nLab
cohesive (infinity,1)-topos

Idea

A cohesive $(\mathrm{\infty },1)$-topos is a big (∞,1)-topos $H$ that provides a context of generalized spaces in which higher geometry makes sense.

It is an $(\mathrm{\infty },1)$-topos whose global section (∞,1)-geometric morphism $(\mathrm{Disc}⊣\Gamma ):H\stackrel{\stackrel{\mathrm{Disc}}{\leftarrow }}{\underset{\Gamma }{\to }}$ ∞Grpd admits a further left adjoint (∞,1)-functor $\Pi$ and a further right adjoint $\mathrm{coDisc}$:

$$(\Pi ⊣\mathrm{Disc}⊣\Gamma ⊣\mathrm{coDisc}):H\to \mathrm{\infty }\mathrm{Grpd}$$(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : \mathbf{H} \to \infty Grpd

with $\mathrm{Disc}$ and $\mathrm{coFisc}$ both full and faithful (∞,1)-functors and such that $\Pi$ moreover preserves finite (∞,1)-products. Here

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

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

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

Conversely, $\mathrm{Disc}$ and $\mathrm{CoDisc}$ send an $\mathrm{\infty }$-groupoid $A$ either to the discrete ∞-groupoid $\mathrm{Disc}(A)$ with discrete cohesive structure (for instance with discrete topology) or to the codiscrete ∞-groupoid $\mathrm{Codisc}(A)$ with the codiscrete cohesive structure (for instance with codiscrete topology).

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

Examples of cohesive $(\mathrm{\infty },1)$-toposes include

• the $(\mathrm{\infty },1)$-topos $H=$Disc∞Grpd of discrete ∞-groupoids;

• the $(\mathrm{\infty },1)$-topos $H=$ ETop∞Grpd of Euclidean-topological ∞-groupoids;

• the $(\mathrm{\infty },1)$-topos $H=$ Smooth∞Grpd of smooth ∞-groupoids;

• the $(\mathrm{\infty },1)$-topos $H=$ SynthDiff∞Grpd of synthetic differential ∞-groupoids;

• the $(\mathrm{\infty },1)$-topos $H=$ Super∞Grpd of super ∞-groupoids;

• the $(\mathrm{\infty },1)$-topos $H=$ SmoothSuper∞Grpd of smooth super ∞-groupoids.

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 $(\mathrm{\infty },1)$-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 $H$ 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 $(\mathrm{\infty },1)$-topos is equivalently the following:

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

$$(\Pi ⊣\mathrm{Disc}⊣\Gamma ⊣\mathrm{coDisc}):H\stackrel{\stackrel{\stackrel{\Pi }{\to }}{\stackrel{\mathrm{Disc}}{\leftarrow }}}{\stackrel{\underset{\Gamma }{\to }}{\underset{\mathrm{coDisc}}{\leftarrow }}}\mathrm{\infty }\mathrm{Grpd}$$(\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 $\Pi$ 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.

$$(\Pi ⊣♭⊣\#):H\stackrel{\stackrel{\Pi }{\to }}{\stackrel{\stackrel{Disc}{\leftarrow }}{\underset{\Gamma }{\to }}}\mathrm{\infty }\mathrm{Grpd}\stackrel{\stackrel{\mathrm{Disc}}{\to }}{\stackrel{\stackrel{\Gamma }{\leftarrow }}{\underset{\mathrm{coDisc}}{\to }}}H.$$( \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 ”$♭$” 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 $X$ in a cohesive $(\mathrm{\infty },1)$-topos $H$ if the morphism

$$\Gamma X\to \Gamma \mathrm{Disc}\Pi X\simeq \Pi X$$\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 $\Gamma$ of the $(\mathrm{Disc}⊣\Gamma )$-unit and the second is an inverse of the $(\Pi ⊣\mathrm{Disc})$-counit (which is invertible because $\mathrm{Disc}$ is full and faithful in a local (∞,1)-topos.)

Definition

We say discrete objects are concrete in $H$ if for all $S\in$∞Grpd the morphism

$$\mathrm{Disc}S\to \mathrm{coDisc}\Gamma \mathrm{Disc}S\stackrel{\simeq }{\to }\mathrm{coDisc}S$$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 $(\mathrm{\infty },1)$-topos without references to functors on it, and instead entirely in terms of structures in it.

