geometry of physics - basic notions of topos theory

## Basic notions of Topos theory
  {#BasicNotionsOfToposTheory}

We have explained in Remark \ref{PresaheavesAsGeneralizedSpaces} how [[presheaves]] on a [[category]] $\mathcal{C}$ may be thought of as _[[generalized spaces]] probe-able by the objects of $\mathcal{C}$_, and that two consistency conditions on this interpretation are provided by the [[Yoneda lemma]] (Prop. \ref{YonedaLemma}) and the resulting [[Yoneda embedding]] (Prop. \ref{YonedaEmbedding}). Here we turn to a third consistency condition that one will want to impose, namely a _locality_ or _gluing condition_ (Remark \ref{SheafConditionAsLocality} below), to be called the _[[sheaf]]_ condition (Def. \ref{SheafConditionAsLocality} below).

More in detail, we had seen that any [[category of presheaves]] $[\mathcal{C}^{op}, Set]$ is the [[free cocompletion]] of the given [[small category]] $\mathcal{C}$ (Prop. \ref{FreeCocompletion}) and hence exhibits [[generalized spaces]] $\mathbf{X} \in [\mathcal{C}^{op}, Set]$ as being glued or _[[generators and relations|generated]]_ form the "ordinary spaces" $X \in \mathcal{C}$. Further conditions to be imposed now will impose _[[generators and relations|relations]]_ among these generators, such as the locality relation embodied by the [[sheaf]]-condition.

It turns out that these relations are reflected by special properties of an [[adjunction]] (Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}) that relates [[generalized spaces]] to ordinary [[spaces]]:

**[[generalized spaces]] via [[generators and relations]]:**

| $\phantom{A}$[[free cocompletion]]$\phantom{A}$ <br/> $\phantom{A}=$[[category of presheaves|presheaves]]$\phantom{A}$  | $\phantom{A}$[[locally presentable category|loc. presentable category]]$\phantom{A}$ | $\phantom{A}$[[sheaf topos]]$\phantom{AAAA}$ |
|-----|--|------|
| $\phantom{A}\mathbf{H} \underoverset{\underset{\phantom{AAA}}{\longrightarrow}}{\overset{}{\longleftarrow}}{\simeq} [\mathcal{C}^{op},Set]$ |  $\phantom{A}\mathbf{H} \underoverset{\underset{\text{accessible}}{\hookrightarrow}}{\overset{}{\longleftarrow}}{\bot} [\mathcal{C}^{op}, Set]$ | $\phantom{A}\mathbf{H} \underoverset{\underset{\text{accessible}}{\hookrightarrow}}{\overset{\text{left exact}}{\longleftarrow}}{\bot} [\mathcal{C}^{op}, Set]$ |
| $\phantom{A}$Prop. \ref{FreeCocompletion}$\phantom{A}$ | $\phantom{A}$Def. \ref{LocallyPresentableCategory}$\phantom{A}$ | $\phantom{A}$Prop. \ref{SheafToposViaLexReflection}$\phantom{A}$ |
|  |   |    |
| $\phantom{A}$**[[simplicial presheaves]]$\phantom{A}$** |  **$\phantom{A}$[[combinatorial model category]]$\phantom{A}$** |   **$\phantom{A}$[[model topos]]$\phantom{A}$** |
| $\phantom{A}\mathbf{H} \underoverset{\underset{\phantom{AAA}}{\longrightarrow}}{\overset{}{\longleftarrow}}{\simeq_{Qu}} [\mathcal{C}^{op},sSet_{Qu}]_{proj}$ |  $\phantom{A}\mathbf{H} \underoverset{\underset{\text{accessible}}{\hookrightarrow}}{\overset{}{\longleftarrow}}{\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj}$ | $\phantom{A}\mathbf{H} \underoverset{\underset{\text{accessible}}{\hookrightarrow}}{\overset{\text{left exact}}{\longleftarrow}}{\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj}$ |
| $\phantom{A}$Example \ref{CategoriesOfSimplicialPresheaves} | $\phantom{A}$Def. \ref{BousfieldLocalizationOfModelCategories} |  $\phantom{A}$Def. \ref{ModelTopos} |

$\,$

+-- {: .num_remark #SheafConditionAsLocality}
###### Remark
**([[sheaf|sheaf condition]] as [[local-to-global principle]] for [[generalized spaces]])**

If the [[objects]] of $\mathcal{C}$ are thought of as [[spaces]] of sorts, as in Remark \ref{PresaheavesAsGeneralizedSpaces}, then there is typically a notion of _locality_ in these spaces, reflected by a notion of what it means to _[[cover]]_ a given space by ("smaller") spaces (a _[[coverage]]_, Def. \ref{Coverage} below).

But if a space $X \in \mathcal{C}$ is covered, say by two other spaces $U_1, U_2 \in \mathcal{C}$, via morphisms

$$
  \array{
    U_1 && && U_2
    \\
    & {}_{\mathllap{i_1}}\searrow && \swarrow_{\mathrlap{i_2}}
    \\
    && X
  }
$$

then this must be reflected in the behaviour of the probes of any generalized space $\mathbf{Y}$ (in the sense of Remark \ref{PresaheavesAsGeneralizedSpaces}) by these test spaces:

For ease of discussion, suppose that there is a sense in which these two patches above [[intersection|intersect]] in $X$ to form a space $U_1 \cap_X U_2 \in \mathcal{C}$. Then locality of probes should imply that the ways of mapping $U_1$ and $U_2$ into $\mathbf{Y}$ such that these maps agree on the intersection $U_1 \cap_X U_2$, should be equivalent to the ways of mapping all of $X$ into $\mathbf{Y}$.

$$
  \text{locality}
  \;:\;
  \left\{
    \array{
       \text{maps from}\,U_1\,\text{and}\,U_2\,\text{to}\,\mathbf{Y}
       \\
       \text{that coincide on}\,U_1 \cap_X U_2
    }
  \right\}
  \;\simeq\;
  \left\{
    \text{maps from}\,X\,\text{into}\,\mathbf{Y}
  \right\}
$$

One could call this the condition of _locality of probes of generalized spaces probeable by objects of $\mathcal{C}$_. But the established terminology is that this is the _[[sheaf|sheaf condition]] (eq:SheafCondition) on [[presheaves]] over $\mathcal{C}$_. Those presheaves which satisfy this condition are called the _[[sheaves]]_ (Def. \ref{Sheaf} below).

=--

+-- {: .num_remark}
###### Remark
**Warning**

Most (if not all) introductions to [[sheaf theory]] insist on motivating the concept from the special case of [[sheaves on topological spaces]] (Example \ref{SheafOnATopologicalSpace} below). This is good motivation for what Grothendieck called "[[petit topos]]"-theory. The motivation above, instead, naturally leads to the "[[gros topos]]"-perspective, as in Example \ref{SmoothSetAnnounced} below, which is more useful for discussing the [[synthetic differential geometry|synthetic]] [[higher differential geometry|higher]] [[geometry of physics -- supergeometry|supergeometry]] of [[physics]]. In fact, this is the perspective of _[[functorial geometry]]_ that has been highlighted since [Grothendieck 65](functorial+geometry#Grothendieck65), but which has maybe remained underappreciated.

=--

$\,$

We now first introduce the [[sheaf]]-condition (Def. \ref{Sheaf}) below in its traditional form via "[[matching families]]" (Def. \ref{CompatibleElements} below). Then we show (Prop. \ref{CechGroupoidCoRepresents} below) how this is equivalently expressed in terms of _[[Cech groupoids]]_ (Example \ref{CechGroupoid} below). This second formulation is convenient for understanding and handling various constructions in ordinary [[topos theory]] (for instance the definition of [[cohesive sites]]) and it makes immediate the generalization to [[higher topos theory]].

$\,$

### Descent

Here we introduce the [[sheaf]]-condition (Def. \ref{Sheaf} below) in its component-description via [[matching families]] (Def. \ref{CompatibleElements} below). Then we consider some of the general key properties of the resulting [[categories of sheaves]], such as notably their "convenience", in the technical sense of Prop. \ref{PropertiesOfSheafToposes} below.

$\,$

+-- {: .num_defn #Coverage}
###### Definition
**([[coverage]] and [[site]])**

Let $\mathcal{C}$ be a [[small category]] (Def. \ref{SmallCategory}). Then a _[[coverage]]_ on $\mathcal{C}$ is

* for each [[object]] $X \in \mathcal{C}$ a [[set]] of [[indexed sets]] of [[morphisms]] into $X$

  $$
    \left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I}
  $$

  called the _[[coverings]]_ of $X$,

such that

* for every [[covering]] $\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I}$ of $X$ and every [[morphism]] $Y \overset{f}{\longrightarrow} X$ there exists a _refining covering_ $\left\{ V_j \overset{\iota_j}{\to} Y \right\}_{j \in J}$ of $Y$, meaning that for each $j \in J$ there exists $i \in I$ and a morphism $V_j \overset{\iota_{j,i}}{\to} U_i$ such that

  \[
    \label{ConditionOnCoverage}
    f \circ \iota_j
     \;=\;
    \iota_i \circ \iota_{j,i}
    \phantom{AAAAAAA}
    \array{
      V_j
        &\overset{\iota_{j,i}}{\longrightarrow}&
      U_i
      \\
      {}^{\mathllap{ \iota_j }}\big\downarrow
        &&
      \big\downarrow{}^{\mathrlap{ \iota_i}}
      \\
      Y &\underset{f}{\longrightarrow}& X
    }
  \]

A [[small category]] $\mathcal{C}$ equipped with a [[coverage]] is called a _[[site]]_.
=--

+-- {: .num_example #CanonicalCoverageOnTopologicalSpaces}
###### Example
**(canonical [[coverage]] on [[topological spaces]])**

The [[category]] [[Top]] of (small) [[topological spaces]] (Example \ref{ExamplesOfConcreteCategories}) carries a [[coverage]] (Def. \ref{Coverage}) whose [[coverings]] are the usal [[open covers]] of topological spaces.

The condition (eq:ConditionOnCoverage) on a coverage is met, since the [[preimages]] of [[open subsets]] under a [[continuous function]] $f$ are again [[open subsets]], so that the preimages of an open cover consistitute an open cover of the [[domain]], such that the [[commuting diagram]]-condition (eq:ConditionOnCoverage) is immediage.

Similarly, for $X \in Top$ a fixed topological space, there is the [[site]] $Op(X)$ whose underlying [[category]] is the _[[category of opens]]_ of $X$, which is the [[thin category]] (Example \ref{PartiallyOrderedSetsAsSmallCategories}) of [[open subsets]] of $X$ and subset inclusions, and whose [[coverings]] are again the [[open covers]].

