresses an intuition not unrelated to that at cohesive (∞,1)-presheaf on E-∞ rings but the definitions are unrelated and apply in somewhat disjoint contexts.

Context

Cohesive \infty-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

Definition

Presentation over a site

Structures in a cohesive (,1)(\infty,1)-topos

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

Discrete and concrete objects

Contents

Idea

A cohesive (,1)(\infty,1)-topos is a gros (∞,1)-topos H\mathbf{H} that provides a context of generalized spaces in which higher geometry makes sense, in particular higher differential geometry. See also at motivation for cohesive toposes for a non-technical discussion.

Technically, it is an (,1)(\infty,1)-topos whose global section (∞,1)-geometric morphism (DiscΓ):HΓDisc(Disc \dashv \Gamma): \mathbf{H} \stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}} ∞Grpd admits a further left adjoint (∞,1)-functor Π\Pi and a further right adjoint coDisccoDisc:

(ΠDiscΓcoDisc):HGrpd (\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : \mathbf{H} \to \infty Grpd

with DiscDisc and coDisccoDisc both full and faithful (∞,1)-functors and such that Π\Pi moreover preserves finite (∞,1)-products. Here

  1. the existence of coDisccoDisc induces a sub-(∞,1)-quasitopos Conc(H)HConc(\mathbf{H}) \hookrightarrow \mathbf{H} of concrete objects that behave like ∞-groupoids equipped with extra cohesive structure , such as with continuous structure, smooth structure, etc.

  2. the existence of Π\Pi induces a notion of geometric fundamental ∞-groupoid, hence under ||:Grpd|-| : \infty Grpd \simeq Top of geometric realization |Π()||\Pi(-)| of objects in H\mathbf{H}.

The functor Γ\Gamma itself may be thought of as sending a cohesive ∞-groupoid XX to its underlying bare \infty-groupoid Γ(X)\Gamma(X). This is XX with all cohesion forgotten (for instance with the continuous or the smooth structure forgotten).

Conversely, DiscDisc and CoDiscCoDisc send an \infty-groupoid AA either to the discrete ∞-groupoid Disc(A)Disc(A) with discrete cohesive structure (for instance with discrete topology) or to the codiscrete ∞-groupoid Codisc(A)Codisc(A) with the codiscrete cohesive structure (for instance with codiscrete topology).

This kind of adjoint quadruple, directly analgous to the structure introduced by William Lawvere for cohesive toposes, induces two adjoint modalities which in turn are adjoint to each other, to yield the adjoint string shape modality \dashv flat modality \dashv sharp modality \int \dashv \flat \dashv \sharp, which may be thought of as expressing “contiuum” and “quantity” in the sense of Georg Hegel’s Science of Logic (as explained in detail there.)

The existence of such an adjoint quadruple of adjoint (,1)(\infty,1)-functors alone implies a rich internal higher geometry in H\mathbf{H} that comes with its internal notion of Galois theory, Lie theory, differential cohomology, Chern-Weil theory.

Examples of cohesive (,1)(\infty,1)-toposes include

In ETop∞Grpd and those contexts containing it, the internal notions of geometric realization, geometric homotopy and Galois theory subsume the usual ones (over well-behaved topological spaces). In Smooth∞Grpd also the notions of Lie theory, differential cohomology and Chern-Weil theory subsume the usual ones. In SynthDiff∞Grpd the internal notion of Lie algebra and Lie algebroid subsumes the traditional one — and generalizes them to higher smooth geometry.

Definition

We state the definition in several equivalent ways.

  1. externally in the ambient context;

  2. internally to the cohesive (,1)(\infty,1)-topos itselfs;

  3. internally and formulated in homotopy type theory

Externally

The definition is the immediate analog of the definition of a cohesive topos.

Definition

An (∞,1)-topos H\mathbf{H} is cohesive if

  1. it is a strongly ∞-connected (∞,1)-topos;

  2. it is a local (∞,1)-topos.

Remark

As always in topos theory and higher topos theory, such definitions can be made sense of over any base . Here: over any base (∞,1)-topos.

Over the canonical base ∞Grpd of ∞-groupoids, the definition of a cohesive (,1)(\infty,1)-topos is equivalently the following:

the global section (∞,1)-geometric morphism Γ:HGrpd\Gamma : \mathbf{H} \to \infty Grpd lifts to an adjoint quadruple of adjoint (∞,1)-functors

(ΠDiscΓcoDisc):HcoDiscΓDiscΠGrpd (\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : \mathbf{H} \stackrel{\stackrel{\overset{\Pi}{\to}}{\overset{Disc}{\leftarrow}}}{\stackrel{\underset{\Gamma}{\to}}{\underset{coDisc}{\leftarrow}}} \infty Grpd \;

where Π\Pi preserves finite (∞,1)-products.

Often we will tacitly assume to work over ∞Grpd. But most statements and constructions have straightforward generalizations to arbitrary bases. In particular, below in the internal definition of cohesion, it takes more to fix the base topos than to leave it arbitrary.

Every adjoint quadruple induces an adjoint triple of endofunctors.

(Π#):HΓDiscΠGrpdcoDiscΓDiscH. ( \mathbf{\Pi} \dashv \mathbf{\flat} \dashv \mathbf{#} ) : \mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{\Disc}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd \stackrel{\overset{Disc}{\to}}{\stackrel{\overset{\Gamma}{\leftarrow}}{\underset{coDisc}{\to}}} \mathbf{H} \,.