Lemma

A full sub-(∞,1)-category $B\hookrightarrow H$ is

• reflectively embedded precisely if for every object $X\in H$ there is a morphism

  $${\mathrm{loc}}_X:X\to LX$$loc_X : X \to L X

  (the unit) with $LX\in B\hookrightarrow H$, such that for all $Y\in B\hookrightarrow H$ the value of the (∞,1)-categorical hom-space-functor

  $$H({\mathrm{loc}}_X,Y):H(LX,Y)\stackrel{\simeq }{\to }H(X,Y)$$\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 $Y\in H$ there is a morphism

  $${\mathrm{coloc}}_Y:RY\to Y$$coloc_Y : R Y \to Y

  (the counit) with $RY\in B\hookrightarrow H$ such that for all $X\in B\hookrightarrow H$ the value of the (∞,1)-categorical hom-space-functor

  $$H(X,{\mathrm{coloc}}_Y):H(X,RY)\stackrel{\simeq }{\to }H(X,Y)$$\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

$$\mathrm{coDisc}:B_{\mathrm{cod}}\stackrel{\stackrel{\tilde{\Gamma }}{\leftarrow }}{\underset{\mathrm{coDisc}}{\hookrightarrow }}H$$coDisc : \mathbf{B}_{cod} \stackrel{\overset{\tilde \Gamma}{\leftarrow}}{\underset{coDisc}{\hookrightarrow}} \mathbf{H}

and a coreflective embedding

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

fit into a single adjoint triple

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

(hence there is an equivalence $B_{\mathrm{disc}}\simeq B_{\mathrm{cod}}$ that moreover makes the coreflector $\tilde{\Gamma }$ of $\mathrm{Disc}$ coincide with the reflector $\Gamma$ of $\mathrm{coDisc}$) precisely if for the unit and counit given by lemma 1 we have that the morphisms on the left of

(1)$$\mathrm{coDisc}\tilde{\Gamma }(\mathrm{Disc}\Gamma X\to X)=:(\mathrm{coDisc}\tilde{\Gamma }\mathrm{Disc}\Gamma X\stackrel{\simeq }{\to }\mathrm{coDisc}\tilde{\Gamma }X)$$coDisc \tilde \Gamma (Disc \Gamma X \to X) \; =: \; (coDisc \tilde \Gamma Disc \Gamma X \stackrel{\simeq}{\to} coDisc \tilde \Gamma X)

(2)$$\mathrm{Disc}\Gamma (X\to \mathrm{coDisc}\tilde{\Gamma }X)=:(\mathrm{Disc}\Gamma X\stackrel{\simeq }{\to }\mathrm{Disc}\Gamma \mathrm{coDisc}\tilde{\Gamma })$$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 $X\in H$, 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

$$B_{\mathrm{disc}}\stackrel{\stackrel{\mathrm{Disc}}{\hookrightarrow }}{\underset{\Gamma }{\leftarrow }}H\stackrel{\stackrel{\tilde{\Gamma }}{\to }}{\underset{\mathrm{coDisc}}{\hookleftarrow }}{B}_{\mathrm{cod}}.$$\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 $X:=\mathrm{coDisc}A$ gives that $\mathrm{coDisc}$ of the counit of this composite adjunction is an equivalence

$$\mathrm{coDisc}\tilde{\Gamma }\mathrm{Disc}\Gamma \mathrm{coDisc}A\stackrel{\simeq }{\to }\mathrm{coDisc}\tilde{\Gamma }\mathrm{coDisc}A\stackrel{\simeq }{\to }\mathrm{coDisc}A$$coDisc \tilde \Gamma Disc \Gamma coDisc A \stackrel{\simeq}{\to} coDisc \tilde \Gamma coDisc A \stackrel{\simeq}{\to} coDisc A

for all $A$, and since $\mathrm{coDisc}$ is full and faithful, so is the composite counit

$$\tilde{\Gamma }\mathrm{Disc}\Gamma \mathrm{coDisc}A\stackrel{\simeq }{\to }\tilde{\Gamma }\mathrm{coDisc}A\stackrel{\simeq }{\to }A$$\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 $B_{\mathrm{disc}}\simeq B_{\mathrm{cod}}$.

