nLab geometry of physics - cohesive toposes

Gros toposes

We have seen roughly two different kinds of sheaf toposes:

• categories of sheaves on a given space$X$ (Example ), which, by localic reflection (Prop. ), really are just a reflection of the space $X$ in the category of toposes,

  these are called petit toposes;

• categories of sheaves whose objects are generalized spaces (Example )

  these are called gros 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).

$\phantom{A}$gros topos$\phantom{A}$		$\phantom{A}$generalized spaces obey…$\phantom{A}$	$\phantom{A}$example:$\phantom{A}$
$\phantom{A}$cohesion$\phantom{A}$	Def.	$\phantom{A}$principles of differential topology$\phantom{A}$	$\phantom{A}$SmoothSet$\phantom{A}$
$\phantom{A}$elasticity	Def.	$\phantom{A}$principles of differential geometry$\phantom{A}$	$\phantom{A}$FormallSmoothset$\phantom{A}$
$\phantom{A}$solidity$\phantom{A}$	Def.	$\phantom{A}$principles of supergeometry$\phantom{A}$	$\phantom{A}$SuperFormalSmoothSet$\phantom{A}$

$\,$

Cohesive toposes

Definition

(cohesive topos)

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

(1)$$\Pi \dashv Disc \dashv \Gamma \dashv coDisc \;\;\colon\;\; \mathbf{H} \array{ \overset{\phantom{AAA} \Pi \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } Set$$

such that:

1. $Disc$ and $coDisc$ are full and faithful functors (Def. )

2. $\Pi$ preserves finite products.

Example

(adjoint quadruple of presheaves over site with finite products)

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

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

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

such that:

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

   (3)$$\begin{aligned} \Gamma \mathbf{Y} & = \underset{\underset{\mathcal{C}}{\longleftarrow}}{\lim} \mathbf{Y} \\ & \simeq \mathbf{Y}(\ast) \end{aligned}$$

2. $Disc$ and $coDisc$ are full and faithful functors (Def. ).

3. $\Pi$ preserves finite products:

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

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

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

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

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 $[\mathcal{C}^{op}, Set]$ and $[\ast, Set] \simeq Set$, as shown, where

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

2. $Disc$ is the left Kan extension along the inclusion $\ast \hookrightarrow \mathcal{C}$ 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 $Disc$ is fully faithful (Prop. ). This implies that also $coDisc$ is fully faithful, by (Prop. ).

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

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

The fact that this indeed preserves products follows from the assumption that $\mathcal{C}$ 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 $\mathcal{C}$ (Def. ) cohesive if the following conditions are satisfied:

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

2. For every covering family $\{U_i \to X\}_i$ in the given coverage on $\mathcal{C}$ the induced Cech groupoid $C(\{U_i\}_i) \in [C^{op}, Grpd]$ (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:

      $$\pi_0 \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} C(\{U_i\}) \;\simeq\; \ast$$

   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 $X$, regarded as a groupoid:

      $$\pi_0 \underset{\underset{\mathcal{C}^{op}}{\longleftarrow}}{\lim} C(\{U_i\}) \simeq Hom_{\mathcal{C}}(\ast,X) \,.$$

This definition is designed to make the following true:

Proposition

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

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

(5)$$Sh(\mathcal{C}) \array{ \overset{\phantom{AAA} \Pi \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } Set$$

Proof

By example we alreaday have the analogous statement for the categories of presheaves. Hence it is sufficient to show that the functors $Disc$ and $coDisc$ from Example factor through the definition inclusion of the category of sheaves, hence that for each set $S$ the presheaves $Disc(S)$ and $coDisc(S)$ 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 $\{U_i \to X\}$ be a covering family.

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

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

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

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

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

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

$$\left[ \pi_0 \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \left( C(\{U_i\}) \right) \to \ast \,,\, S \right]$$

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

$$\pi_0 \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \left( C(\{U_i\}) \right) \to \ast$$

is already an isomorphism (here: bijection) itself.

