nLab
geometry of physics - cohesive toposes

Gros toposes

Gros toposes

We have seen roughly two different kinds of sheaf toposes:

Remark

(cohesive generalized spaces as foundations of geometry)

If we aim to lay foundations for geometry, then we are interested in isolating those kinds of generalized spaces which have foundational a priori meaning, independent of an otherwise pre-configured notion of space. Hence we would like to first characterize suitable gros toposes, extract concepts of space from these, and only then, possibly, consider the petit topos-reflections of these (Prop. below).

The gros toposes of such foundational generalized spaces ought to have an internal logic that knows about modalities of geometry such as discreteness or concreteness. Via the formalization of modalities in Def. this leads to the definiton of cohesive toposes (Def. , Prop. below, due to Lawvere 91, Lawvere 07).

A\phantom{A}gros toposA\phantom{A}A\phantom{A}generalized spaces obey…A\phantom{A}A\phantom{A}example:A\phantom{A}
A\phantom{A}cohesionA\phantom{A}Def. A\phantom{A}principles of differential topologyA\phantom{A}A\phantom{A}SmoothSetA\phantom{A}
A\phantom{A}elasticityDef. A\phantom{A}principles of differential geometryA\phantom{A}A\phantom{A}FormallSmoothsetA\phantom{A}
A\phantom{A}solidityA\phantom{A}Def. A\phantom{A}principles of supergeometryA\phantom{A}A\phantom{A}SuperFormalSmoothSetA\phantom{A}

\,

Cohesive toposes

Definition

(cohesive topos)

A sheaf topos H\mathbf{H} (Def. ) is called a cohesive topos if there is a quadruple (Remark ) of adjoint functors (Def. ) to the category of sets (Example )

(1)ΠDiscΓcoDisc:HAAAΠAAA AADiscAA AAAΓAAA AAcoDiscAASet \Pi \dashv Disc \dashv \Gamma \dashv coDisc \;\;\colon\;\; \mathbf{H} \array{ \overset{\phantom{AAA} \Pi \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } Set

such that:

  1. DiscDisc and coDisccoDisc are full and faithful functors (Def. )

  2. Π\Pi preserves finite products.

Example

(adjoint quadruple of presheaves over site with finite products)

Let 𝒞\mathcal{C} be a small category (Def. ) with finite products (hence with a terminal object *𝒞\ast \in \mathcal{C} and for any two objects X,Y𝒞X,Y \in \mathcal{C} their Cartesian product X×Y𝒞X \times Y \in \mathcal{C}).

Then there is an adjoint quadruple (Remark ) of functors between the category of presheaves over 𝒞\mathcal{C} (Example ), and the category of sets (Example )

(2)[𝒞 op,Set]AAAΠAAA AADiscAA AAAΓAAA AAcoDiscAASet [\mathcal{C}^{op}, Set] \array{ \overset{\phantom{AAA} \Pi \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } Set

such that:

  1. the functor Γ\Gamma sends a presheaf Y\mathbf{Y} to its set of global sections, which here is its value on the terminal object:

    (3)ΓY =lim𝒞Y Y(*) \begin{aligned} \Gamma \mathbf{Y} & = \underset{\underset{\mathcal{C}}{\longleftarrow}}{\lim} \mathbf{Y} \\ & \simeq \mathbf{Y}(\ast) \end{aligned}
  2. DiscDisc and coDisccoDisc are full and faithful functors (Def. ).

  3. Π\Pi preserves finite products:

    for X,Y[𝒞 op,Set]\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set], we have a natural bijection

    Π(X×Y)Π(X)×Π(Y). \Pi(\mathbf{X} \times \mathbf{Y}) \;\simeq\; \Pi(\mathbf{X}) \times \Pi(\mathbf{Y}) \,.

Hence the category of presheaves over a small category with finite products, hence the category of sheaves for the trivial coverage (Example ) is a cohesive topos (Def. ).

Proof

The existence of the terminal object in 𝒞\mathcal{C} means equivalently (by Example ) that there is an adjoint pair of functors between 𝒞\mathcal{C} and the terminal category (Example ):

*p𝒞 \ast \underoverset {\underset{}{\hookrightarrow}} {\overset{p}{\longleftarrow}} {\bot} \mathcal{C}

whose right adjoint takes the unique object of the terminal category to that terminal object.

From this it follows, by Example , that Kan extension produces an adjoint quadruple (Remark ) of functors between the category of presheaves [𝒞 op,Set][\mathcal{C}^{op}, Set] and [*,Set]Set[\ast, Set] \simeq Set, as shown, where

  1. Γ\Gamma is the operation of pre-composition with the terminal object inclusion *𝒞\ast \hookrightarrow \mathcal{C}

  2. DiscDisc is the left Kan extension along the inclusion *𝒞\ast \hookrightarrow \mathcal{C} of the terminal object.

The former is manifestly the operation of evaluating on the terminal object. Moreover, since the terminal object inclusion is manifestly a fully faithful functor (Def. ), it follows that also its left Kan extension DiscDisc is fully faithful (Prop. ). This implies that also coDisccoDisc is fully faithful, by (Prop. ).

Equivalently, Discp *Disc \simeq p^\ast is the constant diagram-assigning functor. By uniqueness of adjoints (Prop. ) implies that Π\Pi is the functor that sends a presheaf, regarded as a functor Y:𝒞 opSet\mathbf{Y} \;\colon\; \mathcal{C}^{op} \to Set, to its colimit

(4)Π(Y)=lim𝒞 opY. \Pi(\mathbf{Y}) \;=\; \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{Y} \,.