Here “\mathbf{\flat}” is meant to be pronounced “flat”. The interpretation of these three functors is discussed in detail at cohesive (∞,1)-topos -- structures.

Sometimes it is desireable to add further axioms, such as the following.

Definition

We say that pieces have points for an object XX in a cohesive (,1)(\infty,1)-topos H\mathbf{H} if the points-to-pieces transform

ΓXΓDiscΠXΠX \Gamma X \to \Gamma Disc \Pi X \simeq \Pi X

is an effective epimorphism in an (∞,1)-category, equivalently (as discussed there) such that this is an epimorphism on connected components.

Here the first morphismism is the image under Γ\Gamma of the (DiscΓ)(Disc \dashv \Gamma)-unit and the second is an inverse of the (ΠDisc)(\Pi \dashv Disc)-counit (which is invertible because DiscDisc is full and faithful in a local (∞,1)-topos.)

Definition

We say discrete objects are concrete in H\mathbf{H} if for all SS \in ∞Grpd the morphism

DiscScoDiscΓDiscScoDiscS Disc S \to coDisc \Gamma Disc S \stackrel{\simeq}{\to} coDisc S

induces monomorphisms on all homotopy sheaves.

Internally

We reformulate the above axioms for a cohesive (,1)(\infty,1)-topos without references to functors on it, and instead entirely in terms of structures in it.

Lemma

A full sub-(∞,1)-category BH\mathbf{B} \hookrightarrow \mathbf{H} is

  • reflectively embedded precisely if for every object XHX \in \mathbf{H} there is a morphism

    loc X:XLX loc_X : X \to L X

    (the unit) with LXBHL X \in \mathbf{B} \hookrightarrow \mathbf{H}, such that for all YBHY \in \mathbf{B} \hookrightarrow \mathbf{H} the value of the (∞,1)-categorical hom-space-functor

    H(loc X,Y):H(LX,Y)H(X,Y) \mathbf{H}(loc_X, Y) : \mathbf{H}(L X, Y) \stackrel{\simeq}{\to} \mathbf{H}(X, Y)

    is an equivalence (of ∞-groupoids).

  • coreflectively embedded precisely if for every object YHY \in \mathbf{H} there is a morphism

    coloc Y:RYY coloc_Y : R Y \to Y

    (the counit) with RYBHR Y \in \mathbf{B} \hookrightarrow \mathbf{H} such that for all XBHX \in \mathbf{B} \hookrightarrow \mathbf{H} the value of the (∞,1)-categorical hom-space-functor

    H(X,coloc Y):H(X,RY)H(X,Y) \mathbf{H}(X, coloc_Y) : \mathbf{H}(X, R Y) \stackrel{\simeq}{\to} \mathbf{H}(X, Y)

    is an equivalence (of ∞-groupoids).

This is proven here.

Lemma

A reflective embedding

coDisc:B codcoDiscΓ˜H coDisc : \mathbf{B}_{cod} \stackrel{\overset{\tilde \Gamma}{\leftarrow}}{\underset{coDisc}{\hookrightarrow}} \mathbf{H}

and a coreflective embedding

Disc:B discΓDiscH Disc : \mathbf{B}_{disc} \stackrel{\overset{Disc}{\hookrightarrow}}{\underset{\Gamma}{\leftarrow}} \mathbf{H}

fit into a single adjoint triple

HcoDiscΓDiscB \mathbf{H} \stackrel{ \overset{Disc}{\hookleftarrow} }{ \stackrel{ \overset{\Gamma}{\to} }{ \underset{coDisc}{\hookleftarrow} } } \mathbf{B}

(hence there is an equivalence B discB cod\mathbf{B}_{disc} \simeq \mathbf{B}_{cod} that moreover makes the coreflector Γ˜\tilde\Gamma of DiscDisc coincide with the reflector Γ\Gamma of coDisccoDisc) precisely if for the unit and counit given by lemma 1 we have that the morphisms on the left of

(1)coDiscΓ˜(DiscΓXX)=:(coDiscΓ˜DiscΓXcoDiscΓ˜X) coDisc \tilde \Gamma (Disc \Gamma X \to X) \; =: \; (coDisc \tilde \Gamma Disc \Gamma X \stackrel{\simeq}{\to} coDisc \tilde \Gamma X)
(2)DiscΓ(XcoDiscΓ˜X)=:(DiscΓXDiscΓcoDiscΓ˜) Disc \Gamma (X \to coDisc \tilde \Gamma X) \;\; =: \;\; ( Disc \Gamma X \stackrel{\simeq}{\to} Disc \Gamma coDisc \tilde \Gamma )

are (natural) equivalences for all objects XHX \in \mathbf{H}, as indicated on the right.

Proof

It is clear that if we have an adjoint triple, then (1) and (2) are implied. We discuss now the converse.

First notice that the two embeddings always combine into an adjunction of the form

B discΓDiscHcoDiscΓ˜B cod. \mathbf{B}_{disc} \stackrel{\overset{Disc}{\hookrightarrow}}{\underset{\Gamma}{\leftarrow}} \mathbf{H} \stackrel{\overset{\tilde \Gamma}{\to}}{\underset{coDisc}{\hookleftarrow}} \mathbf{B}_{cod} \,.