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

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

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

$$\left[ \pi_0 \underset{\underset{\mathcal{C}^{op}}{\longleftarrow}}{\lim} C(\{U_i\}) \overset{p_{\{U_i\}_i}}{\to} Hom_{\mathcal{C}}(\ast, X) \,,\, S \right]$$

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

$$\pi_0 \underset{\underset{\mathcal{C}^{op}}{\longleftarrow}}{\lim} C(\{U_i\}) \overset{p_{\{U_i\}_i}}{\to} Hom_{\mathcal{C}}(\ast, X)$$

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)$$\Pi \dashv Disc \dashv \Gamma \dashv coDisc \;\;\colon\;\; \mathbf{H} \array{ \overset{\phantom{AAA} \Pi \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } Set$$

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

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

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

We pronounce these as follows:

$\phantom{A}$ shape modality $\phantom{A}$	$\phantom{A}$ flat modality $\phantom{A}$	$\phantom{A}$ sharp modality $\phantom{A}$
$\phantom{A}$ $ʃ \;\coloneqq\; Disc \circ \Pi$ $\phantom{A}$	$\phantom{A}$ $\flat \;\coloneqq\; Disc \circ \Gamma$ $\phantom{A}$	$\phantom{A}$ $\sharp \;\coloneqq\; coDisc \circ \Gamma$ $\phantom{A}$

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

• a flat-comodal object

  $$\flat X \underoverset{\simeq}{\phantom{A}\epsilon^\flat_X \phantom{A}}{\longrightarrow} X$$

  is called a discrete object;

• a sharp-modal object

  $$X \underoverset{\simeq}{\phantom{A}\eta^\sharp_X\phantom{A}}{\longrightarrow} \sharp X$$

  is called a codiscrete object;

• a sharp-submodal object

  $$X \underoverset{mono}{\phantom{A}\eta^\sharp_X\phantom{A}}{\longrightarrow} \sharp X$$

  is a concrete object.

Proposition

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

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

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

   $$\flat X \overset{ \phantom{AA} \epsilon^\flat_X \phantom{AA} }{\longrightarrow} X \overset{ \phantom{AA} \epsilon^ʃ_X \phantom{AA} }{\longrightarrow} ʃ X$$

   is an epimorphism (Def. )

2. discrete objects are concrete: For every object $X \in \mathbf{H}$, we have that its discrete object $\flat X$ 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 $ptp_{\mathbf{H}}$ is a special case of that discussed in Prop. . First observe, in the notation there, that

$$ptp_{\mathbf{H}} \;\; \text{is epi} \phantom{AAA} \text{iff} \phantom{AAA} ptp_{\mathbf{B}} \;\; \text{is epi} \,.$$

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

In the other direction, assume that $ptp_{\mathbf{H}}$ is an epimorphism. By (?) and (?) we see that $ptp_{\mathbf{B}}$ is re-obtained from this by applying $\Gamma$ and then composition with isomorphisms. But $\Gamma$ 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 $\Pi \epsilon^\flat_{Disc S}$ being an epimorphism, for all $S \in \mathbf{B}$. But this is equivalent to

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

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

Now by Lemma , this is equivalent to

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

being an injection for all $\mathbf{X}$, which, by Def. , is equivalent to $\eta^\sharp_{Disc(S)}$ 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 $\mathcal{C}$ be a cohesive site (Def. ), such that

• for every object $X \in \mathcal{C}$, there is at least one morphism $\ast \overset{\exists}{\to} X$ from the terminal object to $X$, hence such that the hom set $Hom_{\mathcal{C}}(\ast, X)$ is non-empty.

Then the cohesive topos $Sh(\mathcal{C})$, 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 $S \in Set$, the canonical morphism

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

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

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

is an injective function.

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

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

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

Proposition

(quasitopos of concrete objects in a cohesive topos)

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

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

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

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

$$\Gamma \;\dashv\; coDisc \;\;\colon\;\; \mathbf{H} \array{ \overset{\phantom{AA} conc \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA} \iota_{conc} \phantom{AA}}{\hookleftarrow} } \mathbf{H}_{conc} \array{ \overset{\phantom{AA}\Gamma \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA}coDisc\phantom{AA}}{\hookleftarrow} } Set$$

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

$$\array{ \\ \phantom{a} \\ \phantom{A} \\ \Gamma \;\dashv\; coDisc } \;\;\colon\;\; \mathbf{H} \array{ \phantom{\overset{ \phantom{AA} \Pi \phantom{AA} }{\longrightarrow}} \\ \phantom{\overset{ \phantom{AA} Disc \phantom{AA} }{\hookleftarrow}} \\ \overset{\phantom{AA} conc \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA} \iota_{conc} \phantom{AA}}{\hookleftarrow} } \mathbf{H}_{conc} \array{ \overset{ \phantom{AA} \Pi \phantom{AA} }{\longrightarrow} \\ \overset{ \phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AA}\Gamma \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA}coDisc\phantom{AA}}{\hookleftarrow} } Set$$

Proof

For the adjunction on the right, we just need to observe that for every set $S \in Set$, the codiscrete object $coDisc(S)$ 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 $Disc$ factors through $\mathbf{H}_{conc}$.