The fact that this indeed preserves products follows from the assumption that 𝒞\mathcal{C} has finite products, since categories with finite products are cosifted (Prop. )

Example suggests to ask for coverages on categories with finite products which are such that the adjoint quadruple (2) on the category of presheaves (co-)restricts to the corresponding category of sheaves. The following Definition states a sufficient condition for this to be the case:

Definition

(cohesive site)

We call a site 𝒞\mathcal{C} (Def. ) cohesive if the following conditions are satisfied:

  1. The category 𝒞\mathcal{C} has finite products (as in Example ).

  2. For every covering family {U iX} i\{U_i \to X\}_i in the given coverage on 𝒞\mathcal{C} the induced Cech groupoid C({U i} i)[C op,Grpd]C(\{U_i\}_i) \in [C^{op}, Grpd] (Def. ) satisfies the following two conditions:

    1. the set of connected components of the groupoid obtained as the colimit over the Cech groupoid is the singleton:

      π 0lim𝒞 opC({U i})* \pi_0 \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} C(\{U_i\}) \;\simeq\; \ast
    2. the set of connected components of the groupoid obtained as the limit of the Cech groupoid is equivalent to the set of points of XX, regarded as a groupoid:

      π 0lim𝒞 opC({U i})Hom 𝒞(*,X). \pi_0 \underset{\underset{\mathcal{C}^{op}}{\longleftarrow}}{\lim} C(\{U_i\}) \simeq Hom_{\mathcal{C}}(\ast,X) \,.

This definition is designed to make the following true:

Proposition

(category of sheaves on a cohesive site is a cohesive topos)

Let 𝒞\mathcal{C} be a cohesive site (Def. ). Then the adjoint quadruple on the category of presheaves over 𝒞\mathcal{C}, from Example (given that a cohesive site by definition has finite products) (co-)restricts from the category of presheaves over 𝒞\mathcal{C}, to the category of sheaves (Def. ) and hence exhibits Sh(𝒞)Sh(\mathcal{C}) as a cohesive topos (Def. ):

(5)Sh(𝒞)AAAΠAAA AADiscAA AAAΓAAA AAcoDiscAASet Sh(\mathcal{C}) \array{ \overset{\phantom{AAA} \Pi \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } Set
Proof

By example we alreaday have the analogous statement for the categories of presheaves. Hence it is sufficient to show that the functors DiscDisc and coDisccoDisc from Example factor through the definition inclusion of the category of sheaves, hence that for each set SS the presheaves Disc(S)Disc(S) and coDisc(S)coDisc(S) are indeed sheaves (Def. ).

By the formulaton of the sheaf condition via the Cech groupoid (Prop. ), and using the adjunction hom-isomorphisms (here) this is readily seen to be equivalent to the two further conditions on a cohesive site (Def. ):

Let {U iX}\{U_i \to X\} be a covering family.

The sheaf condition (?) for Disc(S)Disc(S) says that

[C({U i})p {U i} iy(X),Disc(S)] \left[ C(\{U_i\}) \overset{p_{\{U_i\}_i}}{\to} y(X) \,,\, Disc(S) \right]

is an isomorphism of groupoids, which by adjunction and using (4) means equivalently that

[lim𝒞 op(C({U i}))*,S] \left[ \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \left( C(\{U_i\}) \right) \to \ast \,,\, S \right]

is an isomorphism of groupoids, where we used that colimits of representables are singletons (Lemma ) to replace lim𝒞 opy(X)*\underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} y(X) \simeq \ast.

But now in this internal hom of groupoids, the set SS is really a groupoid in the image of the reflective embedding of sets into groupoids, whose left adjoint is the connected components-functor π 0\pi_0 (Example ). Hence by another adjunction isomoprhism this is equivalent to

[π 0lim𝒞 op(C({U i}))*,S] \left[ \pi_0 \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \left( C(\{U_i\}) \right) \to \ast \,,\, S \right]

being an isomorphism (a bijection of sets, now). This is true for all SSetS \in Set precisely if (by the Yoneda lemma, if you wish) the morphism

π 0lim𝒞 op(C({U i}))* \pi_0 \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \left( C(\{U_i\}) \right) \to \ast

is already an isomorphism (here: bijection) itself.

Similarly, the sheaf condition (?) for coDisc(S)coDisc(S) says that

[C({U i})p {U i} iy(X),coDisc(S)] \left[ C(\{U_i\}) \overset{p_{\{U_i\}_i}}{\to} y(X) \,,\, coDisc(S) \right]

is an isomorphism, and hence by adjunction and using (3), this is equivalent to

[π 0lim𝒞 opC({U i})p {U i} iHom 𝒞(*,X),S] \left[ \pi_0 \underset{\underset{\mathcal{C}^{op}}{\longleftarrow}}{\lim} C(\{U_i\}) \overset{p_{\{U_i\}_i}}{\to} Hom_{\mathcal{C}}(\ast, X) \,,\, S \right]

being an isomorphism. This holds for all SSetS \in Set if (by the Yoneda lemma, if you wish)

π 0lim𝒞 opC({U i})p {U i} iHom 𝒞(*,X) \pi_0 \underset{\underset{\mathcal{C}^{op}}{\longleftarrow}}{\lim} C(\{U_i\}) \overset{p_{\{U_i\}_i}}{\to} Hom_{\mathcal{C}}(\ast, X)

is an isomorphism.

Definition

(adjoint triple of adjoint modal operators on cohesive topos)

Given a cohesive topos (Def. ), its adjoint quadruple (Remark ) of functors to and from Set

(6)ΠDiscΓcoDisc:HAAAΠAAA AADiscAA AAAΓAAA AAcoDiscAASet \Pi \dashv Disc \dashv \Gamma \dashv coDisc \;\;\colon\;\; \mathbf{H} \array{ \overset{\phantom{AAA} \Pi \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } Set

induce, by composition of functors, an adjoint triple (Remark ) of adjoint modalities (Def. ):

ʃ:HʃDiscΠ DiscΓ coDiscΓH. ʃ \dashv \flat \dashv \sharp \;\;\colon\;\; \mathbf{H} \array{ \overset{ ʃ \;\coloneqq\; Disc \circ \Pi }{\hookleftarrow} \\ \overset{\flat \;\coloneqq\; Disc \circ \Gamma }{\longrightarrow} \\ \overset{ \sharp \;\coloneqq\; coDisc\circ \Gamma }{\hookleftarrow} } \mathbf{H} \,.