The natural equivalence (1) applied to a codiscrete object X:=coDiscAX := coDisc A gives that coDisccoDisc of the counit of this composite adjunction is an equivalence

coDiscΓ˜DiscΓcoDiscAcoDiscΓ˜coDiscAcoDiscA coDisc \tilde \Gamma Disc \Gamma coDisc A \stackrel{\simeq}{\to} coDisc \tilde \Gamma coDisc A \stackrel{\simeq}{\to} coDisc A

for all AA, and since coDisccoDisc is full and faithful, so is the composite counit

Γ˜DiscΓcoDiscAΓ˜coDiscAA \tilde \Gamma Disc \Gamma coDisc A \stackrel{\simeq}{\to} \tilde \Gamma coDisc A \stackrel{\simeq}{\to} A

itself. Analogously, (2) implies that the unit of the composite adjunction is an equivalence. Therefore (1) and (2) together imply that the adjunction itself exhibits an equivalence B discB cod\mathbf{B}_{disc} \simeq \mathbf{B}_{cod}.

Using this we then find for each XHX \in \mathbf{H} a composite natural equivalence

DiscΓ˜XDiscΓcoDiscΓ˜XDiscΓX Disc \tilde \Gamma X \stackrel{\simeq}{\to} Disc \Gamma coDisc \tilde \Gamma X \stackrel{\simeq}{\to} Disc \Gamma X

where the first morphism uses the above equivalence on the codiscrete object Γ˜X\tilde \Gamma X and the second is a choice of natural inverse of (2).

Since DiscDisc is full and faithful, this mean that we have equivalences

Γ˜XΓX \tilde \Gamma X \simeq \Gamma X

natural in XX, hence that Γ˜Γ\tilde \Gamma \simeq \Gamma.

Remark

In the above situation, the defining adjunction equivalence

H(ΠX,A)H(X,A) \mathbf{H}(\mathbf{\Pi} X , A) \simeq \mathbf{H}(X, \flat A)

exhibiting (Π)(\mathbf{\Pi} \dashv \flat) is given by the composite of natural equivalences

H(ΠX,A)H(ΠX,coloc A) 1H(ΠX,A)H(loc X,A)H(X,A) \mathbf{H}(\mathbf{\Pi}X, A) \underoverset{\simeq}{\mathbf{H}(\mathbf{\Pi}X, coloc_A)^{-1}}{\to} \mathbf{H}(\mathbf{\Pi} X, \flat A) \underoverset{\simeq}{\mathbf{H}(loc_X, \flat A)}{\to} \mathbf{H}(X, \flat A)

from lemma 1, using that both ΠX\mathbf{\Pi} X as well as A\flat A are discrete.

Using these lemmas we can now restate cohesiveness internally.

Corollary

For Disc:BHDisc : \mathbf{B} \hookrightarrow \mathbf{H} a sub-(∞,1)-category, the inclusion extends to an adjoint quadruple of the form

HcoDiscΓDiscΠB \mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\hookleftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\hookleftarrow}}}} \mathbf{B}

precisely if there exists for each object XHX \in \mathbf{H}

  1. a morphism XΠ(X)X \to \mathbf{\Pi}(X) with Π(X)BDiscH\mathbf{\Pi}(X) \in \mathbf{B} \stackrel{Disc}{\hookrightarrow} \mathbf{H};

  2. a morphism XX\mathbf{\flat} X \to X with XBDiscH\mathbf{\flat} X \in \mathbf{B} \stackrel{Disc}{\hookrightarrow} \mathbf{H};

  3. a morphism X#XX \to #X with #XBcoDiscH# X \in \mathbf{B} \stackrel{coDisc}{\hookrightarrow} \mathbf{H}

such that for all YBDiscHY \in \mathbf{B} \stackrel{Disc}{\hookrightarrow} \mathbf{H} and Y˜BcoDiscH\tilde Y \in \mathbf{B} \stackrel{coDisc}{\hookrightarrow} \mathbf{H} the induced morphisms

  1. H(ΠX,Y)H(X,Y)\mathbf{H}(\mathbf{\Pi}X , Y) \stackrel{\simeq}{\to} \mathbf{H}(X,Y);

  2. H(Y,X)H(Y,X)\mathbf{H}(Y, \mathbf{\flat}X) \stackrel{\simeq}{\to} \mathbf{H}(Y,X);

  3. H(#X,Y˜)H(X,Y˜)\mathbf{H}(# X , \tilde Y) \stackrel{\simeq}{\to} \mathbf{H}(X,\tilde Y);

  4. #(XX)# (\flat X \to X);

  5. (X#X)\flat (X \to # X)

are equivalences (the first three of ∞-groupoids the last two in H\mathbf{H}).

Moreover, if H\mathbf{H} is a cartesian closed category, then Π\Pi preserves finite products precisely if the DiscDisc-inclusion is an exponential ideal.

The last statement follows from the (,1)(\infty,1)-category analog of the discussion here.

In homotopy type theory

The axioms for cohesion, in the internal version, can be formulated in homotopy type theory, the internal language of an (∞,1)-topos.

The corresponging Coq-HoTT code is in (Shulman).

For more see cohesive homotopy type theory.

Properties

We discuss basic properties implied by the axioms for cohesive (,1)(\infty,1)-toposes in

Then we discuss presentations over special sites in

As a point-like space

Proposition

A nontrivial cohesive (,1)(\infty,1)-topos

  1. has the shape of the point;

  2. has homotopy dimension 0;

  3. has cohomology dimension 0.

Proof

The first holds for every ∞-connected (∞,1)-topos, see there.

