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cohesive topos

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

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

The definition of cohesive topos or category of cohesion aims to axiomatize properties of a topos that make it a gros topos of spaces inside of which geometry may take place. See also at motivation for cohesive toposes for a non-technical discussion.

The idea behind the term is that a geometric space is roughly something consisting of points or pieces that are held together by some cohesion - for instance by topology, by smooth structure, etc.

The canonical global section geometric morphism Γ:Set\Gamma : \mathcal{E} \to Set of a cohesive topos over Set may be thought of as sending a space XX to its underlying set of points Γ(X)\Gamma(X). Here Γ(X)\Gamma(X) is XX with all cohesion forgotten (for instance with the topology or the smooth structure forgotten)

Conversely, the left adjoint and right adjoint of Γ\Gamma

CodiscΓDiscSet \mathcal{E} \stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{Codisc}{\leftarrow}}} Set

send a set SS either to the discrete space Disc(S)Disc(S) with discrete cohesive structure (for instance with discrete topology) or to the codiscrete space Codisc(S)Codisc(S) with the codiscrete cohesive structure (for instance with codiscrete topology). (An instance of an adjoint cylinder/unity of opposites, a “category of being”).

Moreover, the idea is that cohesion makes points lump together to connected pieces . This is modeled by one more functor Π 0:Set\Pi_0 : \mathcal{E} \to Set left adjoint to DiscDisc. In the context of 1-topos theory this sends a cohesive space to its connected components (Π=π 0)(\Pi = \pi_0). More generally in an (n,1)-topos-theory context, Π\Pi sends a cohesive space XX to the (n1)(n-1)-truncation of its geometric fundamental ∞-groupoid Π(X)\Pi(X). See cohesive (∞,1)-topos.

In total this gives an adjoint quadruple

(Π 0DiscΓCodisc):CodiscΓDiscΠSet (\Pi_0 \dashv Disc \dashv \Gamma \dashv Codisc) : \mathcal{E} \stackrel{\stackrel{\overset{\Pi}{\longrightarrow}}{\overset{Disc}{\leftarrow}}}{\stackrel{\underset{\Gamma}{\longrightarrow}}{\underset{Codisc}{\leftarrow}}} Set \;

A cohesive topos is a topos whose terminal geometric morphism admits an extenson to such a quadruple of adjoints, satisfying some further properties.

Notice that most objects in a cohesive topos are far from being just sets with extra structure: while the functor Γ\Gamma does produce the set of points underlying an object XX in the cohesive topos, it may happen that XX is very non-trivial but that nevertheless Γ(X)\Gamma(X) has very few points (possibly none, with the axioms so far). The subcategory of objects in EE that we may think of as point sets equipped with extra structure is the quasitopos Conc Γ(E)Conc_\Gamma(E) of the concrete sheaves inside EE

SetCodiscΓConc Γ()Conc. Set \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{Codisc}{\hookrightarrow}} Conc_\Gamma(\mathcal{E}) \stackrel{\overset{Conc}{\leftarrow}}{\hookrightarrow} \mathcal{E} \,.

It is the fact that \mathcal{E} is a local topos that allows to identify Conc Γ(E)Conc_\Gamma(E).

To ensure that there is a minimum of points, one can add further axioms. From the above adjunctions one gets a canonical natural morphism

ΓXΠ 0X \Gamma X \to \Pi_0 X

from the points of XX to the set of connected pieces of XX. Demanding this to be an epimorphism is demanding that each piece has at least one point .

Moreover, one can demand that the cohesive pieces of product or power spaces are the products of the cohesive pieces of the factors. For powers of a single space, this is demanding that for all SSetS \in Set the following canonical map is an isomorphism:

Π 0(X S)(Π 0(X)) S. \Pi_0(X^S) \xrightarrow{\simeq} (\Pi_0(X))^S \,.

For more general products, it would be a similar map Π 0( iX i) iΠ 0(X i)\Pi_0(\prod_i X_i) \to \prod_i \Pi_0(X_i). See the examples at cohesive site for concrete illustrations of these ideas.

Definition

Fundamental axioms

Definition

A topos \mathcal{E} over some base topos 𝒮\mathcal{S}, i.e. equipped with a geometric morphism

(f *f *):f *f *𝒮 (f^* \dashv f_*) : \mathcal{E} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\longrightarrow}} \mathcal{S}

is cohesive if it is a

In detail this means that it has the following properties:

  1. it is a locally connected topos: there exists a further left adjoint (f !f *)(f_! \dashv f^*) satisfying a suitable condition;

  2. it is a connected topos: the functor f !f_! preserves the terminal object, or equivalently f *f^* is fully faithful;

  3. it is strongly connected : f !f_! preserves even all finite products;

  4. it is a local topos: there exists a further right adjoint (f *f !)(f_* \dashv f^!) (this is sufficient for ff to be local, since we have already assumed it to be connected);

In summary \mathcal{E} is cohesive over 𝒮\mathcal{S} if we have a quadruple of adjoint functors

(f !f *f *f !):f !f *f *f !𝒮 (f_! \dashv f^* \dashv f_* \dashv f^!) : \mathcal{E} \stackrel{\stackrel{\overset{f_!}{\longrightarrow}}{\overset{f^*}{\hookleftarrow}}}{\stackrel{\underset{f_*}{\longrightarrow}}{\underset{f^!}{\hookleftarrow}}} \mathcal{S}

such that f !f_! preserves finite products.

Remark

With f *f^* being a full and faithful functor also f !f^! is, as indicated (for instance by the discussion at adjoint triple).

Hence the definition of cohesion specifies two full subcategories, equivalent to each other, both coreflective and one also reflective.

Since a topos is a cartesian closed category it follows with the discussion here that both of these are exponential ideals. In fact the condition that the f *f^*-inclusion is an exponential ideal is equivalent to the condition that f !f_! preserves finite products.