Since DiscDisc and coDisccoDisc are fully faithful functors by assumption, these are (co-)modal operators (Def. ) on the cohesive topos, by (Prop. ).

We pronounce these as follows:

A\phantom{A} shape modality A\phantom{A}A\phantom{A} flat modality A\phantom{A}A\phantom{A} sharp modality A\phantom{A}
A\phantom{A} ʃDiscΠʃ \;\coloneqq\; Disc \circ \Pi A\phantom{A}A\phantom{A} DiscΓ\flat \;\coloneqq\; Disc \circ \Gamma A\phantom{A}A\phantom{A} coDiscΓ\sharp \;\coloneqq\; coDisc \circ \Gamma A\phantom{A}

and we refer to the corresponding modal objects (Def. ) as follows:

Proposition

(pieces have points \simeq discrete objects are concrete \simeq Aufhebung of bottom adjoint modality)

Let H\mathbf{H} be a cohesive topos (Def. ). Then the following conditions are equivalent:

  1. pieces have points: For every object XHX \in \mathbf{H}, comparison of extremes-transformation (?) for the (ʃ,)(ʃ, \dashv \flat)-adjoint modality (?), hence the \flat-counit of an adjunction composed with the ʃ-unit

    XAAϵ X AAXAAϵ X ʃAAʃX \flat X \overset{ \phantom{AA} \epsilon^\flat_X \phantom{AA} }{\longrightarrow} X \overset{ \phantom{AA} \epsilon^ʃ_X \phantom{AA} }{\longrightarrow} ʃ X

    is an epimorphism (Def. )

  2. discrete objects are concrete: For every object XHX \in \mathbf{H}, we have that its discrete object X\flat X is a concrete object (Def. ).

  3. Aufhebung of bottom adjoint modality

    The adjoint modality \flat \dashv \sharp exhibits Aufhebung (Def. ) of the bottom adjoint modality (Example ), i.e. the initial object (Def. ) is codiscrete (Def. ):

    . \sharp \emptyset \;\simeq\; \emptyset \,.
Proof

The comparison morphism ptp Hptp_{\mathbf{H}} is a special case of that discussed in Prop. . First observe, in the notation there, that

ptp His epiAAAiffAAAptp Bis epi. ptp_{\mathbf{H}} \;\; \text{is epi} \phantom{AAA} \text{iff} \phantom{AAA} ptp_{\mathbf{B}} \;\; \text{is epi} \,.

In one direction, assume that ptp Bptp_{\mathbf{B}} is an epimorphism. By (?) we have ptp H=Disc(ptp B)ptp_{\mathbf{H}} = Disc(ptp_{\mathbf{B}}), but DiscDisc is a left adjoint and left adjoints preserve monomorphisms (Prop. ).

In the other direction, assume that ptp Hptp_{\mathbf{H}} is an epimorphism. By (?) and (?) we see that ptp Bptp_{\mathbf{B}} is re-obtained from this by applying Γ\Gamma and then composition with isomorphisms. But Γ\Gamma is again a left adjoint, and hence preserves epimorphism by Prop. , as does composition with isomorphisms.

By applying (?) again, we find in particular that pieces have points is also equivalent to Πϵ DiscS \Pi \epsilon^\flat_{Disc S} being an epimorphism, for all SBS \in \mathbf{B}. But this is equivalent to

Hom B(Πϵ X ,S)=Hom H(ϵ X ,Disc(S)) Hom_{\mathbf{B}}(\Pi \epsilon^\flat_{\mathbf{X}}, S) = Hom_{\mathbf{\mathbf{H}}}(\epsilon^\flat_{\mathbf{X}}, Disc(S))

being a monomorphism for all SS (by adjunction isomorphism (?) and definition of epimorphism, Def. ).

Now by Lemma , this is equivalent to

Hom H(X,η Disc(S) ) Hom_{\mathbf{H}}( \mathbf{X}, \eta^\sharp_{Disc(S)} )

being an injection for all X\mathbf{X}, which, by Def. , is equivalent to η Disc(S) \eta^\sharp_{Disc(S)} being a monomorphism, hence to discrete objects are concrete.

This establishes the equivalence between the first two items.

Proposition

(cohesive site such that pieces have points/discrete objects are concrete)

Let 𝒞\mathcal{C} be a cohesive site (Def. ), such that

Then the cohesive topos Sh(𝒞)Sh(\mathcal{C}), according to Prop. , satisfies the equivalent conditions from Prop. :

  1. pieces have points,

  2. discrete objects are concrete.

Proof

By Prop. it is sufficient to show the second condition, hence to check that for each set SSetS \in Set, the canonical morphism

Disc(S)coDisc(S) Disc(S) \longrightarrow coDisc(S)

is a monomorphism. By Prop. this means equivalently that for each object X𝒞X \in \mathcal{C} in the site, the component function

Disc(S)(X)coDisc(S)(X) Disc(S)(X) \longrightarrow coDisc(S)(X)

is an injective function.

Now, by the proof of Prop. , this is the diagonal function

S Hom Set(Hom 𝒞(*,X),S) s const s \array{ S & \longrightarrow& Hom_{Set}\left( Hom_{\mathcal{C}}(\ast, X), S \right) \\ s &\mapsto& const_s }

This function is injective precisely if Hom 𝒞(*,X)Hom_{\mathcal{C}}(\ast, X) is non-empty, which is true by assumption.