The second holds for every local (∞,1)-topos, see there.

The third follows from the second, see homotopy dimension.

Remark

This says that a cohesive (,1)(\infty,1)-topos H\mathbf{H} is, when itself regarded as a little topos, a generalized space, a thickened point . We may think of it as the standard point equipped with a cohesive neighbourhood .

In this sense every space XX modeled on the cohesive structure defined by H\mathbf{H} is an étale space over XX: its petit (,1)(\infty,1)-topos H/X\mathbf{H}/X sits by a locally homeomorphic geometric morphism over H\mathbf{H}

H/XlocalhomeoH. \mathbf{H}/X \stackrel{local\;homeo}{\to} \mathbf{H} \,.

Over an \infty-cohesive site

We discuss a presentation of classes of cohesive (∞,1)-toposes by a model structure on simplicial presheaves over a suitable site.

Proposition

For CC an ∞-cohesive site the (∞,1)-category of (∞,1)-sheaves (,1)Sh(C)(\infty,1)Sh(C) over CC is a cohesive (,1)(\infty,1)-topos satisfying the two axioms pieces have points and discrete objects are concrete .

The detailed discussion is at ∞-cohesive site.

General

Proposition

Every cohesive (,1)(\infty,1)-topos over ∞Grpd is a hypercomplete (∞,1)-topos.

Proof

By the above proposition it has finite homotopy dimension. This implies hypercompeteness. See there.

Proposition

For H\mathbf{H} a cohesive (,1)(\infty,1)-topos, the (1,1)-topos τ 11H\tau_{\leq 1-1} \mathbf{H} of 0-truncated objects is a cohesive topos.

Proposition

For a cohesive (,1)(\infty,1)-topos (ΠDiscΓcoDisc):HGrpd(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : \mathbf{H} \to \infty Grpd over an ∞-cohesive site, the functor Π\Pi preserves (∞,1)-pullbacks over discrete objects.

We first consider a lemma. Notice that for AGrpdA \in \infty Grpd the (∞,1)-Grothendieck construction gives an equivalence of (∞,1)-categories

Grpd /AFunc(A,Grpd) \infty Grpd_{/A} \simeq Func(A, \infty Grpd)

from the over (∞,1)-category of Grpd\infty Grpd over AA to the (∞,1)-functor (∞,1)-category from AA to Grpd\infty Grpd.

Lemma

For H\mathbf{H} a cohesive (,1)(\infty,1)-topos over an ∞-cohesive site and for AGrpdA \in \infty Grpd, we have an equivalence of (∞,1)-categories

H /DiscAFunc(A,H). \mathbf{H}_{/ Disc A} \simeq Func(A, \mathbf{H}) \,.
Proof

We establish this via a presentation of H\mathbf{H} by a model structure on simplicial presheaves.

Let CC be an ∞-cohesive site of definition for H\mathbf{H}. Then by the discussion there we have

H([C op,sSet] proj,loc) . \mathbf{H} \simeq ([C^{op}, sSet]_{proj,loc})^\circ \,.

Moreover, picking a Kan complex presentation for AA, which we shall denote by the same symbol, we have that the constant simplicial presheaf constA[C op,sSet] proj,locconst A \in [C^{op}, sSet]_{proj, loc} is fibrant. Therefore by this proposition the induced model structure on an overcategory on [C op,sSet]/constA[C^{op}, sSet]/const A presents the given over (∞,1)-category

H /DiscA(([C op,sSet]/constA) (proj,loc)/constA) . \mathbf{H}_{/ Disc A} \simeq \left( \left([C^{op}, sSet]/const A\right)_{(proj,loc)/const A} \right)^\circ \,.

Now observe that we have an ordinary equivalence of categories

[C op,sSet]/constA[C op,sSet/A] [C^{op}, sSet]/const A \simeq [C^op, sSet/A]

under which the model structure becomes that of the local projective model structure on functors with values in the model structure (sSet/A) Quillen/A(sSet/A)_{Quillen/A} that presents Grpd /A\infty Grpd_{/ A}.

Let then sSet +/AsSet^+/A denote the model structure for left fibrations. By the discussion there, this also presents Grpd /A\infty Grpd_{/ A}. Hence by this proposition we have an equivalence of (∞,1)-categories

H DiscA ([C op,sSet/A] proj,loc) ([C op,sSet +/A] proj,loc) . \begin{aligned} \mathbf{H}_{Disc A} & \simeq \left( [C^{op}, sSet/A]_{proj,loc} \right)^\circ \\ & \simeq \left( [C^{op}, sSet^+/A]_{proj,loc} \right)^\circ \end{aligned} \,.

This allows now to apply this presentation of the (∞,1)-Grothendieck construction to find

([C op,[w(A),sSet Quillen] proj] proj,loc) , \cdots \simeq \left( [C^{op}, [w(A), sSet_{Quillen}]_{proj}]_{proj,loc} \right)^\circ \,,

where w(A)w(A) is the simplicially enriched category corresponding to AA (as discussed at relation between quasi-categories and simplicial categories ) and [w(A),sSet Quillen] proj[w(A), sSet_{Quillen}]_{proj} is the global model structure on sSet-enriched presheaves.

Then using the cartesian closure of the category of simplicial presheaves (which is a topos) inside the () (-)^\circ we have

([w(A),[C op,sSet Quillen] proj,loc] proj) . \cdots \simeq \left( [w(A), [C^{op}, sSet_{Quillen}]_{proj,loc}]_{proj} \right)^\circ \,.