To reflect the geometric interpretation of these axioms we will here and in related entries often name these functors as follows

(Π 0DiscΓcoDisc):coDiscΓDiscΠ 0𝒮. (\Pi_0 \dashv Disc \dashv \Gamma \dashv coDisc) : \mathcal{E} \stackrel{\stackrel{\overset{\Pi_0}{\longrightarrow}}{\overset{Disc}{\hookleftarrow}}}{\stackrel{\underset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\hookleftarrow}}} \mathcal{S} \,.
Definition

The above adjoint quadruple canonically induces an adjoint triple of endofunctors on \mathcal{E}

(ʃ):ΓDiscΠ 0𝒮coDiscΓDisc. (ʃ \dashv \flat \dashv \sharp) : \mathcal{E} \stackrel{\overset{\Pi_0}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\longrightarrow}}} \mathcal{S} \stackrel{\overset{Disc}{\longrightarrow}}{\stackrel{\overset{\Gamma}{\leftarrow}}{\underset{coDisc}{\longrightarrow}}} \mathcal{E} \,.

Being idempotent monads on \mathcal{E}, there are modalities in the type theory (internal logic) of \mathcal{E}. As such we call them: the

The transformation from points to pieces

Notice the existence of the following canonical natural transformations induced from the structure of a cohesive topos (a special case of the construction at unity of opposites).

Definition

Given a cohesive topos \mathcal{E} with (ʃʃ \dashv \flat) its (shape modality \dashv flat modality)-adjunction of def. 2, then the natural transformation

XXʃX \flat X \longrightarrow X \longrightarrow ʃ X

(given by the composition of the \flat-counit followed by the ʃʃ-unit) may be called the transformation from points to their pieces or the points-to-pieces-transformation, for short.

Remark

The (f *f *)(f^\ast \dashv f_\ast)-adjunct of the transformation from pieces to points, def. 3,

XXʃX \flat X \longrightarrow X \longrightarrow ʃ X

is (by the rule of forming right adjuncts by first applying the right adjoint functor and then precomposing with the unit and by the fact that the adjunct of a unit is the identity) the map

(f *Xf !X)(f *Xf *f *f !Xf !X). (f_\ast X \longrightarrow f_! X) \coloneqq \left( f_\ast X \longrightarrow f_\ast f^\ast f_! X \stackrel{\simeq}{\longrightarrow} f_!X \right) \,.

Observe that going backwards by applying f *f^\ast to this and postcomposing with the (f *f *)(f^\ast \dashv f_\ast)-counit is equivalent to just applying f *f^\ast, since by idempotency of \flat the counit is an isomorphism on the discrete object f *f !Xf^\ast f_! X. Therefore the points-to-pieces transformation and its adjunct are related by

(XXʃX)=f *(f *Xf !X). \left( \flat X \longrightarrow X \longrightarrow ʃ X \right) = f^\ast \left( f_\ast X \longrightarrow f_! X \right).

Observe then finally that since f *f^\ast is a full and faithful left and right adjoint, the points-to-pieces transform is an epimorphism/isomorphism/monomorphism precisely if its adjunct f *Xf !Xf_\ast X \longrightarrow f_! X is, respectively.

Below in Further axioms we discuss further axioms that one may want to impose on the points-to-pieces transform.

Below in Properties – Adjoint quadruples we discuss further properties of the points-top-pieces transformation

Example

In infinitesimal cohesion the points-to-pieces-transform, def. 3, is required to be an equivalence.

Example

In tangent cohesion the points-to-pieces transform, def. 3, is part of the canonical differential cohomology diagram.

Further axioms

In addition to the fundamental axioms of cohesion above, there are several further axioms that one may (or may not) want to impose in order to formalize the concept of cohesion.

Definition

For f:𝒮f : \mathcal{E} \to \mathcal{S} a cohesive topos, we say that pieces have points in \mathcal{E} (or that the cohesion “verifies the Nullstellensatz”) if the points-to-pieces natural transformation from def. 3

f *Xf *f *f !Xf !X f_* X \stackrel{}{\longrightarrow} f_* f^* f_! X \stackrel{\simeq}{\longrightarrow} f_! X

is an epimorphism for all XX \in \mathcal{E}.

This is equivalent to the following condition (see the proposition below):

Definition

We say that discrete objects are concrete in \mathcal{E} if the transformation

f *Sf !f *f *Sf !S f^* S \to f^! f_* f^* S \stackrel{\simeq}{\longrightarrow} f^! S

is a monomorphism for all S𝒮S \in \mathcal{S} .

Definition

We say pieces of powers are powers of pieces if for all S𝒮S \in \mathcal{S} and XEX \in E the natural morphism

f !(X f *S)(f !(X)) S f_! (X^{f^* S}) \stackrel{\simeq}{\longrightarrow} (f_!(X))^S

is an isomorphism.

Remark

This morphism is the internal hom-adjunct of

S×f !(X f *S)f !f *(S)×f !(X f *S)f !(f *(S)×X f *S)f !(X) S \times f_!(X^{f^* S}) \stackrel{\simeq}{\longrightarrow} f_!f^*(S) \times f_!(X^{f^* S}) \stackrel{\simeq}{\longrightarrow} f_!(f^*(S) \times X^{f^* S}) \longrightarrow f_!(X)

where we use that by definition f *f^* is full and faithful and then that f !f_! preserves products).

These extra axioms are proposed in (Lawvere, Axiomatic cohesion).

Definition

For f:𝒮f : \mathcal{E} \to \mathcal{S} a cohesive topos, we say that its subobject classifier is contractible if for the subobject classifier Ω\Omega \in \mathcal{E} we have

f !(Ω)*. f_!(\Omega) \simeq * \,.

This implies that for all XX \in \mathcal{E} also f !Ω X*f_! \Omega^X \simeq *.

This appears as axiom 2 in (Lawvere, Categories of spaces).