Proposition

(quasitopos of concrete objects in a cohesive topos)

For H\mathbf{H} a cohesive topos (Def. ), write

H concAAAAH \mathbf{H}_{conc} \overset{ \phantom{AAAA} }{\hookrightarrow} \mathbf{H}

for its full subcategory (Example ) of concrete objects (Def. ).

Then there is a sequence of reflective subcategory-inclusions (Def. ) that factor the (ΓcoDisc)(\Gamma \dashv coDisc)-adjunction as

ΓcoDisc:HAAconcAA AAι concAAH concAAΓAA AAcoDiscAASet \Gamma \;\dashv\; coDisc \;\;\colon\;\; \mathbf{H} \array{ \overset{\phantom{AA} conc \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA} \iota_{conc} \phantom{AA}}{\hookleftarrow} } \mathbf{H}_{conc} \array{ \overset{\phantom{AA}\Gamma \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA}coDisc\phantom{AA}}{\hookleftarrow} } Set

If in addition discrete objects are concrete (Prop. ), then the full adjoint quadruple factors through the concrete objects:

a A ΓcoDisc:HAAΠAA AADiscAA AAconcAA AAι concAAH concAAΠAA AADiscAA AAΓAA AAcoDiscAASet \array{ \\ \phantom{a} \\ \phantom{A} \\ \Gamma \;\dashv\; coDisc } \;\;\colon\;\; \mathbf{H} \array{ \phantom{\overset{ \phantom{AA} \Pi \phantom{AA} }{\longrightarrow}} \\ \phantom{\overset{ \phantom{AA} Disc \phantom{AA} }{\hookleftarrow}} \\ \overset{\phantom{AA} conc \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA} \iota_{conc} \phantom{AA}}{\hookleftarrow} } \mathbf{H}_{conc} \array{ \overset{ \phantom{AA} \Pi \phantom{AA} }{\longrightarrow} \\ \overset{ \phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AA}\Gamma \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA}coDisc\phantom{AA}}{\hookleftarrow} } Set
Proof

For the adjunction on the right, we just need to observe that for every set SSetS \in Set, the codiscrete object coDisc(S)coDisc(S) is concrete, which is immediate by idempotency of \sharp (Prop. ) and the fact that every isomorphism is also a monomorphism. Similarly, the assumption that discrete objects are concrete says exactly that also DiscDisc factors through H conc\mathbf{H}_{conc}.

For the adjunction on the left we claim that the left adjoint concconc, (to be called concretification), is given by sending each object to the image (Def. ) of its (ΓcoDisc)(\Gamma \dashv coDisc) adjunction unit η \eta^\sharp:

conc:Xim(η X ), conc \;\colon\; X \mapsto im(\eta^\sharp_X) \,,

hence to the object which exhibits the epi/mono-factorization (Prop. ) of η X \eta^\sharp_X

(7)η X :Xepiη X concconcXmonoX. \eta^\sharp_X \;\colon\; X \underoverset{epi}{ \eta^{conc}_X }{\longrightarrow} conc X \underoverset{mono}{}{\longrightarrow} \sharp X \,.

First we need to show that concXconc X, thus defined, is indeed concrete, hence that η im(η X ) \eta^\sharp_{im(\eta^\sharp_X)} is a monomorphism (Def. ). For this, consider the following naturality square (?) of the ΓcoDisc\Gamma \dashv coDisc-adjunction hom-isomorphism

(8)Hom Set(Γim(η X ),Γim(η X )) Hom H(im(η X ),im(η X )) ()Γ(η X conc) ()η X conc Hom Set(ΓX,Γim(η X )) Hom H(X,im(η X ))AAAA{id Γim(η X )} (η X conc)η X {η im(η X ) } (η X conc)η X {Γ(η X conc)} {(η X conc)isoη X =η im(η X ) η X conc} \array{ Hom_{Set}( \Gamma im(\eta^\sharp_X), \Gamma im(\eta^\sharp_X) ) &\simeq& Hom_{\mathbf{H}}( im(\eta^\sharp_X), \sharp im(\eta^\sharp_X) ) \\ {}^{ \mathllap{ (-) \circ \Gamma(\eta^{conc}_X) } }\big\downarrow && \big\downarrow^{ \mathrlap{ (-) \circ \eta^{conc}_X } } \\ Hom_{Set}( \Gamma X, \Gamma im(\eta^\sharp_X) ) &\simeq& Hom_{\mathbf{H}}( X, \sharp im(\eta^\sharp_X) ) } \phantom{AAAA} \array{ \left\{ id_{\Gamma im(\eta^\sharp_X)} \right\} &\longrightarrow& \phantom{\sharp(\eta^{conc}_X) \circ \eta^\sharp_{ X }} \left\{ \eta^{\sharp}_{im(\eta^\sharp_X)} \right\} \\ \big\downarrow && \phantom{\sharp(\eta^{conc}_X) \circ \eta^\sharp_{ X }} \big\downarrow \\ \left\{ \Gamma(\eta^{conc}_X) \right\} &\longrightarrow& \left\{ \underset{ iso }{ \underbrace{ \sharp(\eta^{conc}_X) }} \circ \eta^\sharp_{ X } \;=\; \eta^{\sharp}_{ im(\eta^\sharp_X) } \circ \eta^{conc}_X \right\} }

By chasing the identity morphism on Γim(η X )\Gamma im(\eta^\sharp_X) through this diagram, as shown by the diagram on the right, we obtain the equality displayed in the bottom right entry, where we used the general formula for adjuncts (Prop. ) and the definition coDiscΓ\sharp \coloneqq coDisc \circ \Gamma (Def. ).

