nLab cohesive topos

Idea

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

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

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

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

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

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

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

In total this gives an adjoint quadruple

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

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

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

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

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

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

$$\Gamma X \to \Pi_0 X$$

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

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

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

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

Definition

Fundamental axioms

Definition

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

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

is cohesive if it is a

• strongly connected topos

• and a local topos.

In detail this means that it has the following properties:

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

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

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

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

In summary $\mathcal{E}$ is cohesive over $\mathcal{S}$ if we have an quadruple of adjoint functor of the form

$$(f_! \dashv f^* \dashv f_* \dashv f^!) : \mathcal{E} \array{ \overset{\phantom{AA} f_! \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA} f^\ast \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AA} f_\ast \phantom{AA} }{\longrightarrow} \\ \overset{\phantom{AA} f^! \phantom{AA} }{\hookleftarrow} } \mathcal{S}$$

such that $f_!$ preserves finite products.

Remark

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

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

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

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

$$(\Pi_0 \dashv Disc \dashv \Gamma \dashv coDisc) : \mathcal{E} \array{ \overset{\phantom{AAA} \Pi_0 \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } \mathcal{S} \,.$$

Definition

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

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

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

• shape modality $\dashv$ flat modality $\dashv$ sharp modality.

$\,$

Further axioms

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

1. pieces have points;

   equivalently (Prop. below):

   discrete objects are concrete;

2. equivalent over a Boolean base topos (Prop. below):

   Aufhebung of bottom adjoint modality

3. pieces of powers are powers of pieces (Def. below)

4. the subobject classifier is connected (Def. below).

$\,$

First we need some lemmas:

Lemma

Let

$$\mathcal{C} \array{ \overset{ \phantom{A} \Pi \phantom{A} }{\longrightarrow} \\ \overset{ \phantom{A} Disc \phantom{A} }{\hookleftarrow} \\ \overset{ \phantom{A} \Gamma \phantom{A} }{ \longrightarrow } } \mathcal{D}$$

be an adjoint triple, with $Disc$ a fully faithful functor. Denoting the adjunction units/counits as

$\phantom{A}$ adjunction $\phantom{A}$	$\phantom{A}$ unit $\phantom{A}$	$\phantom{A}$ counit $\phantom{A}$
$\phantom{A}$ $(\Pi \dashv Disc)$ $\phantom{A}$	$\phantom{A}$ $\eta^{ʃ}$ $\phantom{A}$	$\phantom{A}$ $\epsilon^{ʃ}$ $\phantom{A}$
$\phantom{A}$ $(Disc \dashv \Gamma)$ $\phantom{A}$	$\phantom{A}$ $\eta^\flat$ $\phantom{A}$	$\phantom{A}$ $\epsilon^\flat$ $\phantom{A}$

we have that the following composites of unit/counit components are equal:

(1)$$\left( \eta^{\flat}_{\Pi X} \right) \circ \left( \Pi \epsilon^\flat_X \right) \;\;=\;\; \left( \Gamma \eta^{ʃ}_{X} \right) \circ \left( \epsilon^{ʃ}_{\Gamma X} \right) \phantom{AAAAAA} \array{ \Pi Disc \Gamma X &\overset{\epsilon^{ʃ}_{\Gamma X}}{\longrightarrow}& \Gamma X \\ {}^{ \mathllap{ \Pi \epsilon^\flat_X } }\big\downarrow && \big\downarrow^{\mathrlap { \Gamma \eta^{ʃ}_{X} } } \\ \Pi X &\underset{ \eta^\flat_{\Pi X} }{\longrightarrow}& \Gamma Disc \Pi X }$$

(Johnstone 11, lemma 2.1)

Proof

We claim that the following diagram commutes:

$$\array{ && && \Gamma X \\ && & {}^{ \epsilon^ʃ_{\Gamma X} }\nearrow && \searrow^{\mathrlap{ \Gamma \eta^{ʃ}_X }} \\ && \Pi Disc \Gamma X && && \Gamma Disc \Pi X \\ & {}^{ \Pi \epsilon^\flat_X }\swarrow && \searrow^{ \mathrlap{ \Pi Disc \Gamma \eta^{ʃ}_X } } && {}^{\mathllap{ \eta^{ʃ}_{\Gamma Disc \Pi X} }}\nearrow && \nwarrow^{ \mathrlap{ \eta^{\flat}_{\Pi X} } } \\ \Pi X && && \Pi Disc \Gamma Disc \Pi X && && \Pi X \\ & {}_{\mathllap{ \Pi \eta^{ʃ}_X }}\searrow && {}^{\mathllap{iso}}\swarrow_{\mathrlap{ \Pi \epsilon^{\flat}_{Disc \Pi X} }} && {}_{\mathllap{ \Pi Disc \eta^\flat_{\Pi X} }}\nwarrow^{\mathrlap{iso}} && \nearrow_{\mathrlap{ \epsilon^{ʃ}_{\Pi X} }} \\ && \Pi Disc \Pi X && \underset{id_{\Pi Disc \Pi X}}{\longleftarrow} && \Pi Disc \Pi X }$$