Remark

There is some overlap between the structures and conditions appearing here and those considered in the context of Q-categories. See there for more details.

Properties

We discuss properties of cohesive toposes. We start in

with some generalities on situations where a sequence of four adjoint functors is given. Then in

we comment on the interdependency of the collection of axioms on a cohesive topos. In

we discuss the induced notion of concrete objects that comes with every cohesive topos and in

the induced subcategory of objects with one point per piece.

Some of these phenomena have a natural

For a long list of further structures that are canonically present in a cohesive context see

For more structure available with a few more axioms see at

Adjoint quadruples and points-to-pieces

Let (p !p *p *p !):𝒮(p_! \dashv p^* \dashv p_*\dashv p^!) : \mathcal{E} \to \mathcal{S} be an adjoint quadruple of adjoint functors such that p *p^* and p !p^! are full and faithful functors. We record some general properties of such a setup, in particular concerning the induced points-to-pieces transforms, def. 3.

We write

ι:idp *p ! \iota : id \to p^* p_!

etc. for units and

η:p !p *id \eta : p_! p^* \to id

etc. for counits.

Proposition/Definition

We have commuting diagrams, natural in XX \in \mathcal{E}, S𝒮S \in \mathcal{S}

p *X η p *X 1 p !p *p *X p *(i X) θ X p !(η X) p *p *p !X ι p !X 1 p !X \array{ p_*X &\stackrel{\eta_{p^* X}^{-1}}{\longrightarrow}& p_! p^* p_*X \\ {}^{\mathllap{p_*(i_X)}}\downarrow &\searrow^{\mathrlap{\theta_X}}& \downarrow^{\mathrlap{p_!(\eta_X)}} \\ p_* p^* p_! X &\stackrel{\iota_{p_!X}^{-1}}{\longrightarrow}& p_! X }

and

p *S ι p *S p !p *p *S p *ϵ S 1 ϕ X p !(ι S 1) p *p *p !S ϵ p !S p !S. \array{ p^* S &\stackrel{\iota_{p^* S}}{\longrightarrow}& p^! p_* p^* S \\ {}^{\mathllap{p^* \epsilon_S^{-1}}}\downarrow &\searrow^{\mathrlap{\phi_X}}& \downarrow^{\mathrlap{p^!(\iota_S^{-1})}} \\ p^* p_* p^!S &\stackrel{{\epsilon}_{p_!S }}{\longrightarrow}& p^!S } \,.

where the diagonal morphisms

θ X:p *Xp !X \theta_X : p_* X \to p_! X

(the points-to-pieces transform, def. 3) and

ϕ S:p *Sp !S \phi_S : p^* S \to p^! S

are defined to be the equal composites of the sides of these diagrams.

This appears as (Johnstone, lemma 2.1, corollary 2.2).

Proposition

The following conditions are equivalent:

  • for all XX \in \mathcal{E} the morphism θ X:p *Xp !X\theta_X : p_*X \to p_! X is an epimorphism;

  • for all S𝒮S \in \mathcal{S},, the morphism ϕ S:p *Sp !S\phi_S : p^*S \to p^! S is a monomorphism;

  • p *p_* is faithful on morphisms of the form Ap *SA \to p^* S.

This appears as (Johnstone, lemma 2.3).

Proof

By the above definition, ϕ S\phi_S is a monomorphism precisely if ι p *S:p *Sp !p *p *S\iota_{p^* S} : p^* S \to p^! p_* p^* S is. This in turn is so (see monomorphism) precisely if the first function in

(A,p *X)(ι p *X)()(A,p !p *p *S)𝒮(p *A,p *p *S) \mathcal{E}(A,p^* X) \stackrel{(\iota_{p^* X}) \circ (-)}{\longrightarrow} \mathcal{E}(A, p^! p_* p^* S) \stackrel{\simeq}{\longrightarrow} \mathcal{S}(p_* A, p_* p^* S)

and hence the composite is a monomorphism in Set.

By definition of adjunct and using the (p *p !)(p_* \dashv p^!)-zig-zag identity, this is equal to the action of p *p_* on morphisms

(ι p *X)():(Ap *S)p *(Ap *S). (\iota_{p^* X}) \circ (-) : (A \to p^* S) \mapsto p_*(A \to p^* S) \,.

Similarly, by the above definition the morphism θ X\theta_X is an epimorphism precisely if p !(η X):p !p *p *Xp !Xp_!(\eta_X) : p_! p^* p_* X \to p_! X is so, which is the case precisely if the top morphism in

𝒮(p !X,S) ()p !(η X) 𝒮(p !p *p *X,S) (p *p *X,p *S) (X,p *S) p * 𝒮(p *X,p *p *S) \array{ \mathcal{S}(p_! X, S) &\stackrel{(-) \circ p_!(\eta_X)}{\longrightarrow} & \mathcal{S}(p_! p^* p_* X, S) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ && \mathcal{E}(p^* p_* X, p^* S) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ \mathcal{E}(X, p^* S) &\stackrel{p_*}{\longrightarrow}& \mathcal{S}(p_* X, p_* p^* S) }

and hence the bottom morphism is a monomorphism in Set, where again the commutativity of this diagram follows from the definition of adjunct and the (p !p *)(p_! \dashv p^*)-zig-zag identity.

Relations between the axioms

We record some relations between the various axioms characterizing cohesive toposes.

Proposition

The axioms pieces have points and discrete objects are concrete are equivalent.

This is just a reformulation of the above proposition.

The statement of the first items appears as (Johnstone, prop. 1.6). The last item is then a consequence by definition.

Proposition

For a sheaf topos the condition that it

  1. is locally connected;

  2. is connected;

  3. satisfies pieces have points

is equivalent to the condition that it

  1. is locally connected;

  2. is hyperconnected;

  3. is local.

This is (Johnstone, theorem 3.4).

Subcategories of discrete and codiscrete objects

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.