=--

+-- {: .num_example #DifferentiablyGoodOpenCoversOfSmoothManifolds}
###### Example
**(differentiably [[good open covers]] of [[smooth manifolds]])**

The [[category]] [[SmthMfd]] of [[smooth manifold]] (Example \ref{ExamplesOfConcreteCategories}) carries a [[coverage]] (Def. \ref{Coverage}), where for $X \in SmthMfd$ any [[smooth manifold]] of [[dimension]] $D \in \mathbb{N}$, its [[coverings]] are collections of [[smooth functions]] from the [[Cartesian space]] $\mathbb{R}^D$ to $X$ whose [[image]] is the inclusion of an [[open ball]].

Hence these are the usual _[[open covers]]_ of $X$, but with the extra condition that every patch is [[diffeomorphism|diffeomorphic]] to a Cartesian space (hence to a smooth [[open ball]]).

One may further constrain this and ask that also all the non-empty finite [[intersections]] of these open balls are [[diffeomorphism|diffeomorphic]] to open balls. These are the _differentiably [[good open covers]]_.

To see that these coverings satisfy the condition (eq:ConditionOnCoverage): The plain pullback of an [[open cover]] along any continuous function is again an open cover, just not necessarily by patches diffeomorphic to open balls. But every open cover may be _refined_ by one that is (see at _[[good open cover]]_), and this is sufficient for (eq:ConditionOnCoverage).

=--

Example \ref{DifferentiablyGoodOpenCoversOfSmoothManifolds} is further developed in the chapters _[[geometry of physics -- smooth sets|smooth sets]]_ and _[[geometry of physics -- smooth homotopy types|on smooth homotopy types]]_.

+-- {: .num_defn #CompatibleElements}
###### Definition
**([[matching family]] -- [[descent object]])**

Let $\mathcal{C}$ be a [[small category]] equipped with a [[coverage]], hence a [[site]] (Def. \ref{Coverage}) and consider a [[presheaf]] $\mathbf{Y} \in [\mathcal{C}^{op}, Set]$ (Example \ref{CategoryOfPresheaves}) over $\mathcal{C}$.

Given an [[object]] $X \in \mathcal{C}$ and a [[covering]] $\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I}$ of it (Def. \ref{Coverage}) we say that a _[[matching family]]_ (of probes of $\mathbf{Y}$) is a [[tuple]] $(\phi_i \in \mathbf{Y}(U_i))_{i \in I}$ such that for all $i,j \in I$ and [[pairs]] of [[morphisms]] $U_i \overset{\kappa_i}{\leftarrow} V \overset{\kappa_j}{\to} U_j$ satisfying

\[
  \label{MatchingCondition}
  \iota_i \circ \kappa_i
  \;=\;
  \iota_j \circ \kappa_j
  \phantom{AAAAAAAA}
  \array{
    && V
    \\
    & {}^{\mathllap{\kappa_i}}\swarrow && \searrow^{\mathrlap{\kappa_j}}
    \\
    U_i && && U_j
    \\
    & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}}
    \\
    && X
  }
\]

we have

\[
  \label{GluingCondition}
  \mathbf{Y}(\kappa_i)(\phi_i)
  \;=\;
  \mathbf{Y}(\kappa_j)(\phi_j)
  \,.
\]

We write

\[
  \label{SetOfMatching}
  Match\big(
    \{U_i\}_{i \in I}
    \,,\,
    \mathbf{Y}
  \big)
  \subset
  \underset{i}{\prod} \mathbf{Y}(U_i)
  \;\in\;
  Set
\]

for the set of [[matching families]] for the given presheaf and covering.

This is also called the _[[descent object]]_ of $\mathbf{Y}$ for _[[descent]]_ along the [[covering]] $\{U_i \overset{\iota_i}{\to}X\}$.

=--

+-- {: .num_example #MatchingFamiliesThatGlue}
###### Example
**([[matching families]] that glue)**

Let $\mathcal{C}$ be a [[small category]] equipped with a [[coverage]], hence a [[site]] (Def. \ref{Coverage}) and consider a [[presheaf]] $\mathbf{Y} \in [\mathcal{C}^{op}, Set]$ (Example \ref{CategoryOfPresheaves}) over $\mathcal{C}$.

Given an [[object]] $X \in \mathcal{C}$ and a [[covering]] $\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I}$ of it (Def. \ref{Coverage}), then every element

$$
  \phi \;\in\; \mathbf{Y}(X)
$$

induces a [[matching family]] (Def. \ref{CompatibleElements}) by

$$
  \big( \mathbf{Y}(\iota_i)(\phi)  \big)_{i \in I}
  \,.
$$

(That this indeed satisfies the matching condition follows immediately by the [[functor|functoriality]] of $\mathbf{Y}$.)

This construction provides a [[function]] of the form

\[
  \label{SheafComparison}
  \mathbf{Y}(X)
    \longrightarrow
  Match\big(
    \{U_i\}_{i \in I}
    \,,\,
    \mathbf{Y}
  \big)
\]

The matching families in the image of this function are hence those [[tuples]] of probes of $\mathbf{Y}$ by the patches $U_i$ of $X$ which _glue_ to a global probe out of $X$.

=--

+-- {: .num_defn #Sheaf}
###### Definition
**([[sheaves]] and [[sheaf toposes]])**

Let $\mathcal{C}$ be a [[small category]] equipped with a [[coverage]], hence a [[site]] (Def. \ref{Coverage}) and consider a [[presheaf]] $\mathbf{Y} \in [\mathcal{C}^{op}, Set]$ (Example \ref{CategoryOfPresheaves}) over $\mathcal{C}$.

The presheaf $\mathbf{Y}$ is called a _[[sheaf]]_ if for every [[object]] $X \in \mathcal{C}$ and every [[covering]] $\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I}$ of $X$ _all [[matching families]] glue uniquely_, hence if the comparison morphism (eq:SheafComparison) is a [[bijection]]

\[
  \label{SheafCondition}
  \mathbf{Y}(X)
    \overset{\simeq}{\longrightarrow}
  Match\big(
    \{U_i\}_{i \in I}
    \,,\,
    \mathbf{Y}
  \big)
  \,.
\]

The [[full subcategory]] (Example \ref{FullSubcategoryOnClassOfObjects}) of the [[category of presheaves]] over a given [[site]] $\mathcal{C}$, on those that are sheaves is the _[[category of sheaves]]_, denoted

\[
  \label{FullSubcategoryOfSheaves}
  Sh(\mathcal{C})
    \overset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow}
  [\mathcal{C}^{op}, Set]
  \,.
\]

A [[category]] which is [[equivalence of categories|equivalent]] (Def. \ref{EquivalenceOfCategories}) to a [[category of sheaves]] is called a _[[sheaf topos]]_, or often just _[[topos]]_, for short.

For $\mathbf{H}_1$ and $\mathbf{H}_2$ two such sheaf toposes, a [[homomorphism]] $f \;\colon\; \mathbf{H}_1 \to \mathbf{H}_2$ between them, called a _[[geometric morphism]]_ is an [[adjoint pair]] of [[functors]] (Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets})

\[
  \label{GeometricMorphism}
  \mathbf{H}_1
    \underoverset
      {\underset{ \phantom{AA} f_\ast \phantom{AA} }{\longrightarrow}}
      \overset{ \phantom{AA} f^\ast \phantom{AA} }{\longleftarrow}
      {}
  \mathbf{H}_2
\]

such that

* the [[left adjoint]] $f^\ast$, called the _[[inverse image]]_, [[preserved limit|preserves]] [[finite products]].

Hence there is a [[category]] _[[Topos]]_, whose [[objects]] are [[sheaf toposes]] and whose [[morphisms]] are [[geometric morphisms]].

=--

+-- {: .num_example #GlobalSectionsGeometricMorphism}
###### Example
**([[global sections]] [[geometric morphism]])**

Let $\mathbf{H}$ be a [[sheaf topos]] (Def. \ref{Sheaf}). Then there is a [[geometric morphism]] (eq:GeometricMorphism) to the [[category of sets]] (Example \ref{CategoryOfSets}), unique up to [[natural isomorphism]] (Def. \ref{NaturalTransformations}):

$$
  \mathbf{H}
    \underoverset
      {\underset{\phantom{AA}\Gamma\phantom{AA}}{\longrightarrow}}
      {\overset{L}{\hookleftarrow}}
      {\bot}
  Set
  \,.
$$

Here $\Gamma$ is called [[generalized the|the]] _[[global sections]]-[[functor]]_.

=--

+-- {: .proof}
###### Proof

Notice that every [[set]] $S \in Set$ is the [[coproduct]], indexed by itself, of the [[terminal object]] $\ast \in Set$ ([[generalized the|the]] [[singleton]]):

$$
  S \;\simeq\; \underset{s \in S}{\coprod} \ast
  \,.
$$

Since $L$ is a [[left adjoint]], it [[preserved limit|preserves]] this [[coproduct]] (Prop. \ref{AdjointsPreserveCoLimits}). Moreover, since $L$ is assumed to preserve [[finite products]], and since the [[terminal object]] is the empty [[product]] (Example \ref{TerminalObjectIsEmptyLimit}), it also preserves the terminal object. Therefore $L$ is fixed, up to [[natural isomorphism]], to act as

$$
  \array{
    L(S)
    & \simeq
    L \left( \underset{s \in S}{\coprod} \ast  \right)
    \\
    & \simeq
    \underset{s \in S}{\coprod} L(\ast)
    \\
    & \simeq
    \underset{s \in S}{\coprod} \ast
  }
  \,.
$$

This shows that $L$ exists and uniquely so, up to natural isomorphism. This implies the essential uniqueness of $\Gamma$ by uniqueness of adjoints (Prop. \ref{UniquenessOfAdjoints}).

=--

+-- {: .num_example #TrivialCoverage}
###### Example
**([[trivial coverage]])**

For $\mathcal{C}$ a [[small category]] (Def. \ref{SmallCategory}), the _trivial coverage_ on it is the [[coverage]] (Def. \ref{Coverage}) with no [[covering]] families at all, meaning that the [[sheaf|sheaf condition]] (Def. \ref{Sheaf}) over the resulting [[site]] is empty, in that _every_ [[presheaf]] is a [[sheaf]] for this coverage.