This commutes, because:

1. the left square is the image under $\Pi$ of naturality for $\epsilon^\flat$ on $\eta^{ʃ}_X$;

2. the top square is naturality for $\epsilon^{ʃ}$ on $\Gamma \eta^{ʃ}_X$;

3. the right square is naturality for $\epsilon^{ʃ}$ on $\eta^{\flat}_{\Pi X}$;

4. the bottom commuting triangle is the image under $\Pi$ of the zig-zag identity for $(Disc \dashv \Gamma)$ on $\Pi X$.

Moreover, notice that

1. the total bottom composite is the identity morphism $id_{\Pi X}$, due to the zig-zag identity for $(Disc \dashv \Gamma)$;

2. also the other two morphisms in the bottom triangle are isomorphisms, as shown, due to the idempoency of the $(Disc-\Gamma)$-adjunction

Therefore the total composite from $\Pi Disc \Gamma X \to \Gamma Disc \Pi X$ along the bottom part of the diagram equals the left hand side of (1), while the composite along the top part of the diagram clearly equals the right hand side of (1).

Proposition

(points-to-pieces transform)

Consider an adjoint triple of the form

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

(for instance a cohesive topos over some base topos $\mathbf{B}$).

Then for all $X \in \mathbf{H}$ the following two natural transformations, constructed from the adjunction units/counits and their inverse morphisms (using idempotency), are equal:

(2)$$ptp_{\mathbf{B}} \;\;\coloneqq\;\; \left( \Pi \epsilon^\flat_X \right) \circ \left( \eta^{ʃ}_{\Gamma X} \right)^{-1} \;\;=\;\; \left( \eta^\flat_{\Pi X} \right)^{-1} \circ \left( \Gamma \eta^{ʃ}_X \right) \phantom{AAAAAAA} \array{ \Gamma X & \overset{ \Gamma \eta^{ʃ}_X }{\longrightarrow} & \Gamma Disc \Pi X \\ {}^{ \mathllap{ \left( \eta^{ʃ}_{\Gamma X} \right)^{-1} } }\big\downarrow & \searrow^{ \mathrlap{ ptp_{\mathbf{B}} } } & \big\downarrow^{ \mathrlap{ \left( \eta^\flat_{\Pi X} \right)^{-1} } } \\ \Pi Disc \Gamma X &\underset{ \Pi \epsilon^\flat_X }{\longrightarrow}& \Pi X }$$

Moreover, the image of these morphisms under $Disc$ equals the following composite:

(3)$$ptp_{\mathbf{H}} \;\colon\; \flat X \overset{ \phantom{A} \epsilon^{\flat}_X \phantom{A} }{\longrightarrow} X \overset{ \phantom{A} \eta^{ʃ}_X \phantom{A} }{\longrightarrow} ʃ X \,,$$

hence

(4)$$ptp_{\mathbf{H}} \;=\; Disc(ptp_{\mathbf{B}}) \,.$$

Either of these morphisms we call the points-to-pieces transform.

The first statement is due to (Johnstone 11, Corollary 2.2).

Proof

The first statement follows directly from Lemma .

For the second statement, notice that the $(Disc \dashv \Gamma)$-adjunct of

$$ptp_{\mathbf{H}} \;\colon\; Disc \Gamma X \overset{ \phantom{A} \epsilon^{\flat}_X \phantom{A} }{\longrightarrow} X \overset{ \phantom{A} \eta^{ʃ}_X \phantom{A} }{\longrightarrow} Disc \Pi X$$

is

(5)$$\widetilde{ ptp_{\mathbf{H}} } \;\;=\;\; \underset{ = id_{\Gamma X} }{ \underbrace{ \Gamma X \underoverset{iso}{ \phantom{A} \eta^{\flat}_{\Gamma X} \phantom{A} }{ \longrightarrow } \Gamma Disc \Gamma X \underoverset{iso}{ \phantom{A} \Gamma \epsilon^{\flat}_X \phantom{A} }{\longrightarrow} \Gamma X }} \overset{ \phantom{A} \Gamma \eta^{ʃ}_X \phantom{A} }{\longrightarrow} \Gamma Disc \Pi X \,,$$

where under the braces we uses the zig-zag identity.

(As a side remark, for later usage, we observe that the morphisms on the left in (5) are isomorphisms, as shown, by idempotency of the adjunctions.)