Quasitoposes of concrete objects

A cohesive topos comes canonically with various subcategories, sub-quasi-toposes and subtoposes of interest. We discuss some of these.

Observation

For (ΠDiscΓcoDisc):Set(\Pi \dashv \Disc \dashv \Gamma \dashv coDisc) : \mathcal{E} \to Set a cohesive topos, (ΓcoDisc)(\Gamma \dashv coDisc) exhibits SetSet as a subtopos

(ΓcoDisc):Set. (\Gamma \dashv \coDisc) : Set \stackrel{\leftarrow}{\hookrightarrow} \mathcal{E} \,.
Proof

By general properties of local toposes. See there.

Observation

The category SetSet is equivalent to the full subcategory of \mathcal{E} on those objects XX \in \mathcal{E} for which the (ΓcoDisc)(\Gamma \dashv coDisc) unit

XcoDiscΓX X \to coDisc \Gamma X

is an isomorphism.

Proof

By general properties of reflective subcategories. See there for details.

Definition

An object XX of the cohesive topos \mathcal{E} for which XcoDiscΓXX \to coDisc \Gamma X is a monomorphism we call a concrete object.

Write

Conc()Conc(\mathcal{E}) \hookrightarrow \mathcal{E}

for the full subcategory on concrete objects.

Proposition

The functor Γ:Set\Gamma : \mathcal{E} \to Set is a faithful functor on morphisms (XY)(X \to Y) \in \mathcal{E} precisely if YY is a concrete object.

In particular, the restriction Γ:Conc()Set\Gamma : Conc(\mathcal{E}) \to Set makes the category of concrete objects a concrete category.

Proof

Observe that the composite morphism

F:(X,Y)Γ𝒮(ΓX,ΓY)(X,coDiscΓY) F : \mathcal{E}(X,Y) \stackrel{\Gamma}{\longrightarrow} \mathcal{S}(\Gamma X, \Gamma Y) \stackrel{\simeq}{\longrightarrow} \mathcal{E}(X, coDisc \Gamma Y)

is given (see adjunct) by postcomposition with the (ΓcoDisc)(\Gamma \dashv coDisc)-unit ι Y:YcoDiscΓY\iota_Y : Y \to coDisc \Gamma Y

X Y coDiscΓX coDiscΓY. \array{ X &\to& Y \\ \downarrow && \downarrow \\ coDisc \Gamma X &\to& coDisc \Gamma Y } \,.

The condition that YY is a concrete object, hence that YcoDiscΓYY \to coDisc \Gamma Y is a monomorphism is therefore equivalent (see there) to the condition that FF is a monomorphism, which is equivalent to FF being a faithful functor.

Remark

This means that in the formal sense discussed at stuff, structure, property we may regard Conc()Conc(\mathcal{E}) as a category of sets equipped with cohesive structure .

Proposition

The category Conc()Conc(\mathcal{E}) is a quasitopos and a reflective subcategory of \mathcal{E}.

Proof

Let (C,J)(C,J) be a site of definition for \mathcal{E} with coverage JJ, so that =Sh J(C)PSh(C)\mathcal{E} = Sh_J(C) \hookrightarrow PSh(C). Since SetCoDiscSet \stackrel{CoDisc}{\hookrightarrow} \mathcal{E} is a subtopos, we have that SetSet is itself a category of sheaves on CC

SetSh K(C)CoDiscSh J(C)PSh(C) Set \simeq Sh_K(C) \stackrel{CoDisc}{\hookrightarrow} Sh_J(C) \hookrightarrow PSh(C)

which must correspond to another coverage KK: SetSh K(C)Set \simeq Sh_K(C). Since we have this sequence of inclusions, we have an inclusion of coverages JKJ \subset K. We observe that the concrete objects in \mathcal{E} are precisely the (J,K)(J,K)-biseparated presheaves on CC. The claim then follows by standard facts of quasitoposes of biseparated presheaves.

Proposition

Precisely if the cohesive topos \mathcal{E} satisfies the axiom discrete objects are concrete (saying that for all SSetS \in Set the canonical morphism DiscScoDiscΓDiscScoDiscSDisc S \to coDisc \Gamma Disc S \simeq coDisc S is a monomorphism) then Conc()Conc(\mathcal{E}) is a cohesive quasitopos in that we have a quadrupled of adjoint functors.

Conc()coDiscΓDiscΠSet. Conc(\mathcal{E}) \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} Set \,.
Proof

The axiom says precisely that the functor Disc:SetDisc : Set \to \mathcal{E} factors through Conc()Conc(\mathcal{E}). Also coDisc:SetcoDisc : Set \to \mathcal{E} clearly factors through Conc()Conc(\mathcal{E}). Since Conc()Conc(\mathcal{E}) is a full subcategory therefore the restriction of Γ\Gamma and Π\Pi to Conc()Conc(\mathcal{E}) yields a quadruple of adjunctions as indicated.

Since by reflectivity limits in Conc()Conc(\mathcal{E}) may be computed in \mathcal{E}, Π\Pi preserves finite products on Conc()Conc(\mathcal{E}).

Objects with one point per cohesive piece

Theorem

Let (ΠDiscΓCodisc):𝒮(\Pi \dashv Disc \dashv \Gamma \dashv Codisc) : \mathcal{E} \to \mathcal{S} be a cohesive topos with ΓXΠX\Gamma X \to \Pi X an epimorphism for all XX.

Let s *:s^* : \mathcal{L} \hookrightarrow \mathcal{E} be the full subcategory on those objects XX for which ΓXΠX\Gamma X \to \Pi X is an isomorphism. Then

  1. \mathcal{L} is a reflective subcategory and a coreflective subcategory

    (s !s *s *):s *s *s ! (s_! \dashv s^* \dashv s_*) : \mathcal{E} \stackrel{\overset{s_!}{\longrightarrow}}{\stackrel{\overset{s^*}{\leftarrow}}{\underset{s_*}{\longrightarrow}}} \mathcal{L}
  2. s *s_* preserves coproducts.