Hence the [[category of presheaves]] $[\mathcal{C}^{op},Set]$ (Example \ref{CategoryOfPresheaves}) over a site $\mathcal{C}_{triv}$ with trivial coverage is already the corresponding [[category of sheaves]], hence the corresponding [[sheaf topos]]:

$$
  Sh\left( \mathcal{C}_{triv}\right)
  \;\simeq\;
  [\mathcal{C}^{op}, Set]
  \,.
$$

=--

+-- {: .num_example }
###### Example
**([[sheaves]] on the [[terminal category]] are plain [[sets]])**

Consider the [[terminal category]] $\ast$ (Example \ref{InitialCategoryAndTerminalCategory}) equipped with its [[trivial coverage]] (Example \ref{TrivialCoverage}). Then there is a canonical [[equivalence of categories]] (Def. \ref{EquivalenceOfCategories}) between the [[category of sheaves]] on this [[site]] (Def. \ref{Sheaf}) and the [[category of sets]] (Example \ref{CategoryOfSets}):

$$
  Sh(\ast) \;\simeq\; Set
  \,.
$$

Hence the [[category of sets]] is a [[sheaf topos]].

=--

+-- {: .num_example #SheafOnATopologicalSpace}
###### Example
**([[sheaves on a topological space]] -- [[spatial topos|spatial]] [[petit toposes]])**

In the literature, the concept of (pre-)sheaf (Def. \ref{Sheaf}) is sometimes not defined relative to a [[site]], but relative to a [[topological space]]. But the latter is a special case: For $X$ a [[topological space]], consider its [[category of open subsets]] $Op(X)$ from Example \ref{CanonicalCoverageOnTopologicalSpaces}, with [[coverage]] given by the usual [[open covers]]. Then a "[[sheaf on a topological space|sheaf on this topological space]]" is a sheaf, in the sense of Def. \ref{Sheaf}, on this site of opens. One writes

$$
  Sh(X)
    \;\coloneqq\;
  Sh(Op(X))
   \overset{\phantom{AA}}{\hookrightarrow}
  [Op(X)^{op}, Set]
  \,,
$$

for short.  The [[sheaf toposes]] arising this way are also called _[[spatial toposes]]_.

=--

+-- {: .num_prop #LocalicReflection}
###### Proposition
**([[localic reflection]])**

The construction of [[categories of sheaves on a topological space]] (Example \ref{SheafOnATopologicalSpace}) extends to a [[functor]] from the [[category]] _[[Top]]_ of [[topological spaces]] and [[continuous functions]] between them (Example \ref{ExamplesOfConcreteCategories}) to the [[category]] _[[Topos]]_ of [[sheaf toposes]] and [[geometric morphisms]] between them (Example \ref{SheafOnATopologicalSpace}).

$$
  Sh(-)
  \;\colon\;
  Top \longrightarrow Topos
  \,.
$$

Moreover, when restricted to [[sober topological spaces]], this becomes a [[fully faithful functor]], hence a [[full subcategory]]-inclusion (Def. \ref{FullyFaithfulFunctor})

$$
  Sh(-)
  \;\colon\;
  SoberTop \overset{\phantom{AAA}}{\hookrightarrow} Topos
  \,.
$$

More generally, this holds for _[[locales]]_ (i.e. for "[[sober topological spaces]] not necessarily supported on points"), in which case it becomes a [[reflective subcategory]]-inclusion (Def. \ref{ReflectiveSubcategory})

$$
  Locale
    \underoverset
      {\underset{\phantom{AA} Sh(-) \phantom{AA} }{\hookrightarrow}}
      {\overset{\phantom{AAAA}}{\longleftarrow}}
      {\bot}
  Topos
$$

This says that [[categories of sheaves on topological spaces]] are but a reflection of soper topological spaces (generally: locales) and nothing more, whence they are also called _[[petit toposes]]_.

=--

+-- {: .num_example #AbelianSheaves}
###### Example
**([[abelian sheaves]])**

In the literature, sometimes sheaves are understood by default as taking values not in the [[category of sets]], but in the category of [[abelian groups]]. Combined with Example \ref{SheafOnATopologicalSpace} this means that some authors really mean "sheaf of abelian groups of the site of opens of a topological space", when they write just "sheaf".

But for $\mathcal{S}$ any [[mathematical structure]], a sheaf of $\mathcal{S}$-structured sets is equivalently an $\mathcal{S}$-structure [[internalization|internal]] to the [[category of sheaves]] according to Def. \ref{Sheaf}. In particular [[sheaves of abelian groups]] are equivalently abelian [[group objects]] in the category of sheaves of sets as discussed here.

=--

+-- {: .num_example #SmoothSetAnnounced}
###### Example
**([[smooth sets]])**

Consider the [[site]] [[SmthMfd]] of _all_ [[smooth manifolds]], from Example \ref{DifferentiablyGoodOpenCoversOfSmoothManifolds}. The [[category of sheaves]] over this (Def. \ref{Sheaf}) is [[equivalence of categories|equivalent]] to the category of _[[smooth sets]]_, discussed in the chapter _[[geometry of physics -- smooth sets]]_:

$$
  Sh(SmthMfd) \;\simeq\; SmoothSet
  \,.
$$

This is a _[[gros topos]]_, in a sense made precise by Def. \ref{CohesiveTopos} below (a _[[cohesive topos]]_).

=--

+-- {: .num_remark #OrdinarySpacesAreGeneratorsAndRelationsForGeneralizedSpaces}
###### Remark
**(ordinary [[spaces]] and their [[coverings]] are [[generators and relations]] for [[generalized spaces]])**

Given a [[site]] $\mathcal{C}$ (Def. \ref{Coverage}), then its [[presheaf topos]] $[\mathcal{C}^{op}, Set]$ (Example \ref{TrivialCoverage}) is the [[free cocompletion]] of the [[category]] $\mathcal{C}$ (Prop. \ref{FreeCocompletion}), hence the category obtained by [[free construction|freely]] forming [[colimits]] ("gluing") of objects of $\mathcal{C}$.

In contrast, the [[full subcategory]] inclusion $Sh(\mathcal{C}) \hookrightarrow [\mathcal{C}^{op}, Set]$ enforces _relations_ between these free colimits.

Therefore in total we may think of a [[sheaf topos]] $Sh(\mathcal{C})$ as obtained by [[generators and relations]] from the [[objects]] of its [[site]] $\mathcal{C}$:

* the objects of $\mathcal{C}$ are the generators;

* the [[coverings]] of $\mathcal{C}$ are the relations.

=--

+-- {: .num_prop #Sheafification}
###### Proposition
**([[sheafification]] and [[plus construction]])**

Let $\mathcal{C}$ be a [[site]] (Def. \ref{Coverage}). Then the [[full subcategory]]-inclusion (eq:FullSubcategoryOfSheaves) of the [[category of sheaves]] over $\mathcal{C}$ (Def. \ref{Sheaf}) into the [[category of presheaves]] (Example \ref{CategoryOfPresheaves}) has a [[left adjoint]] (Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}) called _[[sheafification]]_

\[
  Sh(\mathcal{C})
   \underoverset
     {\underset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow}}
     {\overset{ L }{\longleftarrow}}
     {\bot}
  [\mathcal{C}^{op}, Set]
  \,.
\]

An explicit formula for [[sheafification]] is given by applying the following "[[plus construction]]" _twice_:

$$
  L(\mathbf{Y}) \simeq (\mathbf{Y}^+)^+
  \,.
$$

Here the [[plus construction]]

$$
  (-)^+
  \;\colon\;
  [\mathcal{C}^{op}, Set]
    \longrightarrow
  [\mathcal{C}^{op}, Set]
$$

is given by forming [[equivalence classes]] of sets of [[matching families]] (Def. \ref{CompatibleElements}) for all possible [[covers]] (Def. \ref{Coverage})

$$
  \mathbf{Y}^+(X)
  \;\coloneqq\;
  \left\{
    \{U_i \overset{\iota_i}{\to} X\}
    \; \text{covering}
    \;,
    \phi \in Match\left( \{U_i\}, \mathbf{Y}  \right)
  \right\}/\sim
$$

under the [[equivalence relation]] which identifies two such [[pairs]] if the two covers have a joint refinement such that the restriction of the two matching families to that joint refinement coincide.

=--

$\,$

+-- {: .num_example #InducedCoverage}
###### Example
**([[induced coverage]])

Let $\mathcal{C}$ be a [[site]] (Def. \ref{Coverage}). Then a [[full subcategory]] (Def. \ref{FullyFaithfulFunctor})

$$
  \mathcal{D}
    \hookrightarrow
  \mathcal{C}
$$

becomes a [[site]] itself, whose [[coverage]] consists of those [[coverings]] $\{U_i \overset{\iota_i}{\to} Y\}$ in $\mathcal{C}$ that happen to be in $\mathcal{D} \hookrightarrow \mathcal{C}$.

=--

+-- {: .num_defn #DenseSubsite}
###### Definition
**([[dense subsite]])**

Let $\mathcal{C}$ and $\mathcal{D}$ be [[sites]] (Def. \ref{Coverage}) with a a [[full subcategory]]-inclusion (Def. \ref{FullyFaithfulFunctor})

$$
  \mathcal{D}
    \hookrightarrow
  \mathcal{C}
$$

and regard $\mathcal{D}$ as equipped with the [[induced coverage]] (Def. \ref{InducedCoverage}).

This is called a _[[dense subsite]]-inclusion_ if every [[object]] $X \in \mathcal{C}$ has a [[covering]] $\{U_i \overset{\iota_i}{\to} X\}_i$ such that for all $i$ the patches are in the subcategory:

$$
  U_i \;\in\; \mathcal{D} \hookrightarrow \mathcal{C}
  \,.
$$

=--

+-- {: .num_prop #ComparisonLemma}
###### Proposition
**([[comparison lemma]])**

Let $\mathcal{D} \overset{\iota}{\hookrightarrow} \mathcal{C}$ be a [[dense subsite]] inclusion (def. \ref{DenseSubsite}). Then [[precomposition]] with $\iota$ induces an [[equivalence of categories]] (Def. \ref{EquivalenceOfCategories}) between their [[categories of sheaves]] (Def. \ref{Sheaf}):

$$
  \iota^\ast
    \;\colon\;
  Sh(\mathcal{C})
    \overset{\simeq}{\longrightarrow}
  Sh(\mathcal{D})
$$

=--

+-- {: .num_prop #RecognitionOfEpimorphisms}
###### Proposition
**(recognition of [[epimorphism|epi-]]/[[monomorphism|mono-]]/[[isomorphisms]] of [[sheaves]])**

Let $\mathcal{C}$ be a [[site]] (Def. \ref{Coverage}) with $Sh(\mathcal{C})$ its [[category of sheaves]] (Def. \ref{Sheaf}).

