nLab
cohesive topos

Context

Cohesive $∞$ -Toposes

Discrete and concrete objects

Topos Theory

Idea

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

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

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

Conversely, the left adjoint and right adjoint of $Γ$

ℰ \overset{\overset{Disc}{\leftarrow}}{\overset{\overset{Γ}{\to}}{\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) .

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

In total this gives an adjoint quadruple

(Π_{0} ⊣ Disc ⊣ Γ ⊣ Codisc) : ℰ \overset{\overset{\overset{Π}{\to}}{\overset{Disc}{\leftarrow}}}{\overset{\underset{Γ}{\to}}{\underset{Codisc}{\leftarrow}}} Set

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

Notice that most objects in a cohesive topos are far from being just sets with extra structure: while the functor $Γ$ 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 $Γ (X)$ has very few points (possibly none, with the axioms so far). The subcategory of objects in $E$ that we may think of as point sets equipped with extra structure is the quasitopos ${Conc}_{Γ} (E)$ of the concrete sheaves inside $E$

Set \overset{\overset{Γ}{\leftarrow}}{\underset{Codisc}{↪}} {Conc}_{Γ} (ℰ) \overset{\overset{Conc}{\leftarrow}}{↪} ℰ .

It is the fact that $ℰ$ is a local topos that allows to identify ${Conc}_{Γ} (E)$ .

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

Γ X \to Π_{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:

Π_{0} (X^{S}) \overset{≃}{\to} (Π_{0} (X))^{S} .

For more general products, it would be a similar map $Π_{0} (\prod_{i} X_{i}) \to \prod_{i} Π_{0} (X_{i})$ . See the examples at cohesive site for concrete illustrations of these ideas.

Definition

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

(f^{*} ⊣ f_{*}) : ℰ \overset{\overset{f^{*}}{\leftarrow}}{\underset{f_{*}}{\to}} 𝒮

is cohesive if it is a

strongly connected topos
and a local topos.

In detail this means that it has the following properties:

it is a locally connected topos: there exists a further left adjoint $(f_{!} ⊣ f^{*})$ satisfying a suitable condition;
it is a connected topos: the functor $f_{!}$ preserves the terminal object, or equivalently $f^{*}$ is fully faithful;
it is strongly connected : $f_{!}$ preserves even all finite products;
it is a local topos: there exists a further right adjoint $(f_{*} ⊣ f^{!})$ (this is sufficient for $f$ to be local, since we have already assumed it to be connected);

In summary $ℰ$ is cohesive over $𝒮$ if we have a quadruple of adjoint functors

(f_{!} ⊣ f^{*} ⊣ f_{*} ⊣ f^{!}) : ℰ \overset{\overset{\overset{f_{!}}{\to}}{\overset{f^{*}}{↩}}}{\overset{\underset{f_{*}}{\to}}{\underset{f^{!}}{↩}}} 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 coreflective and one also reflective.

Since a topos is a cartesian closed category it follows with the discussion here that both of these are exponential ideals. In fact the condition that the $f^{*}$ -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

(Π_{0} ⊣ Disc ⊣ Γ ⊣ coDisc) : ℰ \overset{\overset{\overset{Π_{0}}{\to}}{\overset{Disc}{↩}}}{\overset{\underset{Γ}{\to}}{\underset{coDisc}{↩}}} 𝒮 .

Remark

The above adjoint quadruple canonically induces an adjoint triple of endofunctors on $ℰ$

(Π_{0} ⊣ ♭ ⊣ #) : ℰ \overset{\overset{Π}{\to}}{\overset{\overset{Disc}{\leftarrow}}{\underset{Γ}{\to}}} 𝒮 \overset{\overset{Disc}{\to}}{\overset{\overset{Γ}{\leftarrow}}{\underset{coDisc}{\to}}} ℰ .

These endofunctors play a role in the internal characterization of cohesion.

There are several further axioms that one may want to impose in order to formalize the concept of cohesion.