Finally this implies the claim using this proposition.

With this lemma we can now give the proof of prop. 5.

Proof

By the discussion at adjoint (∞,1)-functors on slices we have that (ΠDisc)(\Pi \dashv Disc) induces an adjoint pair

(Π/DiscADisc/DiscA):H DiscAGrpd /A. (\Pi/Disc A \dashv Disc / Disc A) : \mathbf{H}_{Disc A} \to \infty Grpd_{/A} \,.

Under the equivalence from lemma 3 the functor Π/DiscA\Pi/Disc A maps to

Func(A,Π):Func(A,H)Func(A,Grpd). Func(A, \Pi) : Func(A, \mathbf{H}) \to Func(A, \infty Grpd) \,.

Since products of (,1)(\infty,1)-functor (,1)(\infty,1)-categories are computed objectwise, and since Π\Pi preserves finite products by the axioms of cohesion, also Func(A,Π)Func(A, \Pi) preserves finite products, and hence so does Π/DiscA\Pi/Disc A. But products in the slice over DiscADisc A are (∞,1)-pullbacks over DiscADisc A. So this proves the claim.

Base of discrete and codiscrete objects

In the internal definition the base of discrete/codiscrete objects is not explicitly axiomatized to be an (∞,1)-topos itself (the base (∞,1)-topos), but this is implied by the axioms. We deduce that and related properties in stages.

In the following, let H\mathbf{H} be an (∞,1)-topos equipped with an adjoint quadruple of functors to an (∞,1)-category B\mathbf{B} – the base of cohesion, where DiscDisc and coDisccoDisc are full and faithful.

Proposition

The base B\mathbf{B} of cohesion has all (∞,1)-limits and (∞,1)-colimits.

Proof

This is a general property of a reflectively and coreflectively embedded subcategory. The limits are computed by computing them in H\mathbf{H} and then applying Γ\Gamma and the colimits are computed by computing them in H\mathbf{H} and then applying Π\Pi. For X:IBX : I \to \mathbf{B} any diagram we have

Γlim iDiscX i lim iΓDiscX i lim iX i \begin{aligned} \Gamma \lim_{\leftarrow_i} Disc X_i & \simeq \lim_{\leftarrow_i} \Gamma Disc X_i \\ & \simeq \lim_{\leftarrow_i} X_i \end{aligned}
Πlim iDiscX i lim iΠDiscX i lim iX i. \begin{aligned} \Pi \lim_{\to_i} Disc X_i & \simeq \lim_{\leftarrow_i} \Pi Disc X_i \\ & \simeq \lim_{\leftarrow_i} X_i \end{aligned} \,.
Remark

Since DiscDisc, being both a left adjoint as well as a right adjoint preserves limits and colimits, it follows that a (co)limit of discrete objects computed in H\mathbf{H} is itself again discrete and is the image under DiscDisc of the coresponding (co)limit computed in B\mathbf{B}.

Example

Since loop space objects are (∞,1)-limits it follows that the loop space object of any discrete object is itself again a discrete object.

We have also the following stronger statement.

Proposition

The base of cohesion B\mathbf{B} is a presentable (∞,1)-category and in fact an (∞,1)-topos itself.

Proof

By one of the equivalent characterizations of presentable (∞,1)-categories these are reflective sub-(∞,1)-categories of (∞,1)-categories of (∞,1)-presheaves where the embedding is by an accessible (∞,1)-functor.

Since H\mathbf{H} is itself accessibly and reflectively embedded into the presheaves PSh(C)PSh(C) on a (∞,1)-site of definition, we have a composite reflective inclusions

BDiscHPSh(C). \mathbf{B} \stackrel{Disc}{\hookrightarrow} \mathbf{H} \hookrightarrow PSh(C) \,.

Since DiscDisc even preserves all (∞,1)-colimits, it is in particular an accessible (∞,1)-functor, hence so is the above composite.

Finally, since Γ\Gamma preserves all (∞,1)-limits, hence in particular the finite limits, (Γ,coDisc)(\Gamma, coDisc) is a geometric embedding that exhibits an sub-(∞,1)-topos.

Notice that the reflection (ΠDisc)(\Pi \dashv Disc) does not in general constitute a geometric embedding, since Π\Pi is only required to preserve finite products (and in interesting examples rarely preserves more limits than that).

The following statement and its proof about cohesive 1-toposes should hold verbatim also for cohesive (,1)(\infty,1)-toposes.

Proposition

The reflective subcategories of discrete objects and of codiscrete objects are both exponential ideals.

Proof

By the discussion at exponential ideal a reflective subcategories of a cartesian closed category is an exponential ideal precisely if the reflector preserves products. For the codiscrete objects the reflector Γ\Gamma preserves even all limits and for the discrete objects the reflector Π\Pi does so by assumotion of strong connectedness.

Factorization systems associated to Π\mathbf{\Pi}

We discuss orthogonal factorization systems in a cohesive (,1)(\infty,1)-topos that characterize or are characterized by the reflective subcategory of dicrete objects, with reflector Π:HΠGrpdDiscH\mathbf{\Pi} : \mathbf{H} \stackrel{\Pi}{\to} \infty Grpd \stackrel{Disc}{\hookrightarrow} \mathbf{H}.