Then a [[morphisms]] $f \;\colon\; \mathbf{X} \to \mathbf{Y}$ in $Sh(\mathcal{C})$ is

1. a [[monomorphism]] (Def. \ref{Monomorphism}) or [[isomorphism]] (Def. \ref{Isomorphism}) precisely if it is so _globally_ in that for each object $U \in \mathcal{C}$ in the site, then the component $f_U \colon \mathbf{X}(U) \to \mathbf{Y}(U)$ is an [[injection]] or [[bijection]] of [[sets]], respectively.

1.  an [[epimorphism]] (Def. \ref{Monomorphism}) precisely if it is so _locally_, in that: for all $U \in C$ there is a [[covering]] $\{p_i : U_i \to U\}_{i \in I}$ such that for all $i \in I$ and every element $y \in \mathbf{Y}(U)$ the element $f(p_i)(y)$ is in the image of $f(U_i) : \mathbf{X}(U_i) \to \mathbf{Y}(U_i)$.

=--

+-- {: .num_prop #SheafToposEpiMonoFactorization}
###### Proposition
**([[(epi, mono) factorization system|epi/mono-factorization]] through [[image]])**

Let $Sh(\mathcal{C})$ be a [[category of sheaves]] (Def. \ref{Sheaf}). Then every [[morphism]] $f \;\colon\; \mathbf{X} \to \mathbf{Y}$ factors as an [[epimorphism]] followed by a [[monomorphism]] (Def. \ref{Monomorphism}) uniquely up to unique [[isomorphism]]:

$$
  f
  \;\colon\;
  \mathbf{X}
    \overset{epi}{\longrightarrow}
  im(f)
    \overset{mono}{\longrightarrow}
  \mathbf{Y}
  \,.
$$

[[generalized the|The]] [[object]] $im(f)$, as a [[subobject]] of $\mathbf{Y}$, is called the _[[image]]_ of $f$.

In fact this is an [[orthogonal factorization system]], in that for every [[commuting square]] where the left morphism is an [[epimorphism]], and the right one a [[monomorphism]], there exists a unique [[lift]]:

\[
  \label{EpiMonoOrthogonalLifting}
  \array{
    A &\overset{\phantom{AAA}}{\longrightarrow}& B
    \\
    {}^{\mathllap{epi}}\big\downarrow
      &{}^{\exists!}\nearrow&
    \big\downarrow^{\mathrlap{mono}}
    \\
    C &\underset{\phantom{AAA}}{\longrightarrow}& D
  }
\]

This implies that this is a [[functorial factorization]], in that for every [[commuting square]]

$$
  \array{
    \mathbf{X}_1 &\overset{f_1}{\longrightarrow}& \mathbf{Y}_1
    \\
    \big\downarrow && \big\downarrow
    \\
    \mathbf{X}_2 &\underset{f_2}{\longrightarrow}& \mathbf{Y}_2
  }
$$

there is an induced morphism of [[images]] such that the resulting rectangular [[commuting diagram|diagram commutes]]:

$$
  \array{
    \mathbf{X}_1 &\overset{epi}{\longrightarrow}& im(f_1) &\overset{mono}{\longrightarrow}& \mathbf{Y}_1
    \\
    \big\downarrow && \big\downarrow && \big\downarrow
    \\
    \mathbf{X}_2 &\overset{epi}{\longrightarrow}& im(f_2) &\overset{mono}{\longrightarrow}& \mathbf{Y}_2
  }
$$

=--

$\,$

We discuss some of the key properties of [[sheaf toposes]]:

+-- {: .num_prop #PropertiesOfSheafToposes}
###### Proposition
**([[sheaf toposes]] are [[cosmoi]])**

Let $\mathcal{C}$ be a [[site]] (Def. \ref{Coverage}) and  $Sh(\mathcal{C})$ its [[sheaf topos]] (Def. \ref{Sheaf}). Then:

1. All [[limits]] exist in $Sh(\mathcal{C})$ (Def. \ref{Limits}), and they are computed as limits of presheaves, via Example \ref{LimitsOfPresheavesAreComputedObjectwise}:

   $$
     \iota\left(
       \underset{\underset{d}{\longleftarrow}}{\lim} \mathbf{X}_d
     \right)
     \;\simeq\;
       \underset{\underset{d}{\longleftarrow}}{\lim} \iota(\mathbf{X}_d)
   $$

1. All [[colimits]] exist in $Sh(\mathcal{C})$ (Def. \ref{Limits}) and they are given by the [[sheafification]] (Def. \ref{Sheafification}) of the same colimits computed in the [[category of presheaves]], via Example \ref{LimitsOfPresheavesAreComputedObjectwise}:

   $$
     \underset{\underset{d}{\longrightarrow}}{\lim} \mathbf{X}_d
     \;\simeq\;
     L\left(
       \underset{\underset{d}{\longleftarrow}}{\lim} \iota(\mathbf{X}_d)
     \right)
   $$

1. The [[cartesian monoidal category|cartesian]] (Example \ref{CartesianMonoidalCategory}) [[closed monoidal category]]-structure (Def. \ref{ClosedMonoidalCategory}) on the [[category of presheaves]] $[\mathcal{C}^{op}, Set]$ from Example \ref{CartesianClosedMonoidalnessOfCategoriesOfPresheaves} restricts to sheaves:

   $$
     Sh(\mathcal{C})
       \underoverset
         {\underset{[\mathbf{X}, -]}{\longrightarrow}}
         {\overset{\mathbf{X} \times (-)}{\longleftarrow}}
         {}
     Sh(\mathcal{C})
   $$

   In particular, for $\mathbf{X}, \mathbf{Y} \in Sh(\mathcal{C})$ two [[sheaves]], their [[internal hom]] $[\mathbf{X}, \mathbf{Y}] \in Sh(\mathcal{C})$ is a [[sheaf]] given by

   $$
     [\mathbf{X}, \mathbf{Y}]
     \;\colon\;
     U
     \;\mapsto\;
     Hom_{Sh(\mathcal{C})}( y(U) \mathbf{X}, \mathbf{Y} )
     \,,
   $$

   where $y(U)$ is the [[representable presheaf|presheaf represented]] by $U \in \mathcal{C}$ (Example \ref{RepresentablePresheaves}).

This may be summarized by saying that every [[sheaf topos]] (in particular every [[category of presheaves]], by Example \ref{TrivialCoverage}) is a [[cosmos]] for [[enriched category theory]] (Def. \ref{Cosmos}).

=--

+-- {: .num_defn #LocalEpimorphism}
###### Definition
**([[local epimorphism]])**

Let $\mathcal{C}$ be a [[site]] (Def. \ref{Coverage}). Then a [[morphism]] of [[presheaves]] over $\mathcal{C}$ (Example \ref{CategoryOfPresheaves})

$$
  \mathbf{Y} \overset{\phantom{AA}f\phantom{AA}}{\longrightarrow} \mathbf{X}
  \;\;\;
  \in [\mathcal{S}^{op}, Set]
$$

is called a  _[[local epimorphism]]_ if for every [[object]] $U \in \mathcal{C}$, every [[morphism]] $y(U) \longrightarrow \mathbf{X}$
out of its [[representable presheaf|represented presheaf]] (Example \ref{RepresentablePresheaves}) has the _local [[lifting property]]_ through $f$ in that there is a [[covering]] $\big\{ U_i \overset{\iota_i}{\to} U \big\}$ (Def. \ref{Coverage}) and a [[commuting diagram]] of the form

$$
  \array{
    y(U_i)
      &\overset{\phantom{AA}\exists\phantom{AA}}{\longrightarrow}&
    \mathbf{Y}
    \\
    {}^{\mathllap{y(\iota_i)}} \Big\downarrow
      &&
    \Big\downarrow{}^{\mathrlap{ f }}
    \\
    y(U) &\underset{\phantom{AAAA}}{\longrightarrow}& \mathbf{X}
  }
$$

=--

$\,$

### Codescent

In order to understand the sheaf condition (eq:SheafCondition) better, it is useful to consider [[Cech groupoids]] (Def. \ref{CechGroupoid} below). These are really _[[presheaves of groupoids]]_ (Def. \ref{PresheafOfGroupoids} below), a special case of the general concept of [[enriched presheaves]]. The key property of the [[Cech groupoid]] is that it co-represents the [[sheaf|sheaf condition]] (Prop. \ref{CechGroupoidCoRepresents} below). It is in this incarnation that the concept of sheaf seamlessly generalizes to [[homotopy theory]] via "[[higher stacks]]".

$\,$

+-- {: .num_defn #PresheafOfGroupoids}
###### Definition
**([[presheaves of groupoids]])**

For $\mathcal{C}$ a [[small category]] (Def. \ref{SmallCategory}) consider the [[functor category]] (Example \ref{FunctorCategory})
from the [[opposite category]] of $\mathcal{C}$ (Example \ref{OppositeCategory}) to the category [[Grpd]] of [[small groupoid|small]] [[groupoids]] (Example \ref{CategoriesOfSmallCategories})

\[
  [\mathcal{C}^{op}, Grpd]
  \,.
\]

By Example \ref{ExamplesOfCosmoi} we may regard [[Grpd]] as a [[cosmos]] for [[enriched category theory]]. Since the inclusion $Set \hookrightarrow Grpd$ (Example \ref{ReflectiveSubcategoryInclusionOfSetsIntoGroupoids}) is a [[strong monoidal functor]] (Def. \ref{LaxMonoidalFunctor}) of [[cosmoi]] (Example \ref{ExamplesOfCosmoi}), the plain category $\mathcal{C}$ may be thought of as a [[Grpd]]-[[enriched category]] (Def. \ref{TopEnrichedCategory}) and hence a functor $\mathcal{C}^{op} \to Grpd$ is equivalently a [[Grpd]]-[[enriched functor]] (Def. \ref{TopologicallyEnrichedFunctor}).

This means that the plain [[category of functors]] $[\mathcal{C}^{op}, Grpd]$ enriches to [[Grpd]]-[[enriched category]] of [[Grpd]]-[[enriched presheaves]] (Example \ref{EnrichedPresheaf}).

Hence we may speak of _[[presheaves of groupoids]]_.