Using this we then find for each $X\in H$ a composite natural equivalence

$$\mathrm{Disc}\tilde{\Gamma }X\stackrel{\simeq }{\to }\mathrm{Disc}\Gamma \mathrm{coDisc}\tilde{\Gamma }X\stackrel{\simeq }{\to }\mathrm{Disc}\Gamma X$$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 $\tilde{\Gamma }X$ and the second is a choice of natural inverse of (2).

Since $\mathrm{Disc}$ is full and faithful, this mean that we have equivalences

$$\tilde{\Gamma }X\simeq \Gamma X$$\tilde \Gamma X \simeq \Gamma X

natural in $X$, hence that $\tilde{\Gamma }\simeq \Gamma$.

Remark

In the above situation, the defining adjunction equivalence

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

exhibiting $(\Pi ⊣♭)$ is given by the composite of natural equivalences

$$H(\Pi X,A)\underset{\simeq }{\overset{H(\Pi X,{\mathrm{coloc}}_A{)}^{-1}}{\to }}H(\Pi X,♭A)\underset{\simeq }{\overset{H({\mathrm{loc}}_X,♭A)}{\to }}H(X,♭A)$$\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 $\Pi X$ as well as $♭A$ are discrete.

Using these lemmas we can now restate cohesiveness internally.

Corollary

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

$$H\stackrel{\stackrel{\Pi }{\to }}{\stackrel{\stackrel{\mathrm{Disc}}{\hookleftarrow }}{\stackrel{\stackrel{\Gamma }{\to }}{\underset{\mathrm{coDisc}}{\hookleftarrow }}}}B$$\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 $X\in H$

1. a morphism $X\to \Pi (X)$ with $\Pi (X)\in B\stackrel{\mathrm{Disc}}{\hookrightarrow }H$;

2. a morphism $♭X\to X$ with $♭X\in B\stackrel{\mathrm{Disc}}{\hookrightarrow }H$;

3. a morphism $X\to \#X$ with $\#X\in B\stackrel{\mathrm{coDisc}}{\hookrightarrow }H$

such that for all $Y\in B\stackrel{\mathrm{Disc}}{\hookrightarrow }H$ and $\tilde{Y}\in B\stackrel{\mathrm{coDisc}}{\hookrightarrow }H$ the induced morphisms

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

2. $H(Y,♭X)\stackrel{\simeq }{\to }H(Y,X)$;

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

4. $\#(♭X\to X)$;

5. $♭(X\to \#X)$

are equivalences (the first three of ∞-groupoids the last two in $H$).

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

The last statement follows from the $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },1)$-toposes in

• As a point-like space;

• General.

• Over arbitrary bases

Then we discuss presentations over special sites in

• Over an ∞-cohesive site.

As a point-like space

Proposition

A nontrivial cohesive $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },1)$-topos $H$ 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 $X$ modeled on the cohesive structure defined by $H$ is an étale space over $X$: its petit $(\mathrm{\infty },1)$-topos $H/X$ sits by a locally homeomorphic geometric morphism over $H$

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

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

The detailed discussion is at ∞-cohesive site.

General

Proposition

Every cohesive $(\mathrm{\infty },1)$-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 $H$ a cohesive $(\mathrm{\infty },1)$-topos, the (1,1)-topos ${\tau }_{\le 1-1}H$ of 0-truncated objects is a cohesive topos.

Proposition

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

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

$$\mathrm{\infty }{\mathrm{Grpd}}_{/A}\simeq \mathrm{Func}(A,\mathrm{\infty }\mathrm{Grpd})$$\infty Grpd_{/A} \simeq Func(A, \infty Grpd)

from the over (∞,1)-category of $\mathrm{\infty }\mathrm{Grpd}$ over $A$ to the (∞,1)-functor (∞,1)-category from $A$ to $\mathrm{\infty }\mathrm{Grpd}$.

Lemma

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

$$H_{/\mathrm{Disc}A}\simeq \mathrm{Func}(A,H).$$\mathbf{H}_{/ Disc A} \simeq Func(A, \mathbf{H}) \,.

Proof

We establish this via a presentation of $H$ by a model structure on simplicial presheaves.