Definition

For $f : ℰ \to 𝒮$ a cohesive topos, we say that pieces have points in $ℰ$ if the natural transformation

f_{*} X \overset{}{\to} f_{*} f^{*} f_{!} X \overset{≃}{\to} f_{!} X

is an epimorphism for all $X \in ℰ$ .

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

Definition

We say that discrete objects are concrete in $ℰ$ if the transformation

f^{*} S \to f^{!} f_{*} f^{*} S \overset{≃}{\to} f^{!} S

is a monomorphism for all $S \in 𝒮$ .

Definition

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

f_{!} (X^{f^{*} S}) \overset{≃}{\to} (f_{!} (X))^{S}

is an isomorphism.

Remark

This morphism is the internal hom-adjunct of

S \times f_{!} (X^{f^{*} S}) \overset{≃}{\to} f_{!} f^{*} (S) \times f_{!} (X^{f^{*} S}) \overset{≃}{\to} f_{!} (f^{*} (S) \times X^{f^{*} S}) \to 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 $f : ℰ \to 𝒮$ a cohesive topos, we say that its subobject classifier is contractible if for the subobject classifier $Ω \in ℰ$ we have

f_{!} (Ω) ≃ * .

This implies that for all $X \in ℰ$ also $f_{!} Ω^{X} ≃ *$ .

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

Remark

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

Properties

We discuss properties of cohesive toposes. We start in

Adjoint quadruples

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

Adjoint quadruples

Let $(p_{!} ⊣ p^{*} ⊣ p_{*} ⊣ p^{!}) : ℰ \to 𝒮$ be an adjoint quadruple of adjoint functors such that $p^{*}$ and $p^{!}$ are full and faithful functors. We record some general properties of such a setup.

We write

ι : id \to p^{*} p_{!}

etc. for units and

η : p_{!} p^{*} \to id

etc. for counits.

Proposition/Definition

We have commuting diagrams, natural in $X \in ℰ$ , $S \in 𝒮$

\begin{matrix} p_{*} X & \overset{η_{p^{*} X}^{- 1}}{\to} & p_{!} p^{*} p_{*} X \\ ^{p_{*} (i_{X})} ↓ & ↘^{θ_{X}} & ↓^{p_{!} (η_{X})} \\ p_{*} p^{*} p_{!} X & \overset{ι_{p_{!} X}^{- 1}}{\to} & p_{!} X \end{matrix}

and

\begin{matrix} p^{*} S & \overset{ι_{p^{*} S}}{\to} & p^{!} p_{*} p^{*} S \\ ^{p^{*} ϵ_{S}^{- 1}} ↓ & ↘^{ϕ_{X}} & ↓^{p^{!} (ι_{S}^{- 1})} \\ p^{*} p_{*} p^{!} S & \overset{ϵ_{p_{!} S}}{\to} & p^{!} S \end{matrix} .

where the diagonal morphisms

θ_{X} : p_{*} X \to p_{!} X

and

ϕ_{S} : p^{*} S \to p^{!} S

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

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

Proposition

The following conditions are equivalent:

for all $X \in ℰ$ the morphism $θ_{X} : p_{*} X \to p_{!} X$ is an epimorphism;
for all $S \in 𝒮$ ,, the morphism $ϕ_{S} : p^{*} S \to p^{!} S$ is a monomorphism;
$p_{*}$ is faithful on morphisms of the form $A \to p^{*} S$ .

This appears as (Johnstone, lemma 2.3).

Proof

By the above definition, $ϕ_{S}$ is a monomorphism precisely if $ι_{p^{*} S} : p^{*} S \to p^{!} p_{*} p^{*} S$ is. This in turn is so (see monomorphism) precisely if the first function in

ℰ (A, p^{*} X) \overset{(ι_{p^{*} X}) \circ (-)}{\to} ℰ (A, p^{!} p_{*} p^{*} S) \overset{≃}{\to} 𝒮 (p_{*} A, p_{*} p^{*} S)

and hence the composite is a monomorphism in Set.