  3. the components of the reflector Xs !s *XX \to s_! s^* X are epimorphisms.

This is theorem 2 in (Lawvere).

Proof

Since Γ\Gamma is a left adjoint it preserves colimits, as does of course Π\Pi. Therefore the collection of objects for which ΓXΠX\Gamma X \to \Pi X is an isomorphism is closed under colimits and hence \mathcal{L} has all colimits and the inclusion s *:s^* : \mathcal{L} \hookrightarrow \mathcal{E} obviously preserves them.

To apply the adjoint functor theorem to deduce that therefore s *s^* has a right adjoint s *s_* it is sufficient to argue that \mathcal{L} is a locally presentable category. To see this, notice that \mathcal{L} \hookrightarrow \mathcal{E} is the inverter of ΓΠ\Gamma \to \Pi, a certain 2-limit in Cat. Since the 2-category of accessible categories and accessible functors is closed under (non-strict) 2-limits in Cat, it follows that \mathcal{L} is accessible. Since we already know that it is also cocomplete it follows that it must be locally presentable.

Since Π\Pi by assumption preserves finite products and Γ\Gamma preserves all products, it follows that LL is also closed under finite products and in particular contains the terminal object **. Since \mathcal{E}, being a topos, is an extensive category, it follows that s *s_* preserves coproducts.

(…details…)

Using that ΓXΠX\Gamma X \to \Pi X is an epi, we find that \mathcal{L} is also closed under subobjects: if YXY \hookrightarrow X is a monomorphism then if in

ΓY ΓX ΠY ΠX \array{ \Gamma Y &\to& \Gamma X \\ \downarrow && \downarrow^{\mathrlap{\simeq}} \\ \Pi Y &\to& \Pi X }

the right vertical morphism is an iso, then so is the left vertical one.

(…details…).

It then also follows that \mathcal{L} is closed under arbitrary products.

(…details…)

This implies the existence of s !s_! and the fact that Xs *s !XX \to s^* s_! X is epi.

(…details…)

Internal modal logic

Every topos \mathcal{E} comes with its internal logic. From this internal perspective, the existence of extra external functors on \mathcal{E} – such as the Π 0\Pi_0 and the coDisccoDisc on a cohesive topos – is manifested by the existence of extra internal logical operators. These may be understood as modalities equipping the internal logic with a structure of a modal logic.

For the case of local toposes, of which cohesive toposes are a special case, this internal modal interpretation of the extra external functor coDisccoDisc has been noticed in (AwodeyBirkedal, section 4.2). (Beware that in that reference the symbols ”\flat” and “#” are used precisely oppositely to their use here).

Definition

For ϕA\phi \hookrightarrow A a monomorphism in a cohesive topos, hence a proposition of type AA in the internal logic, we say that it is discretely true if the pullback #ϕ AA# \phi|_A \to A in

#ϕ A #ϕ A #A \array{ #\phi |_A &\to& # \phi \\ \downarrow && \downarrow \\ A &\to& # A }

is an isomorphism, where A#AA \to # A is the (ΓcoDisc)(\Gamma\dashv coDisc)-unit on AA.

Remark

If a proposition ϕ\phi is true over all discrete objects, then it is discretely true. More precisely, if for X=XX = \mathbf{\flat} X any discrete object we have that

(X,ϕ)(X,A) \mathcal{E}(X, \phi) \to \mathcal{E}(X, A)

is an isomorphism, then ϕ\phi is discretely true.

Because if so, then

(X,ϕ)(X,A) \mathcal{E}(\flat X , \phi) \to \mathcal{E}(\flat X , A)

is an isomorphism and hence

(ΓX,Γϕ)(ΓX,ΓA) \mathcal{E}(\Gamma X , \Gamma \phi) \to \mathcal{E}(\Gamma X , \Gamma A)

is for all XX. Therefore in this case ΓϕΓA\Gamma \phi \to \Gamma A is an isomorphism and hence so is #ϕ#A# \phi \to # A.

Example

For =Sh(CartSp)\mathcal{E} = Sh(CartSp) the sheaf topos over CartSp smooth{}_{smooth} (smooth spaces) and Ω cl n()Ω n()\Omega^n_{cl}(-) \hookrightarrow \Omega^n(-) the inclusion of all closed nn-differential forms into all nn-forms, the proposition is “the nn-form ω\omega is closed”. This is of course not true generally, but it is discretely true: over a discrete space every form is closed.

Examples

Over a discrete site

We discuss some cohesive toposes over sites CC with trivial coverage/topology, so that the category of sheaves is the category of presheaves

Sh(C)PSh(C). Sh(C) \simeq PSh(C) \,.

Families of sets – the Sierpinksi topos

We discuss an example of a cohesive topos over a cohesive site that is about the simplest non-trivial example that there is: the Sierpinski topos. Simple as it is, it does serve to already illustrate some key points. The following site is in fact also an ∞-cohesive site and hence there is a corresponding example of a cohesive (∞,1)-topos: the Sierpinski (∞,1)-topos.

Consider the site given by the interval category

C={*} C = \{\emptyset \to *\}

equipped with trivial topology. This evidently has an initial object \emptyset (which makes it cosifted) and a terminal object **.

The category of sheaves = presheaves on this is the arrow category

Sh({*})PSh({*})Arr(Set) Sh(\{\emptyset \to *\}) \simeq PSh(\{\emptyset \to *\}) \simeq Arr(Set)

since a presheaf XX on CC is given by a morphism

X:(*)(IS) X : (\emptyset \to *) \mapsto (I \leftarrow S)

in Set. We find

  • Γ:(IS)S\Gamma : (I \leftarrow S) \mapsto S

  • Π 0:(IS)I\Pi_0 : (I \leftarrow S) \mapsto I.