From this we obtain the following commuting diagram:

$$\array{ Disc \Gamma X &\overset{ \phantom{A} Disc \Gamma \eta^{ʃ}_X \phantom{A} }{\longrightarrow}& Disc \Gamma Disc \Pi X &\underoverset{iso}{ \phantom{A} Disc \left(\eta^{ \flat }_{\Pi X}\right)^{-1} \phantom{A} }{ \longrightarrow }& Disc \Pi X \\ &{}_{\mathllap{ ptp_{\mathbf{H}} }}\searrow& {}^{ \mathllap{ \epsilon^{\flat}_{Disc \Pi X} } } \big\downarrow^{\mathrlap{\simeq}} & \nearrow_{\mathrlap{id_{\Pi X}}} \\ && Disc \Pi X }$$

Here:

1. on the left we identified $\widetilde {\widetilde {ptp_{\mathbf{H}}}} \;=\; ptp_{\mathbf{H}}$ by applying the formula for $(Disc \dashv \Gamma)$-adjuncts to $\widetilde {ptp_{\mathbf{H}}} = \Gamma \eta^{ʃ}_X$ (5);

2. on the right we used the zig-zag identity for $(Disc \dashv \Gamma)$.

This proves the second statement.

Lemma

Consider an adjoint triple

$$Disc \dashv \Gamma \dashv coDisc \;\;\colon\;\; \mathbf{H} \array{ \overset{\phantom{AA} Disc \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\longleftarrow} } \mathbf{B}$$

Then application of the functor $\Gamma$ on any morphism $\mathbf{X} \overset{f}{\to} \mathbf{Y} \;\;\in \mathbf{H}$ is equal both to the operations of

1. pre-composition with the $(Disc \dashv \Gamma)$-adjunction counit $\epsilon^\flat_{\mathbf{X}}$, followed by passing to the $(Disc \dashv \Gamma)$-adjunct;

2. post-composition with the $(\Gamma \dashv coDisc)$-adjunction unit $\eta^{ \sharp }_{\mathbf{Y}}$, followed by passing to the $(\Gamma \dashv coDisc)$-adjunct:

(6)$$\widetilde{\eta^\sharp_{\mathbf{Y}} \circ (-)} \;=\; \Gamma_{\mathbf{X}, \mathbf{Y}} \;=\; \widetilde{ (-) \circ \epsilon^\flat_{\mathbf{X}} } \,.$$

Proof

For the first equality, consider the following naturality square for the adjunction hom-isomorphism (this Def.):

$$\array{ Hom_{\mathbf{B}}( \Gamma \mathbf{Y} , \Gamma \mathbf{Y} ) &\overset{\widetilde {(-)}}{\longrightarrow}& Hom_{\mathbf{H}}( \mathbf{Y}, coDisc \Gamma \mathbf{Y} ) \\ {}^{\mathllap{ Hom_{\mathbf{B}}(\Gamma(f), \Gamma \mathbf{Y}) }} \big\downarrow && \!\!\!\!\! \big\downarrow^{\mathrlap{ Hom_{\mathbf{H}}( f, coDisc \Gamma \mathbf{Y} ) }} \\ Hom_{\mathbf{B}}( \Gamma \mathbf{X}, \Gamma \mathbf{Y} ) &\overset{\widetilde{ (-) }}{\longleftarrow}& Hom_{\mathbf{H}}( \mathbf{X}, coDisc \Gamma \mathbf{Y} ) } \phantom{AAAAA} \array{ \{ \Gamma \mathbf{Y} \overset{id_{\Gamma \mathbf{Y}}}{\to} \Gamma \mathbf{Y}\} &\longrightarrow& \{ \mathbf{Y} \overset{\eta^\sharp_{\mathbf{Y}}}{\to} coDisc \Gamma \mathbf{Y} \} \\ \big\downarrow && \big\downarrow \\ \{ \Gamma \mathbf{X} \overset{\Gamma(f)}{\to} \Gamma \mathbf{Y} \} &\longleftarrow& \{ \mathbf{X} \overset{\eta^\sharp_{\mathbf{Y}} \circ f}{\longrightarrow} coDisc \Gamma \mathbf{Y} \} }$$

Chasing the identity morphism $id_{\Gamma \mathbf{Y}}$ through this diagram, yields the claimed equality, as shown on the right. Here we use that the right adjunct of the identity morphism is the adjunction unit, as shown.