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

$$conc \;\colon\; X \mapsto im(\eta^\sharp_X) \,,$$

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

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

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

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

By chasing the identity morphism on $\Gamma im(\eta^\sharp_X)$ 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 $\sharp \coloneqq coDisc \circ \Gamma$ (Def. ).

But observe that $\Gamma (\eta^{conc}_X)$, and hence also $\sharp(\eta^{conc}_X)$, is an isomorphism (Def. ), as indicated above: Since $\Gamma$ 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 $\Gamma \eta^{conc}_X$ is the epimorphism onto the image of $\Gamma( \eta^\sharp_X )$. 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

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

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

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

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

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

$\,$

It remains to show that this construction is left adjoint to the inclusion. We claim that the adjunction unit (Def. ) of $(conc \dashv \iota_{conc})$ is provided by $\eta^{conc}$ (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)$$\array{ X_1 &\underoverset{epi}{\eta^{conc}_{X_1}}{\longrightarrow}& im(\eta^\sharp_{X_1}) &\underset{mono}{\longrightarrow}& \sharp X_1 \\ \big\downarrow && \big\downarrow && \big\downarrow \\ X_2 &\underoverset{epi}{\eta^{conc}_{X_2}}{\longrightarrow}& im(\eta^\sharp_{X_2}) &\underset{mono}{\longrightarrow}& \sharp X_2 }$$

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

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

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 $X_2$. 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 $\mathbf{H}_{red}$ be a cohesive topos (Def. ). Then an elastic topos or differentially cohesive topos over $\mathbf{H}_{red}$ is a sheaf topos $\mathbf{H}$ which is

1. a cohesive topos over Set,

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

   $$\mathbf{H}_{red} \array{ \overset{\phantom{AA} \iota_{inf} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Pi_{inf} \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} Disc_{inf} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Gamma_{inf} \phantom{AA} }{\longleftarrow} } \mathbf{H}$$

Lemma

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

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

$$Set \array{ \overset{\phantom{A} \Pi_{red} \phantom{A} }{\longleftarrow} \\ \overset{\phantom{A} Disc_{red} \phantom{A} }{\hookrightarrow} \\ \overset{\phantom{A} \Gamma_{red} \phantom{A} }{\longleftarrow} \\ \overset{\phantom{A} coDisc_{red} \phantom{A} }{\hookrightarrow} } \mathbf{H}_{red} \array{ \overset{\phantom{AA} \iota_{inf} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Pi_{inf} \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} Disc_{inf} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Gamma_{inf} \phantom{AA} }{\longleftarrow} \\ \phantom{A} \\ \phantom{a} } \mathbf{H}$$

and write

$$Set \array{ \overset{\phantom{AA} \Pi \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Gamma \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookrightarrow} } \mathbf{H}$$

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

Proof

The identification

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

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

Definition

(adjoint modalities on elastic topos)

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

$$\Re \dashv \Im \dashv \& \;\;\colon\;\; \mathbf{H} \array{ \overset{ \Re \;\coloneqq\; \iota_{inf} \circ \Pi_{inf} }{\longleftarrow} \\ \overset{\Im \;\coloneqq\; Disc_{inf} \circ \Pi_{inf} }{\longrightarrow} \\ \overset{ \& \;\coloneqq\; Disc_{inf} \circ \Gamma_{inf} }{\longleftarrow} } \mathbf{H} \,.$$

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

We pronounce these as follows:

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

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

• a reduction-comodal object

  $$\Re X \underoverset{\simeq}{\epsilon^\Re_X}{\longrightarrow} X$$

  is called a reduced object;

• an infinitesimal shape-modal object

  $$X \underoverset{\simeq}{\phantom{A}\eta^\Im_X\phantom{A}}{\longrightarrow} \Im X$$

  is called a coreduced object.