But observe that Γ(η X conc)\Gamma (\eta^{conc}_X), and hence also (η X conc)\sharp(\eta^{conc}_X), is an isomorphism (Def. ), as indicated above: Since Γ\Gamma is both a left adjoint as well as a right adjoint, it preserves both epimorphisms as well as monomorphisms (Prop. ), hence it preserves image factorizations (Prop. ). This implies that Γη X conc\Gamma \eta^{conc}_X is the epimorphism onto the image of Γ(η X )\Gamma( \eta^\sharp_X ). But by idempotency of \sharp, the latter is an isomorphism, and hence so is the epimorphism in its image factorization.

Therefore the equality in (8) says that

η X =(isoη im(η X ) )η X conc =monoη X conc, \begin{aligned} \eta^\sharp_{ X } & = \left( iso \circ \eta^{\sharp}_{ im(\eta^\sharp_X)} \right) \circ \eta^{conc}_X \\ & = mono \circ \eta^{conc}_X \,, \end{aligned}

where in the second line we remembered that η X conc\eta^{conc}_X is, by definition, the epimorphism in the epi/mono-factorization of η X \eta^\sharp_X.

Now the defining property of epimorphisms (Def. ) allows to cancel this commmon factor on both sides, which yields

η im(η X ) =isomono=mono. \eta^{\sharp}_{ im(\eta^\sharp_X) } \;=\; iso \circ mono \;=\; mono.

This shows that concXim(η X )conc X \coloneqq im(\eta^\sharp_X) is indeed concret.

\,

It remains to show that this construction is left adjoint to the inclusion. We claim that the adjunction unit (Def. ) of (concι conc)(conc \dashv \iota_{conc}) is provided by η conc\eta^{conc} (7).

To see this, first notice that, since the epi/mono-factorization (Prop. ) is orthogonal and hence functorial, we have commuting diagrams of the form

(9)X 1 epiη X 1 conc im(η X 1 ) mono X 1 X 2 epiη X 2 conc im(η X 2 ) mono X 2 \array{ X_1 &\underoverset{epi}{\eta^{conc}_{X_1}}{\longrightarrow}& im(\eta^\sharp_{X_1}) &\underset{mono}{\longrightarrow}& \sharp X_1 \\ \big\downarrow && \big\downarrow && \big\downarrow \\ X_2 &\underoverset{epi}{\eta^{conc}_{X_2}}{\longrightarrow}& im(\eta^\sharp_{X_2}) &\underset{mono}{\longrightarrow}& \sharp X_2 }

Now to demonstrate the adjunction it is sufficient, by Prop. , to show that η conc\eta^{conc} is a universal morphism in the sense of Def. . Hence consider any morphism f:X 1X 2f \;\colon\; X_1 \to X_2 with X 2H concHX_2 \in \mathbf{H}_{conc} \hookrightarrow \mathbf{H}. Then we need to show that there is a unique diagonal morphism as below, that makes the following top left triangle commute:

X 1 AAfAA X 2 epi η X 1 conc ! mono im(η X 1 ) X 2 \array{ X_1 &\overset{\phantom{AA} f \phantom{AA}}{\longrightarrow}& X_2 \\ {}^{\mathllap{epi}}\big\downarrow^{\mathrlap{\eta^{conc}_{X_1}}} &{}^{\mathllap{\exists !}}\nearrow& \big\downarrow^{\mathrlap{mono}} \\ im(\eta^\sharp_{X_1}) &\underset{}{\longrightarrow}& \sharp X_2 }

Now, from (9), we have a commuting square as shown. Here the left morphism is an epimorphism by construction, while the right morphism is a monomorphism by assumption on X 2X_2. With this, the epi/mono-factorization in Prop. says that there is a diagonal lift which makes both triangles commute.

It remains to see that the lift is unique with just the property of making the top left triangle commute. But this is equivalently the statement that the left morphism is an epimorphism, by Def. .

The equivalence of the first two follows with (Johnstone, lemma 2.1, corollary 2.2). The equivalence of the first and the last is due to Lawvere-Menni 15, lemma 4.1, lemma 4.2.

\,

Elastic toposes

Definition

(elastic topos)

Let H red\mathbf{H}_{red} be a cohesive topos (Def. ). Then an elastic topos or differentially cohesive topos over H red\mathbf{H}_{red} is a sheaf topos H\mathbf{H} which is

  1. a cohesive topos over Set,

  2. equipped with a quadruple of adjoint functors (Def. ) to H red\mathbf{H}_{red} of the form

    H redAAι infAA AAΠ infAA AADisc infAA AAΓ infAAH \mathbf{H}_{red} \array{ \overset{\phantom{AA} \iota_{inf} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Pi_{inf} \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} Disc_{inf} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Gamma_{inf} \phantom{AA} }{\longleftarrow} } \mathbf{H}
Lemma

(progression of (co-)reflective subcategories of elastic topos)

Let H\mathbf{H} be an elastic topos (Def. ) over a cohesive topos H red\mathbf{H}_{red} (Def. ):

SetAΠ redA ADisc redA AΓ redA AcoDisc redAH redAAι infAA AAΠ infAA AADisc infAA AAΓ infAA A aH Set \array{ \overset{\phantom{A} \Pi_{red} \phantom{A} }{\longleftarrow} \\ \overset{\phantom{A} Disc_{red} \phantom{A} }{\hookrightarrow} \\ \overset{\phantom{A} \Gamma_{red} \phantom{A} }{\longleftarrow} \\ \overset{\phantom{A} coDisc_{red} \phantom{A} }{\hookrightarrow} } \mathbf{H}_{red} \array{ \overset{\phantom{AA} \iota_{inf} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Pi_{inf} \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} Disc_{inf} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Gamma_{inf} \phantom{AA} }{\longleftarrow} \\ \phantom{A} \\ \phantom{a} } \mathbf{H}

and write

SetAAΠAA AADiscAA AAΓAA AAcoDiscAAH Set \array{ \overset{\phantom{AA} \Pi \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Gamma \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookrightarrow} } \mathbf{H}

for the adjoint quadruple exhibiting the cohesion of H\mathbf{H} itself. Then these adjoint functors arrange and decompose as in the following diagram