The second equality is fomally dual:

$$\array{ Hom_{\mathbf{B}}( \Gamma \mathbf{X}, \Gamma \mathbf{X}) &\overset{\widetilde { (-) }}{\longrightarrow}& Hom_{\mathbf{H}}( Disc \Gamma \mathbf{X} , \mathbf{X}) \\ {}^{\mathllap{ Hom_{\mathbf{B}}( \Gamma \mathbf{X}, \Gamma(f) ) }} \big\downarrow && \big\downarrow^{ \mathrlap{ Hom_{\mathbf{X}}( Disc \Gamma \mathbf{X}, f ) } } \\ Hom_{\mathbf{B}}( \Gamma \mathbf{X}, \Gamma \mathbf{Y} ) &\overset{ \widetilde{ (-) } }{\longleftarrow}& Hom_{\mathbf{H}}( Disc \Gamma \mathbf{X}, \mathbf{Y} ) } \phantom{AAAAA} \array{ \{ \Gamma \mathbf{X} \overset{id_{\Gamma \mathbf{X}}}{\to} \Gamma \mathbf{X} \} &\longrightarrow& \{ Disc \Gamma \mathbf{X} \overset{\epsilon^{\flat}_X}{\to} \mathbf{X} \} \\ \big\downarrow && \big\downarrow \\ \{ \Gamma \mathbf{X} \overset{\Gamma(f)}{\to} \Gamma(\mathbf{Y}) \} &\longleftarrow& \{ Disc \Gamma \mathbf{X} \overset{f\circ \epsilon^\flat_{\mathbf{X}} }{\longrightarrow} \mathbf{Y}\} }$$

Proposition

(pieces have points $\simeq$ discrete objects are concrete)

Consider an adjoint quadruple of the form

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

(for instance a cohesive topos over some base topos $\mathbf{B}$).

Then the following are equivalent:

1. pieces have points:

   1. as seen in $\mathbf{B}$: For every object $X \in \mathbf{X}$ the points-to-pieces transform (Prop. ) in $\mathbf{B}$ (2) is an epimorphism:
   $$ptp_{\mathbf{B}} \;\colon\; \Gamma X \overset{ epi }{\longrightarrow} \Pi X$$
   1. equivalently, as seen in $\mathbf{H}$: For every object $X \in \mathbf{X}$ the points-to-pieces transform (Prop. ) in $\mathbf{H}$ (3) is an epimorphism:
   $$ptp_{\mathbf{H}} \;\colon\; \flat X \overset{ epi }{\longrightarrow} ʃ X$$

2. discrete objects are concrete: For every object $S \in \mathbf{B}$ the discrete object $Disc(S)$ is a concrete object, in that the sharp adjunction counit on $Disc(S)$ is a monomorphism:

   $$\eta^\sharp_{Disc(S)} \;\colon\; Disc S \overset{ mono }{\longrightarrow} \sharp Disc S$$

Proof

First observe the equivalence of the first two statements:

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

In one direction, assume that $ptp_{\mathbf{B}}$ is an epimorphism. By (4) we have $ptp_{\mathbf{H}} = Disc(ptp_{\mathbf{B}})$, but $Disc$ is a left adjoint and left adjoints preserve epimorphisms.

In the other direction, assume that $ptp_{\mathbf{H}}$ is an epimorphism. By (2) and (5) 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 epimorphisms, as does composition with isomorphisms.

By applying (2) again, we find in particular that pieces have points is also equivalent to $\Pi \epsilon^\flat_{Disc S}$ being an epimorphism, for all $S \in \mathbf{B}$. But this is equivalent to

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

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

Now by Lemma , this is equivalent to

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

being a monomorphism, which is equivalent to $\eta^\sharp_{Disc(S)}$ being a monomorphism, hence to discrete objects are concrete.

Example

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

Example

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

This condition pieces have points may also be expressed as follows:

Proposition

If $\flat \to \int$ is epi, then there is Aufhebung of the initial opposition, in that $\sharp \emptyset \simeq \emptyset$. Conversely if this holds and the base topos is a Boolean topos, then $\flat \to \int$ is epi.

This is Lawvere-Menni 15, lemma 4.1, 4.2.

Definition

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

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

is an isomorphism.

Remark

This morphism is the internal hom-adjunct of

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

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

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

Definition

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

$$f_!(\Omega) \simeq * \,.$$

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

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

Remark

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

Another axiom to consider, also introduced by Lawvere (see at Aufhebung):

Definition

Say that a cohesive topos, def. , has Aufhebung of becoming if the sharp modality, def. , preserves the initial object

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

Example

The Aufhebungs axiom def. is satisfied by all cohesive toposes with a cohesive site of definition, see at Aufhebung – Over cohesive sites.

Remark

In the Aufhebungs-axiom of def. , the extra exactness condition on the shape modality in def. , which says (in particular) that shape preserves the terminal object

$$ʃ \ast \simeq \ast$$

finds a dual companion. It might make sense to consider the variant of the axioms of cohesion which say

1. there is an adjoint triple of idempotent (co-)monads $ʃ \dashv \flat \dashv \sharp$;

2. such that $\sharp$ satisfies Aufhebung and $ʃ$ satisfies co-Aufhebung.