Trivial as this is, it does provide some insight into the interpretation of cohesiveness: by decomposing SS into its fibers, an object (IS)(I \leftarrow S) is an II-indexed family of sets: S= iS iS = \coprod_i S_i. The “cohesive pieces” are the S iS_i and there are I|I|-many of them. This is what Π 0\Pi_0 computes, which clearly preserves products.

Moreover we find for KSetK \in Set:

  • Disc:K(KIdK)Disc : K \mapsto (K \stackrel{Id}{\leftarrow} K);

  • CoDisc:K(*K)CoDisc : K \mapsto (* \stackrel{}{\leftarrow} K)

(and evidently both these functors are full and faithful).

This matches the interpretation we just found: DiscKDisc K is the collection of elements of KK with no two of them lumped together by cohesion, while CodiscKCodisc K is all elements of KK lumped together.

The canonical morphism

Γ(IS)Π 0(IS) \Gamma (I \leftarrow S) \to \Pi_0 (I \leftarrow S)

is

ΓDiscΓ(IS)Γ(IS)ΓDiscΠ 0(IS). \Gamma Disc \Gamma (I \leftarrow S) \to \Gamma (I \leftarrow S) \to \Gamma Disc \Pi_0 (I \leftarrow S) \,.

Plugging in the above this is just

SI S \to I

itself. Indeed, by the above interpretation, this sends each point to its cohesive component. It is not an epimorphism in general, because the fiber S iS_i over an element ii may be empty: the cohesive component ii may have no points.

Over a site with initial and terminal object

The above example is the simplest special case of a more general but still very simple class of examples.

First notice that for CC any small category, we have the left and right Kan extension of presehaves F:C opSetF : C^{op} \to Set along the functor C op*C^{op} \to *. By definition, this are the colimit and limit functors

(lim Constlim ):PSh(C)lim Constlim Set. (\lim_\to \dashv Const \dashv \lim_\leftarrow) : PSh(C) \stackrel{\overset{\lim_{\to}}{\longrightarrow}}{\stackrel{\overset{Const}{\leftarrow}}{\underset{\lim_\leftarrow}{\longrightarrow}}} Set \,.
Observation

If CC has a terminal object ** then

  • the colimit is given by evaluation on this object;

  • there is a further right adjoint (lim CoConst)(\lim_\leftarrow \dashv CoConst)

    given by

    CoConst:S(cSet(C(*,c),S). CoConst : S \mapsto (c \mapsto Set(C(*,c), S) \,.
Proof

The terminal object of CC is the initial object of the opposite category C opC^{op} and therefore the limit over any functor F:C opSetF : C^{op} \to Set is given by evaluation on this object

lim (C opFSet)F(*). \lim_\leftarrow (C^{op} \stackrel{F}{\to} Set) \simeq F(*) \,.

To see that we have a pair of adjoint functors (lim CoConst)(\lim_\to \dashv CoConst) we check the natural hom-set equivalence PSh C(F,CoConstS)Set(lim F,S)PSh_C(F, CoConst S ) \simeq Set(\lim_\to F , S) by computing

PSh C(F,CoConstS) cCSet(F(c),Set(C(*,c),S)) cCSet(F(c)×C(*,c),S) Set( cCF(c)×C(*,c),S) Set(F(*),S), \begin{aligned} PSh_C(F , CoConst S) & \simeq \int_{c \in C} Set(F(c), Set(C(*,c), S)) \\ & \simeq \int_{c \in C} Set(F(c) \times C(*,c), S) \\ & \simeq Set( \int^{c \in C} F(c) \times C(*,c) , \; S) \\ & \simeq Set(F(*), S) \end{aligned} \,,

Here the first step is the expression of natural transformations by end-calculus, the second uses the fact that Set is a cartesian closed category, the third uses that any hom-functor sends coends in the first argument to ends, and the last one uses the co-Yoneda lemma.

The formal dual of this statement is the following.

Corollary

If CC has an initial object \emptyset then

  • the limit is given by evaluation on this object;

  • there is a further left adjoint (Llim )(L \dashv \lim_\to),

    so that lim \lim_\to preserves all small limits and in particular all finite products.

In summary we have

Proposition

If CC has both an initial object \emptyset as well as a terminal object ** then there is a quadruple of adjoint functors

(Π 0DiscΓCoDisc):PSh(C)Set, (\Pi_0 \dashv Disc \dashv \Gamma \dashv CoDisc) : PSh(C) \to Set \,,

where

  • Γ\Gamma is given by evaluation on **;

  • Π 0\Pi_0 is given by evaluation of \emptyset and preserves products.

The above interpretation of the cohesiveness encoded by C={*}C = \{\emptyset \to *\} still applies to the general case: a general object XPSh(C)X \in PSh(C) is, by restriction to the unique morphism *\emptyset \to * in CC a set-indexed family of sets

X(*)=(Π 0(X)=X()X(*)=Γ(X)) X(\emptyset \to *) = (\Pi_0(X) = X(\emptyset) \leftarrow X(*) = \Gamma(X))

and Γ\Gamma picks out the total set of points, while Π 0\Pi_0 picks of the indexing set (“of cohesive components”). The extra information for general CC with initial and terminal object is that for every object cCc \in C these cohesive lumps of points are refined to a hierarchy of lumps and lumps-of-lumps

X(c*)=(Π 0(X)X(c)Γ(X)). X(\emptyset \to c \to *) = (\Pi_0(X) \leftarrow X(c) \leftarrow \Gamma(X)) \,.

Reflexive directed graphs

Proposition

The category RDGraphRDGraph of reflexive directed graphs is a cohesive topos.

The category DGraphDGraph of just directed graphs, not necessarily reflexive, is not a cohesive topos.

This example was considered in (Lawvere, Categories of spaces) as a simple test case for two very similar toposes, one of which is cohesive, the other not.

We spell out some details on the cohesive topos of reflexive directed graphs.