Proof

The identification

(DiscΓ)(Disc infDisc redΓ redΓ inf) (Disc \dashv \Gamma) \;\simeq\; ( Disc_{inf} \circ Disc_{red} \,\dashv\, \Gamma_{red} \circ \Gamma_{inf} )

follows from the essential uniqueness of the global section-geometric morphism (Example ). This implies the identifications ΠΠ redΠ inf\Pi \simeq \Pi_{red} \circ \Pi_{inf} by essential uniqueness of adjoints (Prop. ).

Definition

(adjoint modalities on elastic topos)

Given an elastic topos (differentially cohesive topos) H\mathbf{H} over H red\mathbf{H}_{red} (Def. ), composition of the functors in Lemma yields, via Prop. , the following adjoint modalities (Def. )

&:Hι infΠ inf Disc infΠ inf &Disc infΓ infH. \Re \dashv \Im \dashv \& \;\;\colon\;\; \mathbf{H} \array{ \overset{ \Re \;\coloneqq\; \iota_{inf} \circ \Pi_{inf} }{\longleftarrow} \\ \overset{\Im \;\coloneqq\; Disc_{inf} \circ \Pi_{inf} }{\longrightarrow} \\ \overset{ \& \;\coloneqq\; Disc_{inf} \circ \Gamma_{inf} }{\longleftarrow} } \mathbf{H} \,.

Since ι inf\iota_{inf} and Disc infDisc_{inf} are fully faithful functors by assumption, these are (co-)modal operators (Def. ) on the cohesive topos, by (Prop. ).

We pronounce these as follows:

A\phantom{A} reduction modality A\phantom{A}A\phantom{A} infinitesimal shape modality A\phantom{A}A\phantom{A} infinitesimal flat modality A\phantom{A}
A\phantom{A} ι infΠ inf\Re \;\coloneqq\; \iota_{inf} \circ \Pi_{inf} A\phantom{A}A\phantom{A} Disc infΠ inf\Im \;\coloneqq\; Disc_{inf} \circ \Pi_{inf} A\phantom{A}A\phantom{A} &Disc infΓ inf \& \;\coloneqq\; Disc_{inf} \circ \Gamma_{inf} A\phantom{A}

and we refer to the corresponding modal objects (Def. ) as follows:

Proposition

(progression of adjoint modalities on elastic topos)

Let H\mathbf{H} be an elastic topos (Def. ) and consider the corresponding adjoint modalities which it inherits

  1. for being a cohesive topos, from Def. ,

  2. for being an elastic topos, from Def. :

A\phantom{A} shape modality A\phantom{A}A\phantom{A} flat modality A\phantom{A}A\phantom{A} sharp modality A\phantom{A}
A\phantom{A} ʃDiscΠʃ \;\coloneqq\; Disc \circ \Pi A\phantom{A}A\phantom{A} DiscΓ\flat \;\coloneqq\; Disc \circ \Gamma A\phantom{A}A\phantom{A} coDiscΓ\sharp \;\coloneqq\; coDisc \circ \Gamma A\phantom{A}
A\phantom{A} reduction modality A\phantom{A}A\phantom{A} infinitesimal shape modality A\phantom{A}A\phantom{A} infinitesimal flat modality A\phantom{A}
A\phantom{A} ι infΠ inf\Re \;\coloneqq\; \iota_{inf} \circ \Pi_{inf} A\phantom{A}A\phantom{A} Disc infΠ inf\Im \;\coloneqq\; Disc_{inf} \circ \Pi_{inf} A\phantom{A}A\phantom{A} &Disc infΓ inf \& \;\coloneqq\; Disc_{inf} \circ \Gamma_{inf} A\phantom{A}

Then these arrange into the following progression, via the preorder on modalities from Def.

& ʃ * \array{ \Re &\dashv& \Im &\dashv& \& \\ && \vee && \vee \\ && ʃ &\dashv& \flat &\dashv& \sharp \\ && && \vee && \vee \\ && && \emptyset &\dashv& \ast }

where we display also the bottom adjoint modality *\emptyset \dashv \ast (Example ), for completeness.

Proof

We need to show, for all XHX \in \mathbf{H}, that

  1. X\flat X is an &\&-modal object (Def. ), hence that

    &XX \& \flat X \;\simeq\; X
  2. ʃXʃ X is an \Im-modal object (Def. ), hence that

    ʃXX \Im ʃ X \;\simeq\; X

After unwinding the definitions of the modal operators Def. and Def. , and using their re-identification from Lemma , this comes down to the fact that

Π infDisc infidAAAandAAAΓ infDisc infid, \Pi_{inf} Disc_{inf} \;\simeq\; id \phantom{AAA} \text{and} \phantom{AAA} \Gamma_{inf} Disc_{inf} \;\simeq\; id \,,

which holds by Prop. , since Disc infDisc_{inf} is a fully faithful functor and Π inf\Pi_{inf}, Gamma infGamma_{inf} are (co-)reflectors for it, respectively:

&Disc infΓ infDiscΓ =Disc infΓ infDiscDisc infDisc redΓ =Disc infΓ infDisc infidDisc redDiscΓ Disc infDisc redDiscΓX =DiscΓ = \begin{aligned} \underset{Disc_{inf} \Gamma_{inf}}{\underbrace{\;\;\;\&\;\;\;}} \underset{Disc \Gamma }{\underbrace{\;\;\;\flat\;\;\;}} & = Disc_{inf} \Gamma_{inf} \underset{ Disc_{inf} Disc_{red} }{\underbrace{\;\;\;\Disc\;\;\;}} \Gamma \\ & = \underset{ \simeq Disc }{ \underbrace{ Disc_{inf} \underset{\simeq id}{\underbrace{\Gamma_{inf} Disc_{inf}}} Disc_{red} }} \; \Gamma \\ & \simeq \underset{ Disc }{\underbrace{ Disc_{inf} Disc_{red} }} \Gamma \mathbf{X} \\ & = Disc \Gamma \\ & = \flat \end{aligned}

and

Disc infΠ infʃDiscΠ =Disc infΠ infDiscDisc infDisc redΠ =Disc infΠ infDisc infidDisc redDiscΠ DiscΠ =ʃ \array{ \underset{ Disc_{inf} \Pi_{inf} }{ \underbrace{ \;\;\;\Im\;\;\; }} \underset{ Disc \Pi }{ \underbrace{ \;\;\;ʃ\;\;\; } } & = Disc_{inf} \Pi_{inf} \underset{ Disc_{inf} Disc_{red} }{ \underbrace{ \;\;\;Disc\;\;\; } } \Pi \\ & = \underset{ \simeq Disc }{ \underbrace{ Disc_{inf} \underset{ \simeq id }{ \underbrace{ \Pi_{inf} Disc_{inf} } } Disc_{red} } } \Pi \\ & \simeq Disc \Pi \\ & = ʃ }

\,

Solid toposes

Definition

(solid topos)

Let H bos\mathbf{H}_{bos} be an elastic topos (Def. ) over a cohesive topos H red\mathbf{H}_{red} (Def. ). Then a solid topos or super-differentially cohesive topos over H bos\mathbf{H}_{bos} is a sheaf topos H\mathbf{H}, which is

  1. a cohesive topos over Set (Def. ),

  2. an elastic topos over H red\mathbf{H}_{red},

  3. equipped with a quadruple of adjoint functors (Def. ) to H bos\mathbf{H}_{bos} of the form

    H bosAevenA AAι supAA AAΠ supAA AADisc supAAH \mathbf{H}_{bos} \array{ \overset{\phantom{A} even \phantom{A} }{\longleftarrow} \\ \overset{\phantom{AA} \iota_{sup} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Pi_{sup} \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} Disc_{sup} \phantom{AA} }{\hookrightarrow} } \mathbf{H}

    hence with ι sup\iota_{sup} and Disc supDisc_{sup} being fully faithful functors (Def. ).

Lemma

(progression of (co-)reflective subcategories of solid topos)

Let H\mathbf{H} be a solid topos (Def. ) over an elastic topos H red\mathbf{H}_{red} (Def. ):

SetAΠ redA ADisc redA AΓ redA AcoDisc redAH redAAι infAA AAΠ infAA AADisc infAA AAΓ infAA A AAH bosAAevenAA AAι supAA AAΠ supAA AADisc supAA AAΓ supAA A AA AAH Set \array{ \overset{\phantom{A} \Pi_{red} \phantom{A} }{\longleftarrow} \\ \overset{\phantom{A} Disc_{red} \phantom{A} }{\hookrightarrow} \\ \overset{\phantom{A} \Gamma_{red} \phantom{A} }{\longleftarrow} \\ \overset{\phantom{A} coDisc_{red} \phantom{A} }{\hookrightarrow} } \mathbf{H}_{red} \array{ \overset{\phantom{AA} \iota_{inf} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Pi_{inf} \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} Disc_{inf} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Gamma_{inf} \phantom{AA} }{\longleftarrow} \\ \phantom{A} \\ \phantom{A \atop A} } \mathbf{H}_{bos} \array{ \overset{\phantom{AA} even \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} \iota_{sup} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Pi_{sup} \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} Disc_{sup} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Gamma_{sup} \phantom{AA} }{\longleftarrow} \\ \phantom{A} \\ \phantom{A \atop A} \\ \phantom{A \atop A} } \mathbf{H}

Then these adjoint functors arrange and decompose as shown in the following diagram:

Here the composite adjoint quadruple

SetΠΠ redΠ infΠ sup Disc=Disc supDisc infDisc red Γ=Γ supΓ infΓ red AAcoDiscAAH Set \array{ \overset{\Pi \simeq \Pi_{red}\Pi_{inf} \Pi_{sup} }{\longleftarrow} \\ \overset{Disc = Disc_{sup} Disc_{inf} Disc_{red}}{\hookrightarrow} \\ \overset{\Gamma = \Gamma_{sup} \Gamma_{inf} \Gamma_{red} }{\longleftarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookrightarrow} } \mathbf{H}

exhibits the cohesion of H\mathbf{H} over Set, and the composite adjoint quadruple

H redι supι inf Π infΠ sup Disc infDisc red Γ supH \mathbf{H}_{red} \array{ \overset{\iota_{sup} \iota_{inf}}{\hookrightarrow} \\ \overset{\Pi_{inf} \Pi_{sup} }{\longleftarrow} \\ \overset{Disc_{inf} Disc_{red}}{\hookrightarrow} \\ \overset{ \Gamma_{sup} }{\longleftarrow} } \mathbf{H}

exhibits the elasticity of H\mathbf{H} over H red\mathbf{H}_{red}.

Proof

As in the proof of Prop. , this is immediate by the essential uniqueness of adjoints (Prop. ) and of the global section-geometric morphism (Example ).