=--

+-- {: .num_remark #PresheavesOfGroupoidsAsInternalGroupoidsInPresheaves}
###### Remark
**([[presheaves of groupoids]] as [[internal groupoids]] in [[presheaves]])**

From every [[presheaf of groupoids]] $\mathbf{Y} \in [\mathcal{C}^{op}, Grpd]$ (Def. \ref{PresheafOfGroupoids}), we obtain two ordinary [[presheaves]] of sets (Def. \ref{CategoryOfPresheaves}) called the

* _presheaf of objects_

  $$
    Obj_{\mathbf{Y}(-)}
    \in
    [\mathcal{C}^{op}, Set]
  $$

* the _presheaf of morphisms_

  $$
    Mor_{\mathbf{Y}(-)}
    \;\coloneqq\;
     \underset{x,y \in Obj_{\mathbf{Y}(-)}}{\coprod}
      Hom_{{\mathbf{Y}(-)}}
    \;\colon\;
    [\mathcal{C}^{op}, Set]
  $$

In more abstract language this assignment constitutes an [[equivalence of categories]]

\[
  \label{InternalGroupoidsPresheavesOfGroupoids}
  \array{
    [\mathcal{C}^{op}, Grpd]
      &\overset{\simeq}{\longrightarrow}&
    Grpd\left( [\mathcal{C}^{op}, Grpd]\right)
    \\
    \mathbf{Y}
      &\mapsto&
    \left(
      \phantom{AAA}
      \array{
       \underset{
          Mor_{\mathbf{Y}(-)}
        }{
        \underbrace{
          \underset{x,y \in Obj_{\mathbf{Y}(-)}}{\coprod}
          Hom_{{\mathbf{Y}(-)}}
        }}
        \\
        {}^{\mathllap{
          \array{(x \overset{f}{\to}y) \\ \mapsto\\  x}
         }
        }\Big\downarrow
        \;\;\;\;
        \Big\uparrow^{\mathrlap{
          \array{ x \\ \mapsto \\ x \overset{id_x}{\to} x }
        }}
        \phantom{x \overset{id_x}{\to} }
        \Big\downarrow^{ \mathrlap{
          \array{ (x \overset{f}{\to} y) \\ \mapsto \\ y }
        } }
        \\
        Obj_{\mathbf{Y}(-)}
      }
      \phantom{AAA}
    \right)
  }
  \,.
\]

from [[presheaves of groupoids]] to _[[internal groupoids]]- in the [[category of presheaves]] over $\mathcal{C}$ (Def. \ref{CategoryOfPresheaves}).

=--

+-- {: .num_example #PresaheavesOfSetsReflectiveInPresheavesOfGroupoids}
###### Example
**([[presheaves]] of [[sets]] form [[reflective subcategory]] of [[presheaves of groupoids]])**

Let $\mathcal{C}$ be a [[small category]] (Def. \ref{SmallCategory}). There is the [[reflective subcategory]]-inclusion (Def. \ref{ReflectiveSubcategory}) of the [[category of presheaves]] over $\mathcal{C}$ (Example \ref{CategoryOfPresheaves}) into the category of [[presheaves of groupoids]] over $\mathcal{C}$ (Def. \ref{PresheafOfGroupoids})

$$
  [\mathcal{C}^{op}, Set]
    \underoverset
      {\underset{\phantom{AAAA}}{\hookrightarrow}}
      {\overset{\pi_0}{\longleftarrow}}
      {\bot}
  [\mathcal{C}^{op}, Grpd]
$$

which is given over each object of $\mathcal{C}$ by the reflective inclusion of [[sets]] into [[groupoids]] (Example \ref{ReflectiveSubcategoryInclusionOfSetsIntoGroupoids}).

=--

+-- {: .num_example #CechGroupoid}
###### Example
**([[Cech groupoid]])**

Let $\mathcal{C}$ be a [[site]] (Def. \ref{Coverage}), and $X \in \mathcal{C}$ an [[object]] of that site. For each [[covering]] family $\{ U_i \overset{\iota_i}{\to} X\}$ of $X$ in the given [[coverage]], the _[[Cech groupoid]]_ is the [[presheaf of groupoids]] (Def. \ref{PresheafOfGroupoids})

$$
  C(\{U_i\})
    \;\in\;
  [\mathcal{C}^{op}, Grpd]
  \;\simeq\;
  Grpd\left( [\mathcal{C}^{op}, Set]  \right)
$$

which, regarded as an [[internal groupoid]] in the [[category of presheaves]] over $\mathcal{C}$, via (eq:InternalGroupoidsPresheavesOfGroupoids), has as [[presheaf]] of [[objects]] the [[coproduct]]

$$
  Obj_{C(\{U_i\})} \;\coloneqq\; \underset{i}{\coprod}  y(U_i)
$$

of the [[representable presheaf|presheaves represented]] (under the [[Yoneda embedding]], Prop. \ref{YonedaEmbedding}) by the [[covering]] objects $U_i$, and as [[presheaf]] of [[morphisms]] the [[coproduct]] over all [[fiber products]] of these:

$$
  Mor_{C(\{U_i\})}
    \;\coloneqq\;
  \underset{i,j}{\coprod}  y(U_i) \times_{y(X)} y(U_j)
  \,.
$$

This means equivalently that for any $V \in \mathcal{C}$ the [[groupoid]] assigned by $C(\{U_i\})$ has as set of objects [[pairs]] consisting of an index $i$ and a morphism $V \overset{\kappa_i}{\to} U_i$ in $\mathcal{C}$, and there is a unique morphism between two such objects

$$
  \kappa_i \longrightarrow \kappa_j
$$

precisely if

\[
  \label{CechMatchingCondition}
  \iota_i \circ \kappa_i
  \;=\;
  \iota_j \circ \kappa_j
  \phantom{AAAAAAAA}
  \array{
    && V
    \\
    & {}^{\mathllap{\kappa_i}}\swarrow && \searrow^{\mathrlap{\kappa_j}}
    \\
    U_i && && U_j
    \\
    & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}}
    \\
    && X
  }
\]

=--

Condition (eq:CechMatchingCondition) for [[morphisms]] in the [[Cech groupoid]] to be well-defined is verbatim the condition (eq:MatchingCondition) in the definition of [[matching families]]. Indeed, [[Cech groupoids]] serve to conveniently summarize (and then generalize) the [[sheaf|sheaf condition]] (Def. \ref{Sheaf}):

+-- {: .num_prop #CechGroupoidCoRepresents}
###### Proposition
**([[Cech groupoid]] co-represents [[matching families]] -- [[codescent]])**

For [[Grpd]] regarded as a [[cosmos]] (Example \ref{ExamplesOfCosmoi}), and $\mathcal{C}$ a [[site]] (Def. \ref{Coverage}), let

$$
  \mathbf{Y} \in [\mathcal{C}^{op}, Set] \hookrightarrow [\mathcal{C}^{op}, Grpd]
$$

be a [[presheaf]] on $\mathcal{C}$ (Example \ref{CategoryOfPresheaves}), regarded as a [[Grpd]]-[[enriched presheaf]] via Example \ref{PresaheavesOfSetsReflectiveInPresheavesOfGroupoids}, let $X \in \mathcal{C}$ be any [[object]] and $\{U_i \overset{\iota_i}{\to} X\}_i$ a [[covering]] family (Def. \ref{Coverage}) with induced [[Cech groupoid]] $C(\{U_i\}_i)$ (Example \ref{CechGroupoid}).

Then there is an [[isomorphism]]

$$
  [\mathcal{C}^{op},Grpd]
  \left(
    C\left(\{U_i\}_i\right), \, \mathbf{Y}
  \right)
  \;\simeq\;
  Match\left( \{U_i\}_i, \, \mathbf{Y} \right)
$$

between the [[hom-object|hom-groupoid]] of [[Grpd]]-[[enriched presheaves]] (Def. \ref{PointedTopologicalFunctorCategory}) and the set of [[matching families]] (Def. \ref{CompatibleElements}).

Since hence the Cech-groupoid co-represents the [[descent object]], it is sometimes called the _[[codescent object]]_ along the given covering.

Moreover, under this identification the canonical morphism

\[
  \label{CechResolution}
  C\left( \{U_i\}_i \right)
    \overset{p_{\{U_i\}_i}}{\longrightarrow}
  y(X)
\]

induces the comparison morphism (eq:SheafComparison)

$$
  \array{
    [\mathcal{C}^{op}, Grpd]\left( y(X), \, \mathbf{Y} \right)
    & \simeq &
    \mathbf{Y}(X)
    \\
    {}^{
      \mathllap{
        [\mathcal{C}^{op}, Grpd](p_{\{U_i\}_i}, \mathbf{Y})
      }
    }\downarrow && \downarrow
    \\
    [\mathcal{C}^{op},Grpd]
    \left(
      C\left(\{U_i\}_i\right), \, \mathbf{Y}
    \right)
    &\simeq&
    Match\left( \{U_i\}_i, \, \mathbf{Y} \right)
  }
  \,.
$$

In conclusion, this means that the [[presheaf]] $\mathbf{Y}$ is a [[sheaf]] (Def. \ref{Sheaf}) precisely if homming Cech groupoid projections into it produces an isomorphism:

\[
  \label{SheafIsLocalObjectWithRespectToCechCovers}
  \mathbf{Y}
  \,\text{is a sheaf}
  \phantom{AAAA}
  \Leftrightarrow
  \phantom{AAAA}
  \left[
    C\left( \{U_i\}_i \right)
    \overset{p_{\{U_i\}I}}{\to}
     y(X)
    \;,\;
    \mathbf{Y}
  \right]
  \, \text{is iso, for all covering families} \, \{U_i \to X\}
\]

One also says in this case that $\mathbf{Y}$ is a _[[local object]]
with respect to [[Cech covers]]_/

=--

+-- {: .proof}
###### Proof

By (eq:HomObjectOfEnrichedFunctorCategoryViaEnd) the hom-groupoid is computed as the [[end]]

$$
  [\mathcal{C}^{op},Grpd]
  \left(
    C\left(\{U_i\}_i\right), \, \mathbf{Y}
  \right)
  \;=\;
  \int_{V \in \mathcal{C}}
  \left[
    C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V)
  \right]
  \,,
$$

where, by Example \ref{ExamplesOfCosmoi}, the "integrand" is the [[functor category]] (here: a [[groupoid]]) from the [[Cech groupoid]] at a given $V$ to the set (regarded as a groupoid) assigned by $\mathbf{Y}$ to $V$.

Since $\mathbf{Y}(V)$ is just a set, that functor groupoid, too, is just a set, regarded as a groupoid. Its elements are the [[functors]] $C\left(\{U_i\}_i\right)(V) \longrightarrow \mathbf{Y}(V)$, which are equivalently those  [[functions]] on sets of objects

$$
  \underset{i}{\coprod} y(U_i)(V)
  =
  Obj_{C\left(\{U_i\}_i\right)(V)}
    \longrightarrow
  Obj_{\mathbf{Y}(V)}
  =
  \mathbf{Y}(V)
$$

which respect the [[equivalence relation]] induced by the morphisms in the Cech groupoid at $V$.