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

(ι_{p^{*} X}) \circ (-) : (A \to p^{*} S) \mapsto p_{*} (A \to p^{*} S) .

Similarly, by the above definition the morphism $θ_{X}$ is an epimorphism precisely if $p_{!} (η_{X}) : p_{!} p^{*} p_{*} X \to p_{!} X$ is so, which is the case precisely if the top morphism in

\begin{matrix} 𝒮 (p_{!} X, S) & \overset{(-) \circ p_{!} (η_{X})}{\to} & 𝒮 (p_{!} p^{*} p_{*} X, S) \\ ^{≃} ↓ & ↓^{≃} \\ ℰ (p^{*} p_{*} X, p^{*} S) \\ ^{≃} ↓ & ↓^{≃} \\ ℰ (X, p^{*} S) & \overset{p_{*}}{\to} & 𝒮 (p_{*} X, p_{*} p^{*} S) \end{matrix}

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

Relations between the axioms

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

Proposition

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

This is just a reformulation of the above proposition.

Proposition

A sheaf topos that

is locally connected and connected;
satisfies pieces have points

also is

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

Proposition

For a sheaf topos the condition that it

is equivalent to the condition that it

This is (Johnstone, 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 $Γ$ preserves even all limits and for the discrete objects the reflector $Π$ does so by assumotion of strong connectedness.

Quasitoposes of concrete objects

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

Observation

For $(Π ⊣ Disc ⊣ Γ ⊣ coDisc) : ℰ \to Set$ a cohesive topos, $(Γ ⊣ coDisc)$ exhibits $Set$ as a subtopos

(Γ ⊣ coDisc) : Set \overset{\leftarrow}{↪} ℰ .

Proof

By general properties of local toposes. See there.

Observation

The category $Set$ is equivalent to the full subcategory of $ℰ$ on those objects $X \in ℰ$ for which the $(Γ ⊣ coDisc)$ unit

X \to coDisc Γ X

is an isomorphism.

Proof

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

Definition

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

Write

Conc (ℰ) ↪ ℰ

for the full subcategory on concrete objects.

Proposition

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

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

Proof

Observe that the composite morphism

F : ℰ (X, Y) \overset{Γ}{\to} 𝒮 (Γ X, Γ Y) \overset{≃}{\to} ℰ (X, coDisc Γ Y)

is given (see adjunct) by postcomposition with the $(Γ ⊣ coDisc)$ -unit $ι_{Y} : Y \to coDisc Γ Y$

\begin{matrix} X & \to & Y \\ ↓ & ↓ \\ coDisc Γ X & \to & coDisc Γ Y \end{matrix} .

The condition that $Y$ is a concrete object, hence that $Y \to coDisc Γ Y$ 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 $Conc (ℰ)$ as a category of sets equipped with cohesive structure .

Proposition

The category $Conc (ℰ)$ is a quasitopos and a reflective subcategory of $ℰ$ .

Proof

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

Set ≃ {Sh}_{K} (C) \overset{CoDisc}{↪} {Sh}_{J} (C) ↪ PSh (C)

which must correspond to another coverage $K$ : $Set ≃ {Sh}_{K} (C)$ . Since we have this sequence of inclusions, we have an inclusion of coverages $J \subset K$ . We observe that the concrete objects in $ℰ$ 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 $ℰ$ satisfies the axiom discrete objects are concrete (saying that for all $S \in Set$ the canonical morphism $Disc S \to coDisc Γ Disc S ≃ coDisc S$ is a monomorphism) then $Conc (ℰ)$ is a cohesive quasitopos in that we have a quadrupled of adjoint functors.

Conc (ℰ) \overset{\overset{Π}{\to}}{\overset{\overset{Disc}{\leftarrow}}{\overset{\overset{Γ}{\to}}{\underset{coDisc}{\leftarrow}}}} Set .

Proof

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

Since by reflectivity limits in $Conc (ℰ)$ may be computed in $ℰ$ , $Π$ preserves finite products on $Conc (ℰ)$ .