Let $C$ be an ∞-cohesive site of definition for $H$. Then by the discussion there we have

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

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

$$H_{/\mathrm{Disc}A}\simeq {\left({([C^{\mathrm{op}},\mathrm{sSet}]/\mathrm{const}A)}_{(\mathrm{proj},\mathrm{loc})/\mathrm{const}A}\right)}^{\circ }.$$\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

$$[C^{\mathrm{op}},\mathrm{sSet}]/\mathrm{const}A\simeq [C^{\mathrm{op}},\mathrm{sSet}/A]$$[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 $(\mathrm{sSet}/A{)}_{\mathrm{Quillen}/A}$ that presents $\mathrm{\infty }{\mathrm{Grpd}}_{/A}$.

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

$$\begin{array}{rl}{H}_{\mathrm{Disc}A}& \simeq {([C^{\mathrm{op}},\mathrm{sSet}/A{]}_{\mathrm{proj},\mathrm{loc}})}^{\circ }\\ & \simeq {([C^{\mathrm{op}},{\mathrm{sSet}}^{+}/A{]}_{\mathrm{proj},\mathrm{loc}})}^{\circ }\end{array}.$$\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

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

where $w(A)$ is the simplicially enriched category corresponding to $A$ (as discussed at relation between quasi-categories and simplicial categories ) and $[w(A),{\mathrm{sSet}}_{\mathrm{Quillen}}{]}_{\mathrm{proj}}$ 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

$$\cdots \simeq {([w(A),[C^{\mathrm{op}},{\mathrm{sSet}}_{\mathrm{Quillen}}{]}_{\mathrm{proj},\mathrm{loc}}{]}_{\mathrm{proj}})}^{\circ }.$$\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 $(\Pi ⊣\mathrm{Disc})$ induces an adjoint pair

$$(\Pi /\mathrm{Disc}A⊣\mathrm{Disc}/\mathrm{Disc}A):H_{\mathrm{Disc}A}\to \mathrm{\infty }{\mathrm{Grpd}}_{/A}.$$(\Pi/Disc A \dashv Disc / Disc A) : \mathbf{H}_{Disc A} \to \infty Grpd_{/A} \,.

Under the equivalence from lemma 3 the functor $\Pi /\mathrm{Disc}A$ maps to

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

Since products of $(\mathrm{\infty },1)$-functor $(\mathrm{\infty },1)$-categories are computed objectwise, and since $\Pi$ preserves finite products by the axioms of cohesion, also $\mathrm{Func}(A,\Pi )$ preserves finite products, and hence so does $\Pi /\mathrm{Disc}A$. But products in the slice over $\mathrm{Disc}A$ are (∞,1)-pullbacks over $\mathrm{Disc}A$. 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 $H$ be an (∞,1)-topos equipped with an adjoint quadruple of functors to an (∞,1)-category $B$ – the base of cohesion, where $\mathrm{Disc}$ and $\mathrm{coDisc}$ are full and faithful.

Proposition

The base $B$ 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 $H$ and then applying $\Gamma$ and the colimits are computed by computing them in $H$ and then applying $\Pi$. For $X:I\to B$ any diagram we have

$$\begin{array}{rl}\Gamma \underset{{\leftarrow }_i}{\lim }\mathrm{Disc}{X}_i& \simeq \underset{{\leftarrow }_i}{\lim }\Gamma \mathrm{Disc}{X}_i\\ & \simeq \underset{{\leftarrow }_i}{\lim }{X}_i\end{array}$$\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}

$$\begin{array}{rl}\Pi \underset{{\to }_i}{\lim }\mathrm{Disc}{X}_i& \simeq \underset{{\leftarrow }_i}{\lim }\Pi \mathrm{Disc}{X}_i\\ & \simeq \underset{{\leftarrow }_i}{\lim }{X}_i\end{array}.$$\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 $\mathrm{Disc}$, 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 $H$ is itself again discrete and is the image under $\mathrm{Disc}$ of the coresponding (co)limit computed in $B$.

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

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

Since $\mathrm{Disc}$ even preserves all (∞,1)-colimits, it is in particular an accessible (∞,1)-functor, hence so is the above composite.