Definition

For f:XYf : X \to Y a morphism in H\mathbf{H}, write c ΠfYc_{\mathbf{\Pi}} f \to Y for the (∞,1)-pullback in

c Πf ΠX Πf Y ΠY, \array{ c_{\mathbf{\Pi}} f &\to & \mathbf{\Pi} X \\ \downarrow && \downarrow^{\mathrlap{\mathbf{\Pi} f}} \\ Y &\to& \mathbf{\Pi} Y } \,,

where the bottom morphism is the (ΠDisc)(\Pi \dashv Disc)-unit. We say that c Πfc_{\mathbf{\Pi}} f is the Π\mathbf{\Pi}-closure of ff, and that ff is Π\mathbf{\Pi}-closed if Xc ΠfX \simeq c_{\mathbf{\Pi}} f.

Proposition

If H\mathbf{H} has an ∞-cohesive site of definition, then every morphism f:XYf : X \to Y in H\mathbf{H} factors as

X f Y c Πf, \array{ X &&\stackrel{f}{\to}&& Y \\ & \searrow && \nearrow \\ && c_{\mathbf{\Pi}}f } \,,

such that Xc ΠfX \to c_{\mathbf{\Pi}} f is a Π\mathbf{\Pi}-equivalence in that it is inverted by Π\mathbf{\Pi}.

Proof

The factorization is given by the naturality of Π\mathbf{\Pi} and the universal property of the (,1)(\infty,1)-pullback in def. 4.

X c Πf ΠX f Πf Y ΠY. \array{ X &\to & c_{\mathbf{\Pi}} f &\to & \mathbf{\Pi} X \\ &{}_{\mathllap{f}}\searrow & \downarrow && \downarrow^{\mathrlap{\mathbf{\Pi} f}} \\ && Y &\to& \mathbf{\Pi} Y } \,.

Then by prop. 5 the functor Π\mathbf{\Pi} preserves the (,1)(\infty,1)-pullback over the discrete object ΠY\mathbf{\Pi}Y and since Π(XΠX)\mathbf{\Pi}(X \to \mathbf{\Pi}X) is an equivalence, it follows that Π(Xc Πf)\mathbf{\Pi}(X \to c_{\mathbf{\Pi}f}) is an equivalence.

Proposition

The pair of classes

(Πequivalences,Πclosedmorphisms) (\mathbf{\Pi}-equivalences, \mathbf{\Pi}-closed morphisms)

is an orthogonal factorization system in H\mathbf{H}.

Proof

This follows by the general reasoning discussed at reflective factorization system:

By prop. 9 we have the required factorization. It remains to check the orthogonality.

So let

A X B Y \array{ A &\to& X \\ \downarrow && \downarrow \\ B &\to& Y }

be a square diagram in H\mathbf{H} where the left morphism is a Π\mathbf{\Pi}-equivalence and the right morphism is Π\mathbf{\Pi}-closed. Then by assumption there is a pullback square on the right in

A X ΠX B Y ΠY.. \array{ A &\to& X &\to& \mathbf{\Pi}X \\ \downarrow && \downarrow && \downarrow \\ B &\to& Y &\to& \mathbf{\Pi}Y \,. } \,.

By naturality of the adjunction unit, the total rectangle is equivalent to

A ΠA ΠY B ΠB ΠX. \array{ A &\to& \mathbf{\Pi} A &\to & \mathbf{\Pi} Y \\ \downarrow && \downarrow^{\mathrlap{\simeq}} && \downarrow \\ B &\to& \mathbf{\Pi} B &\to& \mathbf{\Pi}X } \,.

Here by assumption the middle morphism is an equivalence. Therefore there is an essentially unique lift in the square on the right and hence a lift in the total square. Again by the universality of the adjunction, any such lift factors through ΠB\mathbf{\Pi} B and hence also this lift is essentially unique.

Finally by universality of the pullback, this induces an essentially unique lift σ\sigma in

A X ΠX σ B Y ΠY.. \array{ A &\to& X &\to& \mathbf{\Pi}X \\ \downarrow &{}^{\mathllap{\sigma}}\nearrow& \downarrow && \downarrow \\ B &\to& Y &\to& \mathbf{\Pi}Y \,. } \,.
Observation

For f:XYf : X \to Y a Π\mathbf{\Pi}-closed morphism and y:*Yy : * \to Y a global element, the homotopy fiber X y:=y *XX_y := y^* X is a discrete object.

Proof

By the def. 4 and the pasting law we have that y *Xy^* X is equivalently the \infty-pullback in

y *X ΠX * y Y ΠY. \array{ y^* X &\to& &\to& \mathbf{\Pi} X \\ \downarrow && && \downarrow \\ * &\stackrel{y}{\to}& Y &\stackrel{}{\to}& \mathbf{\Pi}Y } \,.

Since the terminal object is discrete, and since the right adjoint DiscDisc preserves \infty-pullbacks, this exhibits y *Xy^* X as the image under DiscDisc of an \infty-pullback of \infty-groupoids.

Structures in a cohesive (,1)(\infty,1)-topos

A cohesive (,1)(\infty,1)-topos is a general context for higher geometry with good cohomology and homotopy properties. We list fundamental structures and constructions that exist in every cohesvive (,1)(\infty,1)-topos.