Hence the hom-groupoid is a subset of the [[end]] of these [[function sets]]:

$$
  \begin{aligned}
    \int_{V \in \mathcal{C}}
    \left[
      C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V)
    \right]
    & \hookrightarrow
    \int_{V \in \mathcal{C}}
    \left[
      \underset{i}{\coprod} y(U_i)(V), \, \mathbf{Y}(V)
    \right]
    \\
    & \simeq
    \int_{V \in \mathcal{C}}
    \underset{i}{\prod}
    \left[
       y(U_i)(V), \, \mathbf{Y}(V)
    \right]
    \\
    & \simeq
    \underset{i}{\prod}
    \int_{V \in \mathcal{C}}
    \left[
       y(U_i)(V), \, \mathbf{Y}(V)
    \right]
    \\
    & \simeq
    \underset{i}{\prod}
    \mathbf{Y}(U_i)
  \end{aligned}
$$

Here we used: first that the [[internal hom]]-functor turns colimits in its first argument into limits (Prop. \ref{InternalHomPreservesLimits}), then that [[limits commute with limits]] (Prop. \ref{LimitsCommuteWithLimits}), hence that in particular [[ends]] commute with [[products]] , and finally the [[enriched Yoneda lemma]] (Prop. \ref{YonedaReductionTopological}), which here is, via Example \ref{NaturalTransformationsViaEnds}, just the plain [[Yoneda lemma]] (Prop. \ref{YonedaLemma}). The end result is hence the same [[Cartesian product]] set that also the set of matching families is defined to be a subset of, in (eq:SetOfMatching).

This shows that an element in
$ \int_{V \in \mathcal{C}}
    \left[
      C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V)
    \right]
$ is a [[tuple]] $(\phi_i \in \mathbf{Y}(U_i))_i$, subject to some condition. This condition is that for each $V \in \mathcal{C}$ the assignment

$$
  \array{
    C\left(\{U_i\}_i\right)(V)
      & \longrightarrow &
    \mathbf{Y}(V)
    \\
    (V \overset{\kappa_i}{\to} U_i)
    &\mapsto&
    \kappa_i^\ast \phi_i
    =
    \mathbf{Y}(\kappa_i)(\phi_i)
  }
$$

constitutes a [[functor]] of [[groupoids]].

By definition of the [[Cech groupoid]], and since the [[codomain]] is a just [[set]] regarded as a [[groupoid]], this is the case precisely if

$$
  \mathbf{Y}(\kappa_i)(\phi_i)
  \;=\;
  \mathbf{Y}(\kappa_j)(\phi_j)
  \phantom{AAAA}
  \text{for all}\, i,j
  \,,
$$

which is exactly the condition (eq:GluingCondition) that makes $(\phi_i)_i$ a matching family.

=--

$\,$

### Local presentation

We now discuss a more abstract characterization of [[sheaf toposes]], in terms of properties enjoyed by the [[adjunction]] that relates them to the corresponding [[categories of presheaves]].

+-- {: .num_defn #LocallyPresentableCategory}
###### Definition
**([[locally presentable category]])**

A [[category]] $\mathbf{H}$ (Def. \ref{Categories}) is called _[[locally presentable category|locally presentable]]_ if there exists a [[small category]] $\mathcal{C}$ (Def. \ref{SmallCategory}) and a [[reflective subcategory]]-inclusion of $\mathcal{C}$ into its [[category of presheaves]] (Example \ref{CategoryOfPresheaves})

$$
  \mathbf{H}
    \underoverset
      {\underset{\text{acc}}{\hookrightarrow}}
      {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}}
      {\bot}
  [\mathcal{C}^{op}, Set]
$$

such that the inclusion functor is an _[[accessible functor|accessible functor]]_ in that it [[preserved limit|preserves]] $\kappa$-[[filtered colimits]] for some [[regular cardinal]] $\kappa$.

=--

+-- {: .num_prop #GiraudTheorem}
###### Proposition
**([[Giraud's theorem]])**

A [[sheaf topos]] (Def. \ref{Sheaf}) is equivalently a [[locally presentable category]] (Def. \ref{LocallyPresentableCategory}) with

1. [[universal colimits]],

1. [[effective quotients]],

1. [[disjoint coproducts]].

=--

+-- {: .num_prop #SheafToposViaLexReflection}
###### Proposition
**([[sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes]])**

Let $(\mathcal{C}, \tau)$ be a [[site]] (Def. \ref{Coverage}).
Then the [[full subcategory]] inclusion $i \colon Sh(\mathcal{C},\tau) \hookrightarrow PSh(\mathcal{C})$ of its [[sheaf topos]] (Def. \ref{Sheaf}) into its [[category of presheaves]] is a [[reflective subcategory]] inclusion (Def. \ref{ReflectiveSubcategory})

$$
  Sh(\mathcal{C},\tau)
   \underoverset
     {\underset{\iota}{\hookrightarrow}}
     {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}}
     {\bot}
  PSh(\mathcal{C})
$$

such that:

1. the inclusion $\iota$ is an [[accessible functor]], thus exhibiting $Sh(\mathcal{C},\tau)$ as a [[locally presentable category]] (Def. \ref{LocallyPresentableCategory})

1. the reflector $L \colon PSh(\mathcal{C})  \to Sh(\mathcal{C})$ (which is [[sheafification]], Prop. \ref{Sheafification}) is [[left exact functor|left exact]] ("lex") in that it [[preserved limit|preserves]] [[finite limits]].

Conversely, every [[sheaf topos]] arises this way. Hence [[sheaf toposes]] $\mathbf{H}$ are equivalently the [[left exact functor|left exact]]-[[reflective subcategory|reflectively]] [[full subcategories]] of [[presheaf toposes]] over some [[small category]] $\mathcal{C}$:

\[
  \label{SheafToposAsLexReflection}
  \mathbf{H}
   \underoverset
     {\underset{\phantom{AA}acc\phantom{AA}}{\hookrightarrow}}
     {\overset{\phantom{AA}lex\phantom{AA}}{\longleftarrow}}
     {\bot}
  PSh(\mathcal{C})
\]

=--

(e.g. [Borceux 94, prop. 3.5.4, cor. 3.5.5](sheaf+toposes+are+equivalently+the+left+exact+reflective+subcategories+of+presheaf+toposes#Borceux94), [Johnstone, C.2.1.11](sheaf+toposes+are+equivalently+the+left+exact+reflective+subcategories+of+presheaf+toposes#Johnstone))

+-- {: .num_example}
###### Remark
**(left exact reflections of [[categories of presheaves]] are [[locally presentable categories]])**

In the characterization of [[sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes|sheaf toposes as left exact reflections of categories of presheaves]] in Prop. \ref{SheafToposViaLexReflection}, the [[accessible functor|accessibility]] of the inclusion, equivalently the [[locally presentable category|local presentability]] (Def. \ref{LocallyPresentableCategory}) is automatically implied (using the [[adjoint functor theorem]]), as indicated in (eq:SheafToposAsLexReflection).

=--


geometry of physics - cohesive toposes

## Gros toposes
 {#GrosToposes}

We have seen roughly two different kinds of [[sheaf toposes]]:

* [[categories of sheaves on a topological space|categories of sheaves on a given space]] $X$ (Example \ref{SheafOnATopologicalSpace}), which, by [[localic reflection]] (Prop. \ref{LocalicReflection}), 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 \ref{SmoothSetAnnounced})

  these are called _[[gros toposes]]_.

+-- {: .num_remark #CohesiveGeneralizedSpacesAsFoundations}
###### Remark
**([[cohesion|cohesive]] [[generalized spaces]] as [[foundations|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 [[localic reflection|petit topos-reflections]] of these (Prop. \ref{LocalicReflection} below).

The [[gros toposes]] of such foundational [[generalized spaces]] ought to  have an _[[internal logic]]_ that knows about _[[modalities]] of [[geometry]]_ such as _[[discrete object|discreteness]]_ or _[[concrete object|concreteness]]_. Via the formalization of [[modalities]] in Def. \ref{ModalOperator} this leads to the definiton of [[cohesive toposes]] (Def. \ref{CohesiveTopos}, Prop. \ref{PiecesHavePoints} below, due to [[Some Thoughts on the Future of Category Theory|Lawvere 91]], [Lawvere 07](#cohesive+topos#LawvereAxiomatic)).

=--

| $\phantom{A}$[[gros topos]]$\phantom{A}$ |  | $\phantom{A}$[[generalized spaces]] obey...$\phantom{A}$ | $\phantom{A}$example:$\phantom{A}$ |
|-----|---|-------|---|
| $\phantom{A}$[[cohesion]]$\phantom{A}$ |  Def. \ref{CohesiveTopos} |  $\phantom{A}$principles of [[differential topology]]$\phantom{A}$ | $\phantom{A}$[[geometry of physics -- smooth sets|SmoothSet]]$\phantom{A}$ |
| $\phantom{A}$[[elastic topos|elasticity]] | Def. \ref{DifferentialCohesion} |$\phantom{A}$principles of [[differential geometry]]$\phantom{A}$ | $\phantom{A}$[[geometry of physics -- manifolds and orbifolds|FormallSmoothset]]$\phantom{A}$ |
| $\phantom{A}$[[solid topos|solidity]]$\phantom{A}$ | Def. \ref{SuperDifferentialCohesion} | $\phantom{A}$principles of [[supergeometry]]$\phantom{A}$ | $\phantom{A}$[[geometry of physics -- supergeometry|SuperFormalSmoothSet]]$\phantom{A}$ |
{: style='margin:auto'}

$\,$

### Cohesive toposes
 {#Cohesive}

+-- {: .num_defn #CohesiveTopos}
###### Definition
**([[cohesive topos]])**

A [[sheaf topos]] $\mathbf{H}$ (Def. \ref{Sheaf}) is called a _[[cohesive topos]]_ if there is a [[adjoint quadruple|quadruple]] (Remark \ref{AdjointTriples}) of [[adjoint functors]] (Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}) to the [[category of sets]] (Example \ref{CategoryOfSets})

\[
  \label{CohesopmAdjointQuadruple}
  \Pi \dashv Disc \dashv \Gamma \dashv coDisc
  \;\;\colon\;\;
  \mathbf{H}
    \array{
      \overset{\phantom{AAA} \Pi \phantom{AAA}}{\longrightarrow}
      \\
      \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow}
      \\
      \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow}
      \\
      \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow}
    }
  Set
\]

such that:

1. $Disc$ and $coDisc$ are [[full and faithful functors]] (Def. \ref{FullyFaithfulFunctor})

1. $\Pi$ [[preserved limit|preserves]] [[finite products]].