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

Notice that the reflection $(\Pi ⊣\mathrm{Disc})$ does not in general constitute a geometric embedding, since $\Pi$ 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 $(\mathrm{\infty },1)$-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 $\Gamma$ preserves even all limits and for the discrete objects the reflector $\Pi$ does so by assumotion of strong connectedness.

Factorization systems associated to $\Pi$

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

Definition

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

$$\begin{array}{ccc}{c}_{\Pi }f& \to & \Pi X\\ \downarrow & & {\downarrow }^{\Pi f}\\ Y& \to & \Pi Y\end{array},$$\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 $(\Pi ⊣\mathrm{Disc})$-unit. We say that $c_{\Pi }f$ is the $\Pi$-closure of $f$, and that $f$ is $\Pi$-closed if $X\simeq c_{\Pi }f$.

Proposition

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

$$\begin{array}{ccccc}X& & \stackrel{f}{\to }& & Y\\ & \searrow & & \nearrow \\ & & c_{\Pi }f\end{array},$$\array{ X &&\stackrel{f}{\to}&& Y \\ & \searrow && \nearrow \\ && c_{\mathbf{\Pi}}f } \,,

such that $X\to c_{\Pi }f$ is a $\Pi$-equivalence in that it is inverted by $\Pi$.

Proof

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

$$\begin{array}{ccccc}X& \to & c_{\Pi }f& \to & \Pi X\\ & {}_f\searrow & \downarrow & & {\downarrow }^{\Pi f}\\ & & Y& \to & \Pi Y\end{array}.$$\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 $\Pi$ preserves the $(\mathrm{\infty },1)$-pullback over the discrete object $\Pi Y$ and since $\Pi (X\to \Pi X)$ is an equivalence, it follows that $\Pi (X\to c_{\Pi f})$ is an equivalence.

Proposition

The pair of classes

$$(\Pi -\mathrm{equivalences},\Pi -\mathrm{closed}\mathrm{morphisms})$$(\mathbf{\Pi}-equivalences, \mathbf{\Pi}-closed morphisms)

is an orthogonal factorization system in $H$.

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

$$\begin{array}{ccc}A& \to & X\\ \downarrow & & \downarrow \\ B& \to & Y\end{array}$$\array{ A &\to& X \\ \downarrow && \downarrow \\ B &\to& Y }

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

$$\begin{array}{ccccc}A& \to & X& \to & \Pi X\\ \downarrow & & \downarrow & & \downarrow \\ B& \to & Y& \to & \Pi Y.\end{array}.$$\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

$$\begin{array}{ccccc}A& \to & \Pi A& \to & \Pi Y\\ \downarrow & & {\downarrow }^{\simeq }& & \downarrow \\ B& \to & \Pi B& \to & \Pi X\end{array}.$$\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 $athbf\Pi B$ and hence also this lift is essentially unique.

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

$$\begin{array}{ccccc}A& \to & X& \to & \Pi X\\ \downarrow & {}^{\sigma }\nearrow & \downarrow & & \downarrow \\ B& \to & Y& \to & \Pi Y.\end{array}.$$\array{ A &\to& X &\to& \mathbf{\Pi}X \\ \downarrow &{}^{\mathllap{\sigma}}\nearrow& \downarrow && \downarrow \\ B &\to& Y &\to& \mathbf{\Pi}Y \,. } \,.

Observation

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

Proof

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

$$\begin{array}{ccccc}{y}^{*}X& \to & & \to & \Pi X\\ \downarrow & & & & \downarrow \\ *& \stackrel{y}{\to }& Y& \stackrel{}{\to }& \Pi Y\end{array}.$$\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 $\mathrm{Disc}$ preserves $\mathrm{\infty }$-pullbacks, this exhibits $y^{*}X$ as the image under $\mathrm{Disc}$ of an $\mathrm{\infty }$-pullback of $\mathrm{\infty }$-groupoids.

Structures in a cohesive $(\mathrm{\infty },1)$-topos

A cohesive $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },1)$-topos.

This section is at

• cohesive (∞,1)-topos -- structures

Infinitesimal cohesion

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

This section is at

• cohesive (∞,1)-topos -- infinitesimal cohesion

Locally ringed cohesion

Every cohesive $(\mathrm{\infty },1)$-topos $H$ equipped with infinitesimal cohesion comes canonically equipped with a notion of formally étale morphisms (as discussed there). Combined with the canonical interpretation of $H$ 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

• cohesive (∞,1)-topos -- structure ∞-sheaves.