This section is at

Types of cohesion

Infinitesimal cohesion

Differential cohesion

We discuss extra structure on a cohesive (,1)(\infty,1)-topos that encodes a refinement of the corresponding notion of cohesion to infinitesimal cohesion . More precisely, we consider inclusions HH th\mathbf{H} \hookrightarrow \mathbf{H}_{th} of cohesive (,1)(\infty,1)-toposes that exhibit the objects of H th\mathbf{H}_{th} as infinitesimal cohesive neighbourhoods of objects in H\mathbf{H}.

This section is at

Locally ringed cohesion

Every cohesive (,1)(\infty,1)-topos H\mathbf{H} equipped with differential cohesion comes canonically equipped with a notion of formally étale morphisms (as discussed there). Combined with the canonical interpretation of H\mathbf{H} as the classifying topos of a theory of local T-algebras, this caninically induces a notion of locally algebra-ed (∞,1)-toposes with cohesive structure, generalizing the notion of locally ringed spaces and locally ringed toposes.

This section is at

Examples of cohesive \infty-toposes

We list examples of cohesive (,1)(\infty,1)-toposes, both specific ones as well as classes of examples constructed in a certan way.

Cohesive diagram (,1)(\infty,1)-toposes

Cohesive diagrams in a cohesive (,1)(\infty,1)-topos

Proposition

Let H\mathbf{H} be an cohesive (,1)(\infty,1)-topos.

Let DD be a small category (diagram) with initial object \bottom and terminal object \top, or else a presentable (∞,1)-category. Write

(p):Dp* (\bottom \dashv p \dashv \top) : D \stackrel{\overset{\bottom}{\hookleftarrow}}{\stackrel{\overset{p}{\to}}{\underset{\top}{\hookleftarrow}}} *

for the triple of adjoint (∞,1)-functors given by including \bottom and \top.

Then the (∞,1)-functor category H D\mathbf{H}^D is again a cohesive (,1)(\infty,1)-topos, exhibited by the adjoint quadruple which is the composite

H D * *p * *HcoDiscΓDiscΠGrpd, \mathbf{H}^D \stackrel{\overset{\top^*}{\to}}{\stackrel{\overset{p^*}{\hookleftarrow}}{\stackrel{\overset{\bottom^*}{\to}}{\underset{\bottom_*}{\hookleftarrow}}}} \mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\hookleftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\hookleftarrow}}}} \infty Grpd \,,

where the adjoint quadruple on the left is that induced under (∞,1)-Kan extension from (p)(\bottom \dashv p \dashv \top).

Proof

By the discussion at (∞,1)-Kan extension each of the original three functors induces [adjoint triples etc, as indicated. In particular is a right adjoint and therefore preserves finite products (and all small (∞,1)-limits, even).]

By the original adjunctions one finds that !p *\bottom_! \simeq p^* and p ! *p_! \simeq \top^*, which implies the adjoint quadruple as indicated above by essential uniqueness of adjoints.

Finally it is clear that *p *Id\top^* p^* \simeq Id, which implies that p *p^* is a full and faithful (∞,1)-functor (and hence so is *\bottom_*).

In particular we have

Corollary

For H\mathbf{H} a cohesive (,1)(\infty,1)-topos, also its arrow (∞,1)-category H Δ[1]\mathbf{H}^{\Delta[1]} is cohesive.

Example

For H=\mathbf{H} = ∞Grpd (“discrete cohesion”, see below) the corresponding cohesive (,1)(\infty,1)-topos Grpd Δ[1]\infty Grpd^{\Delta[1]} is known as the Sierpinski (∞,1)-topos.

Simplicial objects in a cohesive (,1)(\infty,1)-topos

For H\mathbf{H} a cohesive (.1)(\infty.1)-topos its (∞,1)-category of simplicial objects H Δ op\mathbf{H}^{\Delta^{op}} is cohesive over H\mathbf{H}

H Δ opcoDisc IΓ IDisc IΠ IH. \mathbf{H}^{\Delta^{op}} \stackrel{\Pi_I}{\stackrel{\to}{\stackrel{\overset{Disc_I}{\leftarrow}}{\stackrel{\overset{\Gamma_I}{\to}}{\underset{coDisc_I}{\leftarrow}}}}} \mathbf{H} \,.

Here

  • Π I\Pi_I sends a simplicial object to the homotopy colimit over its components, hence to its “geometric realization” as seen in H\mathbf{H}.

  • Γ I\Gamma_I evaluates on the 0-simplex;

  • Disc IDisc_I sends an object in H\mathbf{H} to the simplicial object which is simplicially constant on AA.

Hence cohesion of H Δ op\mathbf{H}^{\Delta^{op}} relative to H\mathbf{H} expresses the existence of a discrete and directed notion of path.

Notice that there is an inclusion

Grpd(H)Cat(H)H Δ op Grpd(\mathbf{H}) \hookrightarrow Cat(\mathbf{H}) \hookrightarrow \mathbf{H}^{\Delta^{op}}

of the groupoid objects internal to H\mathbf{H} and of the category objects internal to H\mathbf{H} inside H Δ op\mathbf{H}^{\Delta^{op}}.

Here H Δ op\mathbf{H}^{\Delta^{op}} is also the classifying topos for linear intervals. Its homotopy type theory internal language is equipped with an interval type.

For more see at simplicial object in an (∞,1)-category.

Bundles of cohesive spectra

The tangent (∞,1)-category THT\mathbf{H} to a cohesive \infty-topos is itself cohesive, (by the discussion at tangent ∞-category – Examples – Of an ∞-topos), the tangent cohesive (∞,1)-topos.