=--

+-- {: .num_example #PresheavesAdjointQuadrupleOnSiteWithTerminalObject}
###### Example
**([[adjoint quadruple]] of [[presheaves]] over [[site]] with [[finite products]])**

Let $\mathcal{C}$ be a [[small category]] (Def. \ref{SmallCategory}) 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 \ref{AdjointTriples}) of [[functors]] between the [[category of presheaves]] over $\mathcal{C}$ (Example \ref{CategoryOfPresheaves}), and the [[category of sets]] (Example \ref{CategoryOfSets})

\[
  \label{PreaheafAdjointQuadruple}
  [\mathcal{C}^{op}, Set]
    \array{
      \overset{\phantom{AAA} \Pi \phantom{AAA}}{\longrightarrow}
      \\
      \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow}
      \\
      \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow}
      \\
      \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow}
    }
  Set
\]

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]]:

   \[
     \label{CohesiveGlobalSectionsGivenByPointEvaluation}
     \begin{aligned}
       \Gamma \mathbf{Y}
       & =
       \underset{\underset{\mathcal{C}}{\longleftarrow}}{\lim}
       \mathbf{Y}
       \\
       & \simeq
       \mathbf{Y}(\ast)
     \end{aligned}
   \]

1. $Disc$ and $coDisc$ are [[full and faithful functors]] (Def. \ref{FullyFaithfulFunctor}).

1. $\Pi$ 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 \ref{TrivialCoverage}) is a [[cohesive topos]] (Def. \ref{CohesiveTopos}).

=--

+-- {: .proof}
###### Proof

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

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

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

From this it follows, by Example \ref{KanExtensionOfAdjointPairIsAdjointQuadruple}, that [[Kan extension]] produces an [[adjoint quadruple]] (Remark \ref{AdjointTriples}) 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 $\ast \hookrightarrow \mathcal{C}$

1. $Disc$ 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. \ref{FullyFaithfulFunctor}), it follows that also its [[left Kan extension]] $Disc$ is fully faithful  (Prop. \ref{LeftKanExtensionAlongFullyFaithfulFunctor}). This implies that also $coDisc$ is fully faithful, by (Prop. \ref{FullyFaithfulAdjointTriple}).

Equivalently, $Disc \simeq p^\ast$ is the [[constant functor|constant]] [[diagram]]-assigning functor. By uniqueness of adjoints (Prop. \ref{UniquenessOfAdjoints}) 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]]

\[
  \label{Pi0IsColimit}
  \Pi(\mathbf{Y})
  \;=\;
  \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{Y}
  \,.
\]

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. \ref{CategoriesWithFiniteProductsAreCosifted})

=--

Example \ref{PresheavesAdjointQuadrupleOnSiteWithTerminalObject} suggests to  ask for [[coverages]] on categories with [[finite products]] which are such that the [[adjoint quadruple]]  (eq:PreaheafAdjointQuadruple) on the  [[category of presheaves]] ([[corestriction|co-]])[[restriction|restricts]] to the corresponding [[category of sheaves]]. The following Definition \ref{OneCohesiveSite} states a sufficient condition for this to be the case:

+-- {: .num_defn #OneCohesiveSite}
###### Definition
**([[cohesive site]])**

We call a [[site]] $\mathcal{C}$ (Def. \ref{Coverage}) _[[cohesive site|cohesive]]_ if the following conditions are satisfied:

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

1. 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. \ref{CechGroupoid}) 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]]:

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

   1. the set of [[connected components]] of the [[groupoid]] obtained as the [[limit]] of the [[Cech groupoid]] is [[equivalence of categories|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:

+-- {: .num_prop #CategoriesOfSheavesOnCohesiveSiteIsCohesive}
###### Proposition
**([[category of sheaves]] on a [[cohesive site]] is a [[cohesive topos]])**

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

\[
  \label{SheafToposAdjointQuadruple}
  Sh(\mathcal{C})
    \array{
      \overset{\phantom{AAA} \Pi \phantom{AAA}}{\longrightarrow}
      \\
      \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow}
      \\
      \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow}
      \\
      \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow}
    }
  Set
\]

=--

+-- {: .proof}
###### Proof

By example \ref{PresheavesAdjointQuadrupleOnSiteWithTerminalObject}  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 \ref{PresheavesAdjointQuadrupleOnSiteWithTerminalObject} 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. \ref{Sheaf}).

By the formulaton of the [[sheaf|sheaf condition]] via the [[Cech groupoid]] (Prop. \ref{CechGroupoidCoRepresents}), and using the [[adjunction]] hom-isomorphisms ([here](geometry+of+physics+--+categories+and+toposes#eq:HomIsomorphismForAdjointFunctors)) this is readily seen to be equivalent to the two further conditions on a cohesive site (Def. \ref{OneCohesiveSite}):

Let $\{U_i \to X\}$ be a [[covering]] family.

The sheaf condition (eq:SheafIsLocalObjectWithRespectToCechCovers) 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 (eq:Pi0IsColimit) 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 \ref{ColimitOfRepresentableIsSingleton}) 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 subcategory|reflective embedding]] of sets into groupoids, whose [[left adjoint]] is the [[connected components]]-functor $\pi_0$ (Example \ref{ReflectiveSubcategoryInclusionOfSetsIntoGroupoids}). 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 (eq:SheafIsLocalObjectWithRespectToCechCovers) 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 (eq:CohesiveGlobalSectionsGivenByPointEvaluation), 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.

=--

+-- {: .num_defn #CohesiveModalities}
###### Definition
**([[adjoint triple]] of [[adjoint modality|adjoint]] [[modal operators]] on [[cohesive topos]])**

Given a [[cohesive topos]] (Def. \ref{CohesiveTopos}), its [[adjoint quadruple]] (Remark \ref{AdjointTriples}) of functors to and from [[Set]]

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

induce, by [[composition]] of functors, an [[adjoint triple]] (Remark \ref{AdjointTriples}) of [[adjoint modalities]] (Def. \ref{AdjointModality}):

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

Since $Disc$ and $coDisc$ are [[fully faithful functors]] by assumption,
these are ([[comodal operator|co-]])[[modal operators]] (Def. \ref{ModalOperator}) on the [[cohesive topos]], by (Prop. \ref{ModalOperatorsEquivalentToReflectiveSubcategories}).

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 $ $\phantom{A}$  |
{: style='margin:auto'}

and we refer to the corresponding [[modal objects]] (Def. \ref{ModalObjects}) as follows:

* a [[flat modality|flat]]-[[comodal object]]

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

  is called a _[[discrete object]]_;

* a [[sharp modality|sharp]]-[[modal object]]

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

  is called a _[[codiscrete object]]_;

* a [[sharp modality|sharp]]-[[modal objects|submodal object]]

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

  is a _[[concrete object]]_.

=--

+-- {: .num_prop #PiecesHavePoints}
###### Proposition
**([[pieces have points]] $\simeq$ [[discrete objects are concrete]] $\simeq$ [[Aufhebung]] of [[bottom]] [[adjoint modality]])**

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

1. **[[pieces have points]]**: For every [[object]] $X \in \mathbf{H}$, _comparison of extremes_-transformation (eq:OppositeExtrmeComparison) for the $(ʃ, \dashv \flat)$-[[adjoint modality]] (eq:OppositeExtrmeComparison), hence the $\flat$-[[counit of an adjunction]] [[composition|composed]] with the ʃ-[[unit of an adjunction|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. \ref{Monomorphism})

1. **[[discrete objects are concrete]]**: For every [[object]] $X \in \mathbf{H}$, we have that its [[discrete object]] $\flat X$ is a [[concrete object]] (Def. \ref{CohesiveModalities}).

1. **[[Aufhebung]] of [[bottom]] [[adjoint modality]]**

   The [[adjoint modality]] $\flat \dashv \sharp$ exhibits [[Aufhebung]] (Def. \ref{Aufhebung}) of the [[bottom]] [[adjoint modality]] (Example \ref{InitialAndFinalAdjointModality}), i.e. the [[initial object]] (Def. \ref{InitialObject}) is [[codiscrete object|codiscrete]] (Def. \ref{CohesiveModalities}):

   $$
     \sharp \emptyset \;\simeq\; \emptyset
     \,.
   $$

=--

+-- {: .proof}
###### Proof

The comparison morphism $ptp_{\mathbf{H}}$ is a special case of that discussed in Prop. \ref{ComparisonMorphismBetweenOppositeExtremes}. 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 (eq:PiecesToPointsFromBase) we have $ptp_{\mathbf{H}} = Disc(ptp_{\mathbf{B}})$, but $Disc$ is a [[left adjoint]] and left adjoints preserve monomorphisms (Prop. \ref{LeftAdjointFunctorPreservesEpi}).

In the other direction, assume that $ptp_{\mathbf{H}}$ is an epimorphism. By (eq:PointsToPiecesInTheBase) and (eq:AdjunctOfptpH) 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. \ref{LeftAdjointFunctorPreservesEpi}, as does composition with isomorphisms.

By applying (eq:PointsToPiecesInTheBase) 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 (eq:HomIsomorphismForAdjointFunctors) and definition of [[epimorphism]], Def. \ref{Monomorphism}).

Now by Lemma \ref{ReExpressingMiddleFunctorInAdjointTriple}, this is equivalent to

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

being an injection for all $\mathbf{X}$, which, by Def. \ref{Monomorphism}, 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.

=--

+-- {: .num_prop #CohesiveSiteSuchThatDiscreteObjectsAreConcrete}
###### Proposition
**([[cohesive site]] such that [[pieces have points]]/[[discrete objects are concrete]])**

Let $\mathcal{C}$ be a [[cohesive site]] (Def. \ref{OneCohesiveSite}), such that

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

Then the [[cohesive topos]] $Sh(\mathcal{C})$, according to Prop. \ref{CategoriesOfSheavesOnCohesiveSiteIsCohesive}, satisfies the equivalent conditions from Prop. \ref{PiecesHavePoints}:

1. [[pieces have points]],

1. [[discrete objects are concrete]].

=--

+-- {: .proof}
###### Proof

By Prop. \ref{PiecesHavePoints} 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. \ref{RecognitionOfEpimorphisms} 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. \ref{CategoriesOfSheavesOnCohesiveSiteIsCohesive}, 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 function|injective]] precisely if $Hom_{\mathcal{C}}(\ast, X)$ is [[inhabited set|non-empty]], which is true by assumption.

=--

+-- {: .num_prop #QuasitoposOfConcreteObjects}
###### Proposition
**([[quasitopos]] of [[concrete objects]] in a [[cohesive topos]])**

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

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

for its [[full subcategory]] (Example \ref{FullSubcategoryOnClassOfObjects}) of [[concrete objects]] (Def. \ref{CohesiveModalities}).