Properties

We discuss properties of cohesive toposes. In

• Relations between the axioms

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

• Quasitopos of concrete objects

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

• Objects with one point per cohesive piece

the induced subcategory of objects with one point per piece.

Some of these phenomena have a natural

• Modality interpretation.

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

• cohesive (∞,1)-topos – structures

For more structure available with a few more axioms see at

• differential cohesion

• tangent cohesion.

$\,$

$\,$

Relations between the axioms

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

Proposition

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

This is just a reformulation of the above proposition.

Proposition

A sheaf topos that

1. is locally connected and connected;

2. satisfies pieces have points

also is

1. local;

2. hyperconnected;

3. strongly locally connected

4. cohesive.

The statement of the first items appears as (Johnstone 11, prop. 1.6). The last item is then a consequence by definition.

Proposition

For a sheaf topos the condition that it

1. is locally connected;

2. is connected;

3. satisfies pieces have points

is equivalent to the condition that it

1. is locally connected;

2. is hyperconnected;

3. is local.

This is (Johnstone 11, theorem 3.4).

Subcategories of discrete and codiscrete objects

Proposition

The reflective subcategories of discrete objects and of codiscrete objects are both exponential ideals.

Proof

By the discussion at exponential ideal a reflective subcategories of a cartesian closed category is an exponential ideal precisely if the reflector preserves products. For the codiscrete objects the reflector $\Gamma$ preserves even all limits and for the discrete objects the reflector $\Pi$ does so by assumption of strong connectedness.

Quasitoposes of concrete objects

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

Observation

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

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

Proof

By general properties of local toposes. See there.

Observation

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

$$X \to coDisc \Gamma X$$

is an isomorphism.

Proof

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

Definition

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

Write

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

for the full subcategory on concrete objects.

Proposition

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

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

Proof

Observe that the composite morphism

$$F : \mathcal{E}(X,Y) \stackrel{\Gamma}{\longrightarrow} \mathcal{S}(\Gamma X, \Gamma Y) \stackrel{\simeq}{\longrightarrow} \mathcal{E}(X, coDisc \Gamma Y)$$

is given (see adjunct) by postcomposition with the $(\Gamma \dashv coDisc)$-unit $\eta_Y : Y \to coDisc \Gamma Y$

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

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

Remark

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

Proposition

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

Proof

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

$$Set \simeq Sh_K(C) \stackrel{CoDisc}{\hookrightarrow} Sh_J(C) \hookrightarrow PSh(C)$$

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

Proposition

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

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

Proof

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

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

Objects with one point per cohesive piece

Theorem

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

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

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

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

2. $s_*$ preserves coproducts.

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

This is theorem 2 in (Lawvere).

Proof

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

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

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

(…details…)

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

$$\array{ \Gamma Y &\to& \Gamma X \\ \downarrow && \downarrow^{\mathrlap{\simeq}} \\ \Pi Y &\to& \Pi X }$$

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

(…details…).

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

(…details…)

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

(…details…)

Internal modal logic

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

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

Definition

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

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

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

Remark

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

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

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

Because if so, then

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

is an isomorphism and hence

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

is for all $X$. Therefore in this case $\Gamma \phi \to \Gamma A$ is an isomorphism and hence so is $# \phi \to # A$.

Example

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

Examples

Over a discrete site

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

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

Families of sets – the Sierpinski topos

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

Consider the site given by the interval category

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

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

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

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

since a presheaf $X$ on $C$ is given by a morphism

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

in Set. We find

• $\Gamma : (I \leftarrow S) \mapsto S$

• $\Pi_0 : (I \leftarrow S) \mapsto I$.

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

Moreover we find for $K \in Set$:

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

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

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

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

The canonical morphism

$$\Gamma (I \leftarrow S) \to \Pi_0 (I \leftarrow S)$$

is

$$\Gamma Disc \Gamma (I \leftarrow S) \to \Gamma (I \leftarrow S) \to \Gamma Disc \Pi_0 (I \leftarrow S) \,.$$

Plugging in the above this is just

$$S \to I$$

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

Over a site with initial and terminal object

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

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

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

Observation

If $C$ has a terminal object $*$ then

• the colimit is given by evaluation on this object;

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

  given by

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

Proof

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

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

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

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

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

The formal dual of this statement is the following.