Proposition

(progression of adjoint modalities on elastic topos)

Let $\mathbf{H}$ 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. :

$\phantom{A}$ shape modality $\phantom{A}$	$\phantom{A}$ flat modality $\phantom{A}$	$\phantom{A}$ sharp modality $\phantom{A}$
$\phantom{A}$ $ʃ \;\coloneqq\; Disc \circ \Pi$ $\phantom{A}$	$\phantom{A}$ $\flat \;\coloneqq\; Disc \circ \Gamma$ $\phantom{A}$	$\phantom{A}$ $\sharp \;\coloneqq\; coDisc \circ \Gamma$ $\phantom{A}$
$\phantom{A}$ reduction modality $\phantom{A}$	$\phantom{A}$ infinitesimal shape modality $\phantom{A}$	$\phantom{A}$ infinitesimal flat modality $\phantom{A}$
$\phantom{A}$ $\Re \;\coloneqq\; \iota_{inf} \circ \Pi_{inf}$ $\phantom{A}$	$\phantom{A}$ $\Im \;\coloneqq\; Disc_{inf} \circ \Pi_{inf}$ $\phantom{A}$	$\phantom{A}$ $\& \;\coloneqq\; Disc_{inf} \circ \Gamma_{inf}$ $\phantom{A}$

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

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

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

Proof

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

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

   $$\& \flat X \;\simeq\; X$$

2. $ʃ X$ is an $\Im$-modal object (Def. ), hence that

   $$\Im ʃ X \;\simeq\; X$$

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

$$\Pi_{inf} Disc_{inf} \;\simeq\; id \phantom{AAA} \text{and} \phantom{AAA} \Gamma_{inf} Disc_{inf} \;\simeq\; id \,,$$

which holds by Prop. , since $Disc_{inf}$ is a fully faithful functor and $\Pi_{inf}$, $Gamma_{inf}$ are (co-)reflectors for it, respectively:

$$\begin{aligned} \underset{Disc_{inf} \Gamma_{inf}}{\underbrace{\;\;\;\&\;\;\;}} \underset{Disc \Gamma }{\underbrace{\;\;\;\flat\;\;\;}} & = Disc_{inf} \Gamma_{inf} \underset{ Disc_{inf} Disc_{red} }{\underbrace{\;\;\;\Disc\;\;\;}} \Gamma \\ & = \underset{ \simeq Disc }{ \underbrace{ Disc_{inf} \underset{\simeq id}{\underbrace{\Gamma_{inf} Disc_{inf}}} Disc_{red} }} \; \Gamma \\ & \simeq \underset{ Disc }{\underbrace{ Disc_{inf} Disc_{red} }} \Gamma \mathbf{X} \\ & = Disc \Gamma \\ & = \flat \end{aligned}$$

and

$$\array{ \underset{ Disc_{inf} \Pi_{inf} }{ \underbrace{ \;\;\;\Im\;\;\; }} \underset{ Disc \Pi }{ \underbrace{ \;\;\;ʃ\;\;\; } } & = Disc_{inf} \Pi_{inf} \underset{ Disc_{inf} Disc_{red} }{ \underbrace{ \;\;\;Disc\;\;\; } } \Pi \\ & = \underset{ \simeq Disc }{ \underbrace{ Disc_{inf} \underset{ \simeq id }{ \underbrace{ \Pi_{inf} Disc_{inf} } } Disc_{red} } } \Pi \\ & \simeq Disc \Pi \\ & = ʃ }$$

$\,$

Solid toposes

Definition

(solid topos)

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

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

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

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

   $$\mathbf{H}_{bos} \array{ \overset{\phantom{A} even \phantom{A} }{\longleftarrow} \\ \overset{\phantom{AA} \iota_{sup} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Pi_{sup} \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} Disc_{sup} \phantom{AA} }{\hookrightarrow} } \mathbf{H}$$

   hence with $\iota_{sup}$ and $Disc_{sup}$ being fully faithful functors (Def. ).

Lemma

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

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

$$Set \array{ \overset{\phantom{A} \Pi_{red} \phantom{A} }{\longleftarrow} \\ \overset{\phantom{A} Disc_{red} \phantom{A} }{\hookrightarrow} \\ \overset{\phantom{A} \Gamma_{red} \phantom{A} }{\longleftarrow} \\ \overset{\phantom{A} coDisc_{red} \phantom{A} }{\hookrightarrow} } \mathbf{H}_{red} \array{ \overset{\phantom{AA} \iota_{inf} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Pi_{inf} \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} Disc_{inf} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Gamma_{inf} \phantom{AA} }{\longleftarrow} \\ \phantom{A} \\ \phantom{A \atop A} } \mathbf{H}_{bos} \array{ \overset{\phantom{AA} even \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} \iota_{sup} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Pi_{sup} \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} Disc_{sup} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Gamma_{sup} \phantom{AA} }{\longleftarrow} \\ \phantom{A} \\ \phantom{A \atop A} \\ \phantom{A \atop A} } \mathbf{H}$$