Then there is a sequence of [[reflective subcategory]]-inclusions (Def. \ref{ReflectiveSubcategory}) 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. \ref{PiecesHavePoints}), 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}
###### 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 object|concrete]], which is immediate by [[idempotent monad|idempotency]] of $\sharp$ (Prop. \ref{ModalOpIdempotent}) 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. \ref{SheafToposEpiMonoFactorization}) 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 system|epi/mono-factorization]] (Prop. \ref{SheafToposEpiMonoFactorization}) of $\eta^\sharp_X$

\[
  \label{ConcretificationUnitFromEpiMonoFactorization}
  \eta^\sharp_X
  \;\colon\;
  X
    \underoverset{epi}{ \eta^{conc}_X }{\longrightarrow}
  conc X
    \underoverset{mono}{}{\longrightarrow}
  \sharp X
  \,.
\]

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

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

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. \ref{GeneralAdjunctsInTermsOfAdjunctionUnitCounit})  and the definition $\sharp \coloneqq coDisc \circ \Gamma$ (Def. \ref{CohesiveModalities}).

But observe that $\Gamma (\eta^{conc}_X)$, and hence also $\sharp(\eta^{conc}_X)$, is an [[isomorphism]] (Def. \ref{Isomorphism}), 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. \ref{LeftAdjointFunctorPreservesEpi}), hence it preserves [[image]] factorizations (Prop. \ref{SheafToposEpiMonoFactorization}). This implies that $\Gamma \eta^{conc}_X$ is the epimorphism onto the image of $\Gamma( \eta^\sharp_X )$. But by [[idempotent monad|idempotency]] of $\sharp$, the latter is an [[isomorphism]], and hence so is the epimorphism in its image factorization.

Therefore the equality in (eq:ConcNatur) 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. \ref{Monomorphism}) 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. \ref{AdjunctionUnitFromHomIsomorphism}) of $(conc \dashv \iota_{conc})$ is provided by $\eta^{conc}$ (eq:ConcretificationUnitFromEpiMonoFactorization).

To see this, first notice that, since the [[(epi, mono) factorization system|epi/mono-factorization]] (Prop. \ref{SheafToposEpiMonoFactorization}) is [[orthogonal factorization system|orthogonal]] and hence [[functorial factorization|functorial]], we have [[commuting diagrams]] of the form

\[
  \label{NaturalitySquareForConcretificationUnit}
  \array{
    X_1
      &\underoverset{epi}{\eta^{conc}_{X_1}}{\longrightarrow}&
    im(\eta^\sharp_{X_1})
      &\underset{mono}{\longrightarrow}&
    \sharp X_1
    \\
    \big\downarrow && \big\downarrow && \big\downarrow
    \\
    X_2
      &\underoverset{epi}{\eta^{conc}_{X_2}}{\longrightarrow}&
    im(\eta^\sharp_{X_2})
      &\underset{mono}{\longrightarrow}&
    \sharp X_2
  }
\]

Now to demonstrate the adjunction it is sufficient, by Prop. \ref{CollectionOfUniversalArrowsEquivalentToAdjointFunctor}, to show that $\eta^{conc}$ is a [[universal morphism]] in the sense of Def. \ref{UniversalArrow}. 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_ [[commuting diagram|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 (eq:NaturalitySquareForConcretificationUnit), 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 system|epi/mono-factorization]] in Prop. \ref{SheafToposEpiMonoFactorization} says that there is a diagonal [[lift]] which makes _both_ triangles [[commuting diagram|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. \ref{Monomorphism}.

=--

The equivalence of the first two follows with ([Johnstone, lemma 2.1, corollary 2.2](cohesive+topos#Johnstone)). The equivalence of the first and the last is due to [Lawvere-Menni 15, lemma 4.1, lemma 4.2](cohesive+topos#LawvereMenni15).

$\,$

### Elastic toposes
 {#ElasticToposes}

+-- {: .num_defn #DifferentialCohesion}
###### Definition
**([[elastic topos]])**

Let $\mathbf{H}_{red}$ be a [[cohesive topos]] (Def. \ref{CohesiveTopos}). 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]],

1. equipped with a [[adjoint quadruple|quadruple]] of [[adjoint functors]] (Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}) to $\mathbf{H}_{red}$ of the form

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

=--

+-- {: .num_lemma #ProgressionOfSubcategoriesOfElasticTopos}
###### Lemma
**(progression of ([[coreflective subcategory|co-]])[[reflective subcategories]] of [[elastic topos]])**

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

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

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]]

<center>
<img src="https://ncatlab.org/nlab/files/ProgressionOfSubcategoriesDifferentialCohesion.jpg" width="400">
</center>

=--

+-- {: .proof}
###### 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 \ref{GlobalSectionsGeometricMorphism}). This implies the identifications $\Pi \simeq \Pi_{red} \circ \Pi_{inf}$ by essential uniqueness of [[adjoints]] (Prop. \ref{UniquenessOfAdjoints}).

=--

+-- {: .num_defn #DiffCohesiveModalities}
###### Definition
**([[adjoint modalities]] on [[elastic topos]])**

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

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

Since $\iota_{inf}$ and $Disc_{inf}$ are [[fully faithful functors]] by assumption, these are ([[comodal operator|co-]])[[modal operators]] (Def. \ref{ModalOperator}) on the [[cohesive topos]], by (Prop. \ref{ModalOperatorsEquivalentToReflectiveSubcategories}).

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} $ $\phantom{A}$  |
{: style='margin:auto'}

and we refer to the corresponding [[modal objects]] (Def. \ref{ModalObjects}) as follows:

* a [[reduction modality|reduction]]-[[comodal object]]

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

  is called a _[[reduced object]]_;

* an [[infinitesimal shape modality|infinitesimal shape]]-[[modal object]]

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

  is called a _[[coreduced object]]_.

=--

+-- {: .num_prop #ProgressionOfModalitiesOnElasticTopos}
###### Proposition
**(progression of [[adjoint modalities]] on [[elastic topos]])**

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

1. for being a [[cohesive topos]], from Def. \ref{CohesiveModalities},

1. for being an [[elastic topos]], from Def. \ref{DiffCohesiveModalities}:

| $\phantom{A}$ [[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} $ $\phantom{A}$  |
{: style='margin:auto'}

Then these arrange into the following progression, via the [[preorder]] on modalities from Def. \ref{PreorderOnModalities}

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

where we display also the [[bottom]] [[adjoint modality]] $\emptyset \dashv \ast$ (Example \ref{InitialAndFinalAdjointModality}), for completeness.

=--

+-- {: .proof}
###### Proof

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

1. $\flat X$ is an $\&$-[[modal object]] (Def. \ref{ModalObjects}), hence that

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

1. $ʃ X$ is an $\Im$-[[modal object]] (Def. \ref{ModalObjects}), hence that

   $$
     \Im ʃ X  \;\simeq\; X
   $$

After unwinding the definitions of the modal operators Def. \ref{CohesiveModalities} and Def. \ref{CohesiveModalities}, and using their re-identification from Lemma \ref{ProgressionOfSubcategoriesOfElasticTopos}, 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. \ref{FullyFaithfulAndInvertibleAdjoints}, since $Disc_{inf}$ is a [[fully faithful functor]] and $\Pi_{inf}$, $Gamma_{inf}$ are ([[coreflective subcategory|co-]])[[reflective subcategory|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
  {#SolidToposes}

+-- {: .num_defn #SuperDifferentialCohesion}
###### Definition
**([[solid topos]])**

Let $\mathbf{H}_{bos}$ be an [[elastic topos]] (Def. \ref{DifferentialCohesion}) over a [[cohesive topos]] $\mathbf{H}_{red}$ (Def. \ref{CohesiveTopos}). 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. \ref{CohesiveTopos}),

1. an [[elastic topos]] over $\mathbf{H}_{red}$,

1. equipped with a [[adjoint quadruple|quadruple]] of [[adjoint functors]] (Def. \ref{AdjointFunctorsInTermsOfNaturalBijectionOfHomSets}) 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. \ref{FullyFaithfulFunctor}).

=--

+-- {: .num_lemma #ProgressionOfSubcategoriesOfSolidTopos}
###### Lemma
**(progression of ([[coreflective subcategory|co-]])[[reflective subcategories]] of [[solid topos]])**

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

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

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

<center>
<img src="https://ncatlab.org/nlab/files/ProgressionSubcategoriesSuperCohesion.jpg" width="540">
</center>

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 [[elastic topos|elasticity]] of $\mathbf{H}$ over $\mathbf{H}_{red}$.

=--

+-- {: .proof}
###### Proof

As in the proof of Prop. \ref{ProgressionOfSubcategoriesOfElasticTopos},  this is immediate by the essential uniqueness of adjoints (Prop. \ref{UniquenessOfAdjoints}) and of the [[global section]]-[[geometric morphism]] (Example \ref{GlobalSectionsGeometricMorphism}).

=--

+-- {: .num_defn #SuperDiffCohesiveModalities}
###### Definition
**([[adjoint modalities]] on [[solid topos]])**

Given a [[solid topos]] $\mathbf{H}$ over $\mathbf{H}_{bos}$ (Def. \ref{SuperDifferentialCohesion}), [[composition]] of the functors in Lemma \ref{ProgressionOfSubcategoriesOfSolidTopos} yields, via Prop. \ref{ModalOperatorsEquivalentToReflectiveSubcategories}, the following [[adjoint modalities]] (Def. \ref{AdjointModality})

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

Since $\iota_{sup}$ and $Disc_{sup}$ are [[fully faithful functors]] by assumption,
these are ([[comodal operator|co-]])[[modal operators]] (Def. \ref{ModalOperator}) on the [[cohesive topos]], by (Prop. \ref{ModalOperatorsEquivalentToReflectiveSubcategories}).

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} $ $\phantom{A}$  |
{: style='margin:auto'}

and we refer to the corresponding [[modal objects]] (Def. \ref{ModalObjects}) 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_;

=--

+-- {: .num_prop #ProgressionOfModalitiesOnSolidTopos}
###### Proposition
**(progression of [[adjoint modalities]] on [[solid topos]])**

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

1. for being a [[cohesive topos]], from Def. \ref{CohesiveModalities},

1. for being an [[elastic topos]], from Def. \ref{DiffCohesiveModalities},

1. for being a [[solid topos]], from Def. \ref{SuperDiffCohesiveModalities}:

| $\phantom{A}$ [[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} $ $\phantom{A}$  |
{: style='margin:auto'}

Then these arrange into the following progression, via the [[preorder]] on modalities from Def. \ref{PreorderOnModalities}:

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

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

=--

+-- {: .proof}
###### Proof

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

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

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

The proof is directly analogous to that of Prop. \ref{ProgressionOfModalitiesOnElasticTopos}, now using the decompositions from Lemma \ref{ProgressionOfSubcategoriesOfSolidTopos}:

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

and

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

=--

(...)