Corollary

If $C$ has an initial object $\emptyset$ then

• the limit is given by evaluation on this object;

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

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

In summary we have

Proposition

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

$$(\Pi_0 \dashv Disc \dashv \Gamma \dashv CoDisc) : PSh(C) \to Set \,,$$

where

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

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

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

$$X(\emptyset \to *) = (\Pi_0(X) = X(\emptyset) \leftarrow X(*) = \Gamma(X))$$

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

$$X(\emptyset \to c \to *) = (\Pi_0(X) \leftarrow X(c) \leftarrow \Gamma(X)) \,.$$

Reflexive directed graphs

Proposition

The category $RDGraph$ of reflexive directed graphs is a cohesive topos.

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

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

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

Definition

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

• $\sigma \circ \sigma = \sigma$

• $\tau \circ \tau = \tau$

• $\tau \circ \sigma = \tau$

• $\sigma \circ \tau = \sigma$

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

$$s := X(\sigma) : X \to X$$

sends each edge to its source and the projection

$$t := X(\tau) : X \to X$$

sends each edge to its target. The identities

$$s \circ t = t$$

and

$$t \circ s = s$$

express the fact that source and target are identity edges.

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

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

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

In summary this shows that

Observation

We have an equivalence of categories

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

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

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

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

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

$$X_1 \stackrel{\overset{t}{\to}}{\underset{s}{\to}} X_0 \to \Pi_0 X \,.$$

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

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

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

Simplicial sets

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

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

Notice that reflexive directed graphs are equivalently skeleta of simplicial sets.

Proposition

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

We have for $X \in sSet$

• $\Gamma : X \mapsto X_0$;

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

And for $S \in Set$:

• $Disc S$ the constant simplicial set on $S$;

• $Codisc S$ the simplicial set which in degree $k$ has the set of $(k+1)$-tuples of elements of $S$.

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

$$Ho_{KanCplx}(X,Y) = \Pi_0(Y^X) \,.$$

Over a cohesive site

We discuss here presentations of cohesive toposes as categories of sheaves over sites equipped with suitable extra properties: “cohesive sites” (Def. ) below. The definition is readily motivated from the following basic Example , which constitutes the special case of cohesive sites with trivial coverage:

Example

(adjoint quadruple of presheaves over site with terminal objects)

Let $\mathcal{C}$ be a small category 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 of functors between the category of presheaves over $\mathcal{C}$ (this def.), and the category of sets (this Def.)

(7)$$[\mathcal{C}, Set] \array{ \overset{\phantom{AAA} \Pi_0 \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } Set$$

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:

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

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

3. $\Pi_0$ preserves finite products:

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

   $$\Pi_0(\mathbf{X} \times \mathbf{Y}) \;\simeq\; \Pi_0(\mathbf{X}) \times \Pi_0(\mathbf{Y}) \,.$$

Hence the category of presheaves over a small category with finite products is a cohesive topos (Def. ).

Proof

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

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

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

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

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

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

The former is manifestly the operation of evaluating on the terminal object. Moreover, since the terminal object inclusion is manifestly a fully faithful functor (this def.), it follows that also its left Kan extension $Disc$ is fully faithful (this prop.). This implies that also $coDisc$ is fully faithful, by this prop..

Equivalently, $Disc \simeq p^\ast$ is the constant diagram-assigning functor. By uniqueness of adjoints (this prop.) implies that $\Pi_0$ is the functor that sends a presheaf, regarded as a functor $\mathbf{Y} \;\colon\; \mathcal{C}^{op} \to Set$, to its colimit

(9)$$\Pi_0(\mathbf{Y}) \;=\; \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{Y} \,.$$

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

Example suggests to ask for coverages on categories with finite products which are such that the adjoint quadruple (7) on the category of presheaves (co-)restricts to the corresponding category of sheaves. The following Definition states a sufficient condition for this to be the case:

Definition

(cohesive site)

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

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

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

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

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

   2. the set of connected components of the groupoid obtained as the limit of the Cech groupoid is equivalent to the set of points of $X$, regarded as a groupoid:

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

This definition is designed to make the following true:

Proposition

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

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

(10)$$Sh(\mathcal{C}) \array{ \overset{\phantom{AAA} \Pi_0 \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } Set$$

Proof

With Example it only remains to be shown that for each set $S$ the presheaves $Disc(S)$ and $coDisc(S)$ are indeed sheaves.

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

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

The sheaf condition (in this form) 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 (9) 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 (this Lemma) to replace $\underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} y(X) \simeq \ast$.

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

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

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

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

is already an isomorphism (here: bijection) itself.

Similarly, the sheaf condition (in this form) 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 (8), 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.

Smooth sets

Example

(smooth sets form a cohesive topos)

The category $SmoothSet$ of smooth sets (this Def.) is a cohesive topos (Def. )

(11)$$SmoothSet \array{ \overset{\phantom{AAA} \Pi_0 \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\longleftarrow} } Set$$

Proof

First of all (by this Prop) smooth sets indeed form a sheaf topos, over the site CartSp of Cartesian spaces $\mathbb{R}^n$ with smooth functions between them, and equipped with the coverage of differentiably-good open covers (this def.)

$$SmoothSet \simeq Sh(CartSp) \,.$$

Hence, by Prop. , it is now sufficient to see that CartSp is a cohesive site (Def. ).