Objects with one point per cohesive piece

Theorem

Let $(Π ⊣ Disc ⊣ Γ ⊣ Codisc) : ℰ \to 𝒮$ be a cohesive topos with $Γ X \to Π X$ an epimorphism for all $X$ .

Let $s^{*} : ℒ ↪ ℰ$ be the full subcategory on those objects $X$ for which $Γ X \to Π X$ is an isomorphism. Then

$ℒ$ is a reflective subcategory and a coreflective subcategory

$(s_{!} ⊣ s^{*} ⊣ s_{*}) : ℰ \overset{\overset{s_{!}}{\to}}{\overset{\overset{s^{*}}{\leftarrow}}{\underset{s_{*}}{\to}}} ℒ$ (s_! \dashv s^* \dashv s_*) : \mathcal{E} \stackrel{\overset{s_!}{\to}}{\stackrel{\overset{s^*}{\leftarrow}}{\underset{s_*}{\to}}} \mathcal{L}
$s_{*}$ preserves coproducts.
the components of the reflector $X \to s_{!} s^{*} X$ are epimorphisms.

This is theorem 2 in (Lawvere).

Proof

Since $Γ$ is a left adjoint it preserves colimits, as does of course $Π$ . Therefore the collection of objects for which $Γ X \to Π X$ is an isomorphism is closed under colimits and hence $ℒ$ has all colimits and the inclusion $s^{*} : ℒ ↪ ℰ$ 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 $ℒ$ is a locally presentable category. To see this, notice that $ℒ ↪ ℰ$ is the inverter of $Γ \to Π$ , 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 $ℒ$ is accessible. Since we already know that it is also cocomplete it follows that it must be locally presentable.

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

(…details…)

Using that $Γ X \to Π X$ is an epi, we find that $ℒ$ is also closed under subobjects: if $Y ↪ X$ is a monomorphism then if in

\begin{matrix} Γ Y & \to & Γ X \\ ↓ & ↓^{≃} \\ Π Y & \to & Π X \end{matrix}

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

(…details…).

It then also follows that $ℒ$ 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 $ℰ$ comes with its internal logic. From this internal perspective, the existence of extra external functors on $ℰ$ – such as the $Π_{0}$ 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 ” $♭$ ” and ”#” are used precisely oppositely to their use here).

Definition

For $ϕ ↪ A$ 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 $# ϕ ∣_{A} \to A$ in

\begin{matrix} # ϕ ∣_{A} & \to & # ϕ \\ ↓ & ↓ \\ A & \to & # A \end{matrix}

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

Remark

If a proposition $ϕ$ is true over all discrete objects, then it is discretely true. More precisely, if for $X = ♭ X$ any discrete object we have that

ℰ (X, ϕ) \to ℰ (X, A)

is an isomorphism, then $ϕ$ is discretely true.

Because if so, then

ℰ (♭ X, ϕ) \to ℰ (♭ X, A)

is an isomorphism and hence

ℰ (Γ X, Γ ϕ) \to ℰ (Γ X, Γ A)

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

Example

For $ℰ = Sh (CartSp)$ the sheaf topos over CartSp $_{smooth}$ (smooth spaces) and $Ω_{cl}^{n} (-) ↪ Ω^{n} (-)$ the inclusion of all closed $n$ -differential forms into all $n$ -forms, the proposition is “the $n$ -form $ω$ 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) ≃ PSh (C) .

Families of sets – the Sierpinksi topos

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

Consider the site given by the interval category

C = {\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 *}) ≃ PSh ({\emptyset \to *}) ≃ Arr (Set)

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

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

in Set. We find

$Γ : (I \leftarrow S) \mapsto S$
$Π_{0} : (I \leftarrow S) \mapsto I$ .

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: $S = ∐_{i} S_{i}$ . The “cohesive pieces” are the $S_{i}$ and there are $∣ I ∣$ -many of them. This is what $Π_{0}$ computes, which clearly preserves products.