Examples

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

Cohesive diagrams in a cohesive $(\mathrm{\infty },1)$-topos

Proposition

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

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

$$(\perp ⊣p⊣\top ):D\stackrel{\stackrel{\perp }{\hookleftarrow }}{\stackrel{\stackrel{p}{\to }}{\underset{\top }{\hookleftarrow }}}*$$(\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 $\perp$ and $\top$.

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

$$H^D\stackrel{\stackrel{{\top }^{*}}{\to }}{\stackrel{\stackrel{p^{*}}{\hookleftarrow }}{\stackrel{\stackrel{{\perp }^{*}}{\to }}{\underset{{\perp }_{*}}{\hookleftarrow }}}}H\stackrel{\stackrel{\Pi }{\to }}{\stackrel{\stackrel{\mathrm{Disc}}{\hookleftarrow }}{\stackrel{\stackrel{\Gamma }{\to }}{\underset{\mathrm{coDisc}}{\hookleftarrow }}}}\mathrm{\infty }\mathrm{Grpd},$$\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 $(\perp ⊣p⊣\top )$.

Proof

By the discussion at (∞,1)-Kan extension each of the original three functors induces adjoint triples $({\perp }_{!}⊣{\perp }^{*}⊣{\perp }_{*})$ etc, as indicated. In particular ${\top }^{*}$ is a right adjoint and therefore preserves finite products (and all small (∞,1)-limits, even).

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

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

In particular we have

Corollary

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

Example

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

Over $\mathrm{\infty }$-Cohesive sites of definition

Proposition

Examples of ∞-cohesive sites are

• any category with finite products and equipped with the trivial coverage.

• the full subcategory ${\mathrm{CartSp}}_{\mathrm{top}}\subset$ Top on open balls with the good open cover coverage;

• the site CartSp of Cartesian spaces with smooth functions between them and good open cover coverage.

• the Cahiers topos-site ThCartSp of infinitesimally thickened Cartesian spaces.

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

Discrete $\mathrm{\infty }$-groupoids

• discrete ∞-groupoid

Topological $\mathrm{\infty }$-groupoids

• Euclidean-topological ∞-groupoid

• locally contractible topological ∞-groupoid

Smooth $\mathrm{\infty }$-groupoids

• smooth ∞-groupoid

• synthetic differential ∞-groupoid

Super $\mathrm{\infty }$-groupoids

• super ∞-groupoid

• smooth super ∞-groupoid

Applications

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

See also

• ∞-Chern-Weil theory

• ∞-Chern-Simons theory

Related concepts

• locally connected topos / locally ∞-connected (∞,1)-topos

  • connected topos / ∞-connected (∞,1)-topos

  • strongly connected topos / strongly ∞-connected (∞,1)-topos

  • totally connected topos / totally ∞-connected (∞,1)-topos

• local topos / local (∞,1)-topos.

• cohesive topos / cohesive (∞,1)-topos

and

• locally connected site, locally ∞-connected site

• connected site

• local site

• cohesive site, (∞,1)-cohesive site

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 $\Pi$ 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 $\mathrm{coDisc}$ 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

• Carlos Simpson, Constantin Teleman, deRham theorem for $\mathrm{\infty }$-stacks (pdf)

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 $(\mathrm{Red}⊣{\Pi }_{\inf }⊣{♭}_{\inf })$ 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

• Carlos Simpson, The topological realization of a simplicial presheaf , (arXiv:q-alg/9609004).

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

• Maxim Kontsevich, Alexander Rosenberg, Noncommutative spaces, preprint MPI-2004-35 (ps, dvi)

in the context of Q-categories .

On cohesive $(\mathrm{\infty },1)$-toposes proper

The material presented here is also in section 2 of

• Urs Schreiber, differential cohomology in a cohesive topos

A commented list of further related references is at

• differential cohomology in a cohesive topos -- references

In homotopy type theory

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

• Mike Shulman, Axiomatic cohesion in HoTT (blog post)

The corresponding Coq-code is in

• Mike Shulman, HoTT/Coq/Subcategories

Exposition is at

• Mike Shulman, Internalizing the External, or The Joys of Codiscreteness (blog post)