It clearly has finite products: The terminal object is the point, given by the 0-dimensional Cartesian space

$$\ast = \mathbb{R}^0$$

and the Cartesian product of two Cartesian spaces is the Cartesian space whose dimension is the sum of the two separate dimensions:

$$\mathbb{R}^{n_1} \times \mathbb{R}^{n_2} \;\simeq\; \mathbb{R}^{ n_1 + n_2 } \,.$$

This establishes the first clause in Def. .

For the second clause, consider a differentiably-good open cover $\{U_i \overset{}{\to} \mathbb{R}^n\}$ (this def.). This being a good cover implies that its Cech groupoid is, as an internal groupoid (via this remark), of the form

(12)$$C(\{U_i\}_i) \;\simeq\; \left( \array{ \underset{i,j}{\coprod} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} y(U_i) } \right) \,.$$

where we used the defining property of good open covers to identify $y(U_i) \times_X y(U_j) \simeq y( U_i \cap_X U_j )$.

The colimit of (12), regarded just as a presheaf of reflexive directed graphs (hence ignoring composition for the moment), is readily seen to be the graph of the colimit of the components (the universal property follows immediately from that of the component colimits):

(13)$$\begin{aligned} \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} C(\{U_i\}_i) & \simeq \left( \array{ \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} \underset{i,j}{\coprod} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} \underset{i}{\coprod} y(U_i) } \right) \\ & \simeq \left( \array{ \underset{i,j}{\coprod} \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} y(U_i) } \right) \\ & \simeq \left( \array{ \underset{i,j}{\coprod} \ast \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} \ast } \right) \end{aligned} \,.$$

Here we first used that colimits commute with colimits, hence in particular with coproducts (this prop.) and then that the colimit of a representable presheaf is the singleton set (this Lemma).

This colimiting graph carries a unique composition structure making it a groupoid, since there is at most one morphism between any two objects, and every object carries a morphism from itself to itself. This implies that this groupoid is actually the colimiting groupoid of the Cech groupoid: hence the groupoid obtained from replacing each representable summand in the Cech groupoid by a point.

Precisely this operation on Cech groupoids of good open covers of topological spaces is what Borsuk's nerve theorem is about, a classical result in topology/homotopy theory. This theorem implies directly that the set of connected components of the groupoid (14) is in bijection with the set of connected components of the Cartesian space $\mathbb{R}^n$, regarded as a topological space. But this is evidently a connected topological space, which finally shows that, indeed

$$\pi_0 \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} C(\{U_i\}_i) \;\simeq\; \ast \,.$$

The second item of the second clause in Def. follows similarly, but more easily: The limit of the Cech groupoid is readily seen to be, as before, the unique groupoid structure on the limiting underlying graph of presheaves. Since $CartSp$ has a terminal object $\ast = \mathbb{R}^0$, which is hence an initial object in the opposite category $CartSp^{op}$, limits over $CartSp^{op}$ yield simply the evaluation on that object:

(14)$$\begin{aligned} \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} C(\{U_i\}_i) & \simeq \left( \array{ \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} \underset{i,j}{\coprod} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} \underset{i}{\coprod} y(U_i) } \phantom{A} \right) \\ & \simeq \left( \array{ \underset{i,j}{\coprod} Hom_{CartSp}\left( \ast, U_i \underset{\mathbb{R}^n}{\cap} U_j \right) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} Hom_{CartSp}( \ast, U_i ) } \right) \end{aligned} \,.$$

Here we used that colimits (here coproducts) of presheaves are computed objectwise, and then the definition of the Yoneda embedding $y$.

But the equivalence relation induced by this graph on its set of objects $\underset{i}{\coprod} Hom_{CartSp}( \ast, U_i )$ precisely identifies pairs of points, one in $U_i$ the other in $U_j$, that are actually the same point of the $\mathbb{R}^n$ being covered. Hence the set of equivalence classes is the set of points of $\mathbb{R}^n$, which is just what remained to be shown:

$$\pi_0 \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} C(\{U_i\}_i) \;\simeq\; Hom_{CartSp}(\ast, \mathbb{R}^n) \,.$$

Diffeological spaces

Proposition

The quasitopos of concrete objects in the cohesive topos of smooth sets (Example ) is the category of diffeological spaces.

$$\array{ Conc(Sh(CartSp)) &\hookrightarrow& Sh(CartSp) \\ = && = \\ DiffeolSp &\hookrightarrow& SmoothSets }$$

Proof

A sheaf $X$ on $CartSp$ is a separated presheaf with respect to the further localization given by $CoDisc$ precisely if the canonical morphism (the unit of $(\Gamma \dashv CoDisc)$)

$$X \to CoDisc \Gamma X$$

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

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

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

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

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

Formally smooth sets

Proposition

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

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

(…)

Cohesive over-toposes

Proposition

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

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

1. $X$ is connected $\Pi_0(X) \simeq *$;

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

1. $X$ is a tiny object.