This THT \mathbf{H} the \infty-topos of parameterized spectra in H\mathbf{H}, hence is context for cohesive stable homotopy theory.

Stab(H) coDisc seqΓ seqDisc seqLΠ seq Stab(Grpd) incl incl TH coDisc seqΓ seqDisc seqLΠ seq TGrpd base 0 base 0 base 0 base 0 H coDiscΓDiscΠ Grpd. \array{ Stab(\mathbf{H}) & \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}} & Stab(\infty Grpd) \\ \downarrow^{\mathrlap{incl}} && \downarrow^{\mathrlap{incl}} \\ T \mathbf{H} & \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}} & T \infty Grpd \\ {}^{\mathllap{base}}\downarrow {}^{\mathllap{0}}\uparrow \downarrow^{\mathrlap{base}} \uparrow^{\mathrlap{0}} && {}^{\mathllap{base}}\downarrow {}^{\mathllap{0}}\uparrow \downarrow^{\mathrlap{base}} \uparrow^{\mathrlap{0}} \\ \mathbf{H} & \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} & \infty Grpd } \,.

Global equivariant homotopy theory

Rezk-global equivariant homotopy theory:

cohesive (∞,1)-toposits (∞,1)-sitebase (∞,1)-toposits (∞,1)-site
global equivariant homotopy theory PSh (Glo)PSh_\infty(Glo)global equivariant indexing category GloGlo∞Grpd PSh (*) \simeq PSh_\infty(\ast)point
sliced over terminal orbispace: PSh (Glo) /𝒩PSh_\infty(Glo)_{/\mathcal{N}}Glo /𝒩Glo_{/\mathcal{N}}orbispaces PSh (Orb)PSh_\infty(Orb)global orbit category
sliced over BG\mathbf{B}G: PSh (Glo) /BGPSh_\infty(Glo)_{/\mathbf{B}G}Glo /BGGlo_{/\mathbf{B}G}GG-equivariant homotopy theory of G-spaces L weGTopPSh (Orb G)L_{we} G Top \simeq PSh_\infty(Orb_G)GG-orbit category Orb /BG=Orb GOrb_{/\mathbf{B}G} = Orb_G

From \infty-Cohesive sites of definition

Proposition

Examples of ∞-cohesive sites are

From this one obtains the following list of examples of cohesive (,1)(\infty,1)-toposes.

Discrete \infty-groupoids

Topological \infty-groupoids

Smooth \infty-groupoids

Complex analytic \infty-groupoids

Synthetic differential \infty-groupoids

Smooth Super \infty-groupoids

Smooth \infty-groupoids over algebraic \infty-stacks

One can consider the tangent (∞,1)-topos of the cohesive (∞,1)-topos

Sh (SmthMfd,Sh (Sch ))coDiscΓDiscΠSh (Sch ) Sh_\infty\left(SmthMfd, Sh_\infty\left(Sch_{\mathbb{Z}}\right)\right) \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} Sh_\infty(Sch_{\mathbb{Z}})

of ∞-stacks on the site of smooth manifolds with values in turn in ∞-stack over a site of arithmetic schemes, hence by smooth ∞-groupoids but over a base (∞,1)-topos of algebraic ∞-stacks.

This leads to differential algebraic K-theory. See there for details.

Smooth cohesion over arbitrary base \infty-toposes

The above examples relativize to arbitrary bases.

For instance smooth cohesion over an (∞,1)-topos over a suitable site of schemes is the natural context for differential algebraic K-theory. See there for more.

Applications

As a context for geometric spaces and paths in geometric spaces, cohesive (,1)(\infty,1)-toposes are a natural context in which to formulate fundamental fundamental physics. See higher category theory and physics for more on this.

See also

and

unrelated is the notion of cohesive ∞-prestack

References

Precursors

The category-theoretic definition of cohesive topos was proposed by Bill Lawvere. See the references at cohesive topos.

The observation that the further left adjoint Π\Pi in a locally ∞-connected (∞,1)-topos defines an intrinsic notion of paths and geometric homotopy groups in an (∞,1)-topos was suggested by Richard Williamson.

The observation that the further right adjoint coDisccoDisc in a local (∞,1)-topos serves to characterize concrete (∞,1)-sheaves was amplified by David Carchedi.

Some aspects of the discussion here are, more or less explicitly, in

For instance something similar to the notion of ∞-connected site and the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos is the content of section 2.16. The infinitesimal path ∞-groupoid adjunction (RedΠ inf inf)(\mathbf{Red} \dashv \mathbf{\Pi}_{inf} \dashv \mathbf{\flat}_{inf}) is essentially discussed in section 3. The notion of geometric realization (see structures in a cohesive (∞,1)-topos – geometric realization), is touched on around remark 2.22, referring to

But, more or less explicitly, the presentation of geometric realization of simplicial presheaves is much older, going back to Artin-Mazur. See geometric homotopy groups in an (∞,1)-topos for a detailed commented list of literature.

A characterization of infinitesimal extensions and formal smoothness by adjoint functors (discussed at infinitesimal cohesion) is considered in

in the context of Q-categories .

On cohesive (,1)(\infty,1)-toposes proper

The material presented here is also in section 3 of

In homotopy type theory

Expositions and discussion of the formalization of cohesion in homotopy type theory is in

The corresponding Coq-code is in

Exposition is at

Revised on June 10, 2014 02:03:16 by Urs Schreiber (89.204.155.99)