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

Here the composite adjoint quadruple

$$Set \array{ \overset{\Pi \simeq \Pi_{red}\Pi_{inf} \Pi_{sup} }{\longleftarrow} \\ \overset{Disc = Disc_{sup} Disc_{inf} Disc_{red}}{\hookrightarrow} \\ \overset{\Gamma = \Gamma_{sup} \Gamma_{inf} \Gamma_{red} }{\longleftarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookrightarrow} } \mathbf{H}$$

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

$$\mathbf{H}_{red} \array{ \overset{\iota_{sup} \iota_{inf}}{\hookrightarrow} \\ \overset{\Pi_{inf} \Pi_{sup} }{\longleftarrow} \\ \overset{Disc_{inf} Disc_{red}}{\hookrightarrow} \\ \overset{ \Gamma_{sup} }{\longleftarrow} } \mathbf{H}$$

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

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

$$\rightrightarrows \;\dashv\; \rightsquigarrow \;\dashv\; Rh \;\;\colon\;\; \mathbf{H} \array{ \overset{ \rightrightarrows \;\coloneqq\; \iota_{sup} \circ even }{\longleftarrow} \\ \overset{\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup} }{\longrightarrow} \\ \overset{ Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup} }{\longleftarrow} } \mathbf{H} \,.$$

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

We pronounce these as follows:

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

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

• a $\rightsquigarrow$-comodal object

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

  is called a bosonic object;

• a $Rh$-modal object

  $$X \underoverset{\simeq}{ \eta^{Rh}_X}{\longrightarrow} Rh X$$

  is called a rheonomic object;

Proposition

(progression of adjoint modalities on solid topos)

Let $\mathbf{H}$ 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. :

$\phantom{A}$ shape modality $\phantom{A}$	$\phantom{A}$ flat modality $\phantom{A}$	$\phantom{A}$ sharp modality $\phantom{A}$
$\phantom{A}$ $ʃ \;\coloneqq\; Disc \Pi$ $\phantom{A}$	$\phantom{A}$ $\flat \;\coloneqq\; Disc \circ \Gamma$ $\phantom{A}$	$\phantom{A}$ $\sharp \;\coloneqq\; coDisc \circ \Gamma$ $\phantom{A}$
$\phantom{A}$ reduction modality $\phantom{A}$	$\phantom{A}$ infinitesimal shape modality $\phantom{A}$	$\phantom{A}$ infinitesimal flat modality $\phantom{A}$
$\phantom{A}$ $\Re \;\coloneqq\; \iota_{sup} \iota_{inf} \circ \Pi_{inf}\Pi_{sup}$ $\phantom{A}$	$\phantom{A}$ $\Im \;\coloneqq\; Disc_{sup} Disc_{inf} \circ \Pi_{inf} \Pi_{sup}$ $\phantom{A}$	$\phantom{A}$ $\& \;\coloneqq\; Disc_{sup} Disc_{inf} \circ \Gamma_{inf}\Gamma_{sup}$ $\phantom{A}$
$\phantom{A}$ fermionic modality $\phantom{A}$	$\phantom{A}$ bosonic modality $\phantom{A}$	$\phantom{A}$ rheonomy modality $\phantom{A}$
$\phantom{A}$ $\rightrightarrows \;\coloneqq\; \iota_{sup} \circ even$ $\phantom{A}$	$\phantom{A}$ $\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup}$ $\phantom{A}$	$\phantom{A}$ $Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup}$ $\phantom{A}$

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

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

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

Proof

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

1. $\Im X$ is an $Rh$-modal object, hence $Rh \Im X \simeq X$,

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

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

$$\begin{aligned} Rh \Im & = Disc_{sup} \underset{ \simeq id }{ \underbrace{ \Pi_{sup} \;\; Disc_{sup} } } Disc_{inf} \Pi_{inf} \Pi_{sup} \\ & \simeq Disc_{sup} Disc_{inf} \Pi_{inf} \Pi_{sup} \\ & = \Im \end{aligned}$$

and

$$\begin{aligned} \rightsquigarrow \Re & = \iota_{sup} \underset{\simeq id}{\underbrace{ \Pi_{sup} \;\; \iota_{sup}}} \iota_{inf} \Pi_{inf}\Pi_{sub} \\ & \simeq \iota_{sup} \iota_{inf} \Pi_{inf} \Pi_{sub} \\ & \simeq \Re \end{aligned}$$

(…)