Definition

Let BEnd(Δ[1])\mathbf{B}End(\Delta[1]) be the one-object category coming from the monoid with three idempotent elements {Id,σ,τ}\{Id, \sigma, \tau\}

  • σσ=σ\sigma \circ \sigma = \sigma

  • ττ=τ\tau \circ \tau = \tau

  • τσ=τ \tau \circ \sigma = \tau

  • στ=σ \sigma \circ \tau = \sigma

A prehseaf X:C opSetX : C^{op} \to Set on this is a reflexive directed graph : the set X()X(\bullet) is the set of all edges and vertices regarded as identity edges, the projection

s:=X(σ):XX s := X(\sigma) : X \to X

sends each edge to its source and the projection

t:=X(τ):XX t := X(\tau) : X \to X

sends each edge to its target. The identities

st=t s \circ t = t

and

ts=s t \circ s = s

express the fact that source and target are identity edges.

Equivalently, this is a presheaf on the full subcategory Δ [1]Δ\Delta_{\leq [1]} \subset \Delta of the simplex category on the objects [0][0] and [1][1]

Δ [1]=([1]στ[0]). \Delta_{\leq [1]} = ([1] \stackrel{\overset{\tau}{\leftarrow}}{\stackrel{\to}{\underset{\sigma}{\leftarrow}}} [0]) \,.

In fact this is the Cauchy completion of BEnd(Δ[1])\mathbf{B}End(\Delta[1]), obtained by splitting the idempotents.

In summary this shows that

Observation

We have an equivalence of categories

RDGraphPSh(BEnd(Δ[1]))PSh(Δ [1]). RDGraph \simeq PSh(\mathbf{B} End(\Delta[1])) \simeq PSh(\Delta_{\leq [1]}) \,.

To see that this presheaf topos is cohesive, notice that the terminal geometric morphism

(DiscΓ):Psh(C)Set (Disc \dashv \Gamma) : Psh(C) \to Set

DiscSDisc S is the reflexive directed graph with set of vertices SS and no non-identity morphisms and ΓX\Gamma X is the set of vertices = identity edges.

The extra left adjoint Π 0:PSh(C)Set\Pi_0 : PSh(C) \to Set sends a graph to its set of connected components, the coequalizer of the source and target maps

X 1stX 0Π 0X. X_1 \stackrel{\overset{t}{\to}}{\underset{s}{\to}} X_0 \to \Pi_0 X \,.

Since this is a reflexive coequalizer (by the existence of the unit map X([1][0])X([1] \to [0])) it does preserve products (as discussed there). This is the property that fails for the topos DGraphDGraph of all directed graphs: a general coequalizer does not preserve products.

And CoDisc:SetPSh(C)CoDisc : Set \to PSh(C) sends a set SS to the reflexive graph with vertices SS and one edge for every ordered pair of vertices (the indiscrete or chaotic graph).

The canonical morphism ΓXΠ 0X\Gamma X \to \Pi_0 X sends each vertex to its connected component. Evidently this is epi, hence in RDGraphsRDGraphs cohesive pieces have points .

Simplicial sets

Reflexive directed graphs are equivalently skeleta of simplicial sets.

Proposition

The category sSet of simplicial sets is a cohesive topos in which cohesive pieces have points .

Let C=ΔC = \Delta be the simplex category, regarded as a site with the trivial coverage.

The corresponding sheaf topos Sh(Δ)Sh(\Delta) is the presheaf topos E=PSh(Δ)=E = PSh(\Delta) = sSet of simplicial sets.

We have for XsSetX \in sSet

  • Γ:XX 0\Gamma : X \mapsto X_0;

  • Π 0:Xπ 0(X)=X 0/X 1\Pi_0 : X \mapsto \pi_0(X) = X_0/X_1, the set of connected components.

And for SSetS \in Set:

  • DiscSDisc S the constant simplicial set on SS;

  • CodiscSCodisc S the simplicial set which in degree kk has the set of (k+1)(k+1)-tuples of elements of SS.

If X,YsSetX, Y \in sSet are Kan complexes, then Π 0(Y X)\Pi_0(Y^X) is the set of simplicial homotopy-classes of maps XYX \to Y. We can therefore write the homotopy category of Kan complexes as

Ho KanCplx(X,Y)=Π 0(Y X). Ho_{KanCplx}(X,Y) = \Pi_0(Y^X) \,.

Over a cohesive site

A class of examples is obtained from toposes over a cohesive site:

Theorem

Let CC be a cohesive site. The sheaf topos Sh(C)Sh(C) is cohesive.

See cohesive site for examples.

Diffeological spaces

Proposition

The site CartSp with the standard open cover coverage is a cohesive site and even an (∞,1)-cohesive site.

Proposition

The quasitopos Conc(Sh(CartSp))Conc(Sh(CartSp)) of concrete objects in the cohesive topos over CartSpCartSp is the category of diffeological spaces.

Proof

A sheaf XX on CartSpCartSp is a separated presheaf with respect to the further localization given by CoDiscCoDisc precisely if the canonical morphism (the unit of (ΓCoDisc)(\Gamma \dashv CoDisc))

XCoDiscΓX X \to CoDisc \Gamma X

is a monomorphism. Monomorphisms of sheaves are tested objectwise, so that this is equivalent to

Hom(U,X)X(U)Hom(U,CoDiscΓX) Hom(U,X) \simeq X(U) \to Hom(U,CoDisc \Gamma X)

being a monomorphism for all UCU \in C (where in the first step we used the Yoneda lemma). By the adjunction relation this is equivalently

X(U)Set(Γ(U),Γ(X)). X(U) \to Set(\Gamma(U), \Gamma(X)) \,.

This being a monomorphism is precisely the condition on XX being a concrete sheaf on CartSpCartSp that singles out diffeological spaces among all sheaves on CartSpCartSp.

Smooth toposes

Proposition

The site ThCartSp with the standard open cover coverage is a cohesive site and even an (∞,1)-cohesive site.