Infinitesimal thickening and formal smoothness

See at differential cohesion. Examples include the Cahiers topos

Example

Consider a full subcategory inclusion

$$Disc^{\mathcal{P}} \;\colon\; \mathcal{S} \hookrightarrow \mathcal{P}$$

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

$$\Pi_0^{\mathcal{P}} \simeq \Gamma^{\mathcal{P}} \,.$$

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

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

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

See at infinitesimal cohesion for more on this.

Counter-examples

Example

Counter-Example

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

$$PSh(\mathbf{B}G) \simeq G Set$$

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

$$(\Pi_0 \dashv Const \dashv \lim_\leftarrow) : G Set \stackrel{\overset{\lim_\to}{\to}}{\stackrel{\overset{Const}{\leftarrow}}{\underset{\lim_\leftarrow}{\to}}} Set \,.$$

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

$$\Pi_0(G) \simeq *$$

but

$$\Pi_0(G \times G) \simeq \bar n \,,$$

where $G$ denotes the fundamental representation of $G$ on itself. Therefore $\Pi_0$ does not preserve products in this case.

Related concepts

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

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

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

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

• local topos / local (∞,1)-topos

  • Pi modality$\dashv$ flat modality $\dashv$ sharp modality

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

and

• locally connected site, locally ∞-connected site

• connected site

• local site

• cohesive site

References

The axioms for a cohesive topos originate around

• William Lawvere, Categories of spaces may not be generalized spaces, as exemplified by directed graphs, Revista Colombiana de Matematicas XX (1986) 179-186, reprinted as: Reprints in Theory and Applications of Categories, 9 (2005) 1-7 [tac:tr9]

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

This appears maybe first in

• William Lawvere, Cohesive Toposes and Cantor's "lauter Einsen", Philosophia Mathematica 2 1 (1994) 5-15 [doi:10.1093/philmat/2.1.5, pdf]

The term “cohesion” and parts of its later axiomatization (p. 245) appears thoughout section C.1 of

• William Lawvere, Robert Rosebrugh, Sets for Mathematics, Cambridge University Press (2003) [doi:10.1017/CBO9780511755460, book homepage, pdf]

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

• William Lawvere, Axiomatic cohesion, Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. (tac:19-03, pdf)

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

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

In

• William Lawvere, Some Thoughts on the Future of Category Theory, Como 1991

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

• Georg Hegel, Science of Logic (see On being and becoming ).

and specifically the term “cohesion” probably from

• Georg Hegel, part II “Philosophy of Nature”, second section “Physics”, B c) “cohesion”, in Encyclopedia of the Philosophical Sciences

In

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

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

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

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

on page 14.

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

• William Lawvere, Cohesive Toposes – Combinatorial and Infinitesimal Cases Como (2008)

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

The notion of “cohesion” was explored earlier in

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

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

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

• Peter Johnstone, Remarks on punctual local connectedness, Theory and Applications of Categories, 25 3 (2011) 51-63 [tac:25-03]

Discussion of “sufficient cohesion” is in

• Matías Menni, Continuous Cohesion over sets, Theory and Applications of Categories, Vol. 29, 2014, No. 20, pp 542-568. (TAC, pdf)

• Matías Menni, Sufficient Cohesion over Atomic Toposes , Cah. Top.Géom. Diff. Cat. LV (2014) (preprint)

Discussion of relation to double negation topology is in

• William Lawvere, Matías Menni, Internal choice holds in the discrete part of any cohesive topos satisfying stable connected codiscreteness, Theory and Applications of Categories, Vol. 30, 2015, No. 26, pp 909-932. (TAC)

And further relations of cohesion to Birkhoff's theorem in universal algebra is given in

• William Lawvere, Birkhoff’s Theorem from a geometric perspective: A simple example, to appear in Categories and General Algebraic Structures with Applications (2016), journal page.

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

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

under the name Q-categories.

The internal logic of local toposes is discussed in

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

Notation and terminology above, and generalization to differential cohesion etc., follows:

• Urs Schreiber, differential cohomology in a cohesive topos (arXiv:1310.7930)

Exposition:

• Fosco Loregian, Cohesion in Rome, talk notes, Dec. 2019 (pdf, pdf)