Definition

(adjoint modalities on solid topos)

Given a solid topos H\mathbf{H} over H bos\mathbf{H}_{bos} (Def. ), composition of the functors in Lemma yields, via Prop. , the following adjoint modalities (Def. )

Rh:Hι supeven ι supΠ sup RhDisc supΠ supH. \rightrightarrows \;\dashv\; \rightsquigarrow \;\dashv\; Rh \;\;\colon\;\; \mathbf{H} \array{ \overset{ \rightrightarrows \;\coloneqq\; \iota_{sup} \circ even }{\longleftarrow} \\ \overset{\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup} }{\longrightarrow} \\ \overset{ Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup} }{\longleftarrow} } \mathbf{H} \,.

Since ι sup\iota_{sup} and Disc supDisc_{sup} are fully faithful functors by assumption, these are (co-)modal operators (Def. ) on the cohesive topos, by (Prop. ).

We pronounce these as follows:

A\phantom{A} fermionic modality A\phantom{A}A\phantom{A} bosonic modality A\phantom{A}A\phantom{A} rheonomy modality A\phantom{A}
A\phantom{A} ι supeven\rightrightarrows \;\coloneqq\; \iota_{sup} \circ even A\phantom{A}A\phantom{A} ι supΠ sup\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup} A\phantom{A}A\phantom{A} RhDisc supΠ sup Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup} A\phantom{A}

and we refer to the corresponding modal objects (Def. ) as follows:

  • a \rightsquigarrow-comodal object

    Xϵ X X \overset{\rightsquigarrow}{X} \underoverset{\simeq}{\epsilon^\rightsquigarrow_X}{\longrightarrow} X

    is called a bosonic object;

  • a RhRh-modal object

    Xη X RhRhX X \underoverset{\simeq}{ \eta^{Rh}_X}{\longrightarrow} Rh X

    is called a rheonomic object;

Proposition

(progression of adjoint modalities on solid topos)

Let H\mathbf{H} be a solid topos (Def. ) and consider the adjoint modalities which it inherits

  1. for being a cohesive topos, from Def. ,

  2. for being an elastic topos, from Def. ,

  3. for being a solid topos, from Def. :

A\phantom{A} shape modality A\phantom{A}A\phantom{A} flat modality A\phantom{A}A\phantom{A} sharp modality A\phantom{A}
A\phantom{A} ʃDiscΠʃ \;\coloneqq\; Disc \Pi A\phantom{A}A\phantom{A} DiscΓ\flat \;\coloneqq\; Disc \circ \Gamma A\phantom{A}A\phantom{A} coDiscΓ\sharp \;\coloneqq\; coDisc \circ \Gamma A\phantom{A}
A\phantom{A} reduction modality A\phantom{A}A\phantom{A} infinitesimal shape modality A\phantom{A}A\phantom{A} infinitesimal flat modality A\phantom{A}
A\phantom{A} ι supι infΠ infΠ sup\Re \;\coloneqq\; \iota_{sup} \iota_{inf} \circ \Pi_{inf}\Pi_{sup} A\phantom{A}A\phantom{A} Disc supDisc infΠ infΠ sup\Im \;\coloneqq\; Disc_{sup} Disc_{inf} \circ \Pi_{inf} \Pi_{sup} A\phantom{A}A\phantom{A} &Disc supDisc infΓ infΓ sup \& \;\coloneqq\; Disc_{sup} Disc_{inf} \circ \Gamma_{inf}\Gamma_{sup} A\phantom{A}
A\phantom{A} fermionic modality A\phantom{A}A\phantom{A} bosonic modality A\phantom{A}A\phantom{A} rheonomy modality A\phantom{A}
A\phantom{A} ι supeven\rightrightarrows \;\coloneqq\; \iota_{sup} \circ even A\phantom{A}A\phantom{A} ι supΠ sup\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup} A\phantom{A}A\phantom{A} RhDisc supΠ sup Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup} A\phantom{A}

Then these arrange into the following progression, via the preorder on modalities from Def. :

id id Rh & ʃ * \array{ id &\dashv& id \\ \vee && \vee \\ \rightrightarrows &\dashv& \rightsquigarrow &\dashv& Rh \\ && \vee && \vee \\ && \Re &\dashv& \Im &\dashv& \& \\ && && \vee && \vee \\ && && ʃ &\dashv& \flat &\dashv& \sharp \\ && && && \vee && \vee \\ && && && \emptyset &\dashv& \ast }

where we are displaying, for completeness, also the adjoint modalities at the bottom *\emptyset \dashv \ast and the top ididid \dashv id (Example ).

Proof

By Prop. , it just remains to show that for all objects XHX \in \mathbf{H}

  1. X\Im X is an RhRh-modal object, hence RhXXRh \Im X \simeq X,

  2. X\Re X is a bosonic object, hence XX\overset{\rightsquigarrow}{\Re X} \simeq \Re X.

The proof is directly analogous to that of Prop. , now using the decompositions from Lemma :

Rh =Disc supΠ supDisc supidDisc infΠ infΠ sup Disc supDisc infΠ infΠ sup = \begin{aligned} Rh \Im & = Disc_{sup} \underset{ \simeq id }{ \underbrace{ \Pi_{sup} \;\; Disc_{sup} } } Disc_{inf} \Pi_{inf} \Pi_{sup} \\ & \simeq Disc_{sup} Disc_{inf} \Pi_{inf} \Pi_{sup} \\ & = \Im \end{aligned}

and

=ι supΠ supι supidι infΠ infΠ sub ι supι infΠ infΠ sub \begin{aligned} \rightsquigarrow \Re & = \iota_{sup} \underset{\simeq id}{\underbrace{ \Pi_{sup} \;\; \iota_{sup}}} \iota_{inf} \Pi_{inf}\Pi_{sub} \\ & \simeq \iota_{sup} \iota_{inf} \Pi_{inf} \Pi_{sub} \\ & \simeq \Re \end{aligned}

(…)

Last revised on September 20, 2018 at 03:47:54. See the history of this page for a list of all contributions to it.