The corresponding cohesive topos is the Cahiers topos Sh(ThCartSp) \simeq Sh(ThCartSp). This is a smooth topos that models the axioms of synthetic differential geometry.

Cohesive over-toposes

Proposition

Let \mathcal{E} be a cohesive topos and XX \in \mathcal{E} an object.

A necessary conditions that the over topos /X\mathcal{E}/X is a connected topos is that

  1. XX is contractible Π 0(X)*\Pi_0(X) \simeq *;

Sufficient condition for /X\mathcal{E}/X to be a local topos is that

  1. XX is a tiny object.

Infinitesimal thickening and formal smoothness

See at differential cohesion. Examples include the Cahiers topos

Example

Consider a full subcategory inclusion

Disc 𝒫:𝒮𝒫 Disc^{\mathcal{P}} \;\colon\; \mathcal{S} \hookrightarrow \mathcal{P}

which has a left adjoint Π 0 𝒫\Pi_0^{\mathcal{P}} and a right adjoint Γ 𝒫\Gamma^{\mathcal{P}} that coincide with each other

Π 0 𝒫Γ 𝒫. \Pi_0^{\mathcal{P}} \simeq \Gamma^{\mathcal{P}} \,.

In (Lawvere 07, def. 1) this situation is said to exhibit \mathcal{E} as a quality type over 𝒮\mathcal{S}.

It follows that there is an infinite sequence of adjoints, in particular that there is coDisc 𝒫Disc 𝒫coDisc^{\mathcal{P}} \coloneqq Disc^{\mathcal{P}} right adjoint to Γ 𝒫\Gamma^{\mathcal{P}}, which by the discussion at adjoint triple is also a full and faithful functor, and that Π 0 𝒫\Pi_0^{\mathcal{P}} preserves finite products (in fact all limits).

So the above adjoints makes 𝒫\mathcal{P} be a cohesive topos over the base topos \mathcal{E} with the special property that Π 0 𝒫Γ 𝒫\Pi_0^{\mathcal{P}} \simeq \Gamma^{\mathcal{P}}. In words this says that in 𝒫\mathcal{P} every cohesive neighbourhood contains precisely one point. This is a characteristic of infinitesimally thickened points.

See at infinitesimal cohesion for more on this.

Counter-examples

Counter-Example

Let GG be a non-trivial finite group of cardinality nn. Write BG={ggG}\mathbf{B}G = \{\bullet \stackrel{g}{\to} \bullet | g \in G\} for its delooping groupoid. The presheaf topos

PSh(BG)GSet PSh(\mathbf{B}G) \simeq G Set

is the category of permutation representations of GG. It comes with a triple of adjoint functors

(Π 0Constlim ):GSetlim Constlim Set. (\Pi_0 \dashv Const \dashv \lim_\leftarrow) : G Set \stackrel{\overset{\lim_\to}{\to}}{\stackrel{\overset{Const}{\leftarrow}}{\underset{\lim_\leftarrow}{\to}}} Set \,.

The colimit over a representation (V,ρ):BGSet(V, \rho) : \mathbf{B}G \to Set is quotient set V/ρ(G)V/\rho(G). So we have

Π 0(G)* \Pi_0(G) \simeq *

but

Π 0(G×G)n¯, \Pi_0(G \times G) \simeq \bar n \,,

where GG denotes the fundamental representation of GG on itself. Therefore Π 0\Pi_0 does not preserve products in this case.

and

References

The axioms for a cohesive topos originate in

  • William Lawvere, Categories of spaces may not be generalized spaces, as exemplified by directed graphs , preprint, State University of New York at Buffalo, (1986) Reprints in Theory and Applications of Categories, No. 9, 2005, pp. 1–7 (pdf)

where however the term “cohesive topos” was not yet used.

This appears maybe first in

Under the name categories of cohesion a formal axiomatization is given in

  • William Lawvere, Axiomatic cohesion Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. (pdf)

(This does demand the conditions that “cohesive piece have points” and “pieces of powers are powers of pieces” as part of the definition of “category of cohesion”.)

This builds on a series of precursors of attempts to identify axiomatics for gros toposes.

In

the term category of Being is used for a notion resembling that of a cohesive topos (with an adjoint quadruple but not considering pieces have points or discrete objects are concrete). Behaviour of objects with respect to the extra left adjoined is interpreted in terms of properties of Becoming. The terminology here is probably inspired from

In

  • William Lawvere, Categories of space and quantity in: J. Echeverria et al (eds.), The Space of mathematics , de Gruyter, Berlin, New York (1992)

a proposal for a general axiomatization of homotopy/homology-like “extensive quantities” and cohomology-like “intensive quantities”) as covariant and contravariant functors out of a distributive category are considered.

The left and right adjoint to the global section functor as a means to identify discrete spaces and codiscrete space is mentioned

  • William Lawvere Taking categories seriously, Reprints in Theory and Applications of Categories, No. 8, 2005, pp. 1–24. (pdf)

on page 14.

The precise term cohesive topos apparently first appeared publically in the lecture

Notes for these lectures are in this pdf, made available on Bob Walters’s Como Category Theory Archive.

The notion of “cohesion” was explored earlier in

  • William Lawvere Volterra’s functionals and covariant cohesion of space Perugia Studies in Mathematics (Proceedings of the May 1997 Meeting in Perugia) (pdf)

where (on p. 9) it is suggested that “almost any” extensive category may be called a “species of cohesion”.

An analysis of the interdependency of the axioms on a cohesive topos is in

A good deal of the structure of cohesive toposes is also considered in

under the name Q-categories .

The internal logic of local toposes is discussed in

  • Steve Awodey, Lars Birkedal, Elementary axioms for local maps of toposes Journal of Pure and Applied Algebra, 177(3):215-230, (2003) (ps)

Revised on February 10, 2014 14:53:01 by Urs Schreiber (82.113.98.166)