Moreover we find for $K \in Set$ :

$Disc : K \mapsto (K \overset{Id}{\leftarrow} K)$ ;
$CoDisc : K \mapsto (* \overset{}{\leftarrow} K)$

(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

Γ (I \leftarrow S) \to Π_{0} (I \leftarrow S)

Γ Disc Γ (I \leftarrow S) \to Γ (I \leftarrow S) \to Γ Disc Π_{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 presehaves $F : C^{op} \to Set$ along the functor $C^{op} \to *$ . By definition, this are the colimit and limit functors

(\lim_{\to} ⊣ Const ⊣ \lim_{\leftarrow}) : PSh (C) \overset{\overset{\lim_{\to}}{\to}}{\overset{\overset{Const}{\leftarrow}}{\underset{\lim_{\leftarrow}}{\to}}} Set .

Observation

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

the colimit is given by evaluation on this object;
there is a further right adjoint $(\lim_{\leftarrow} ⊣ CoConst)$

given by

$CoConst : S \mapsto (c \mapsto Set (C (*, c), S) .$ 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} \overset{F}{\to} Set) ≃ F (*) .

To see that we have a pair of adjoint functors $(\lim_{\to} ⊣ CoConst)$ we check the natural hom-set equivalence ${PSh}_{C} (F, CoConst S) ≃ Set (\lim_{\to} F, S)$ by computing

\begin{aligned} {PSh}_{C} (F, CoConst S) & ≃ \int_{c \in C} Set (F (c), Set (C (*, c), S)) \\ ≃ \int_{c \in C} Set (F (c) \times C (*, c), S) \\ ≃ Set (\int^{c \in C} F (c) \times C (*, c), S) \\ ≃ 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 $(L ⊣ \lim_{\to})$ ,

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

(Π_{0} ⊣ Disc ⊣ Γ ⊣ CoDisc) : PSh (C) \to Set,

where

$Γ$ is given by evaluation on $*$ ;
$Π_{0}$ 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 *) = (Π_{0} (X) = X (\emptyset) \leftarrow X (*) = Γ (X))

and $Γ$ picks out the total set of points, while $Π_{0}$ 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 *) = (Π_{0} (X) \leftarrow X (c) \leftarrow Γ (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 $B End (Δ [1])$ be the one-object category coming from the monoid with three idempotent elements ${Id, σ, τ}$

$σ \circ σ = σ$
$τ \circ τ = τ$
$τ \circ σ = τ$
$σ \circ τ = σ$

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

s : = X (σ) : X \to X

sends each edge to its source and the projection

t : = X (τ) : 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 $Δ_{\leq [1]} \subset Δ$ of the simplex category on the objects $[0]$ and $[1]$

Δ_{\leq [1]} = ([1] \overset{\overset{τ}{\leftarrow}}{\vec{\underset{σ}{\leftarrow}}} [0]) .

In fact this is the Cauchy completion of $B End (Δ [1])$ , obtained by splitting the idempotents.

In summary this shows that

Observation

We have an equivalence of categories

RDGraph ≃ PSh (B End (Δ [1])) ≃ PSh (Δ_{\leq [1]}) .

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

(Disc ⊣ Γ) : Psh (C) \to Set

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

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

X_{1} \overset{\overset{t}{\to}}{\underset{s}{\to}} X_{0} \to Π_{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 $Γ X \to Π_{0} X$ sends each vertex to its connected component. Evidently this is epi, hence in $RDGraphs$ cohesive pieces have points .

Simplicial sets

Reflexive directed graphs are equivalently skeleta of simplicial sets.

Proposition

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

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

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

We have for $X \in sSet$

$Γ : X \mapsto X_{0}$ ;
$Π_{0} : X \mapsto π_{0} (X) = X_{0} / X_{1}$ , 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 $Π_{0} (Y^{X})$ 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) = Π_{0} (Y^{X}) .

Over a cohesive site

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

Theorem

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

See cohesive site for examples.

Diffeological spaces

Proposition

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

Proposition

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

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 $(Γ ⊣ CoDisc)$ )

X \to CoDisc Γ X

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

Hom (U, X) ≃ X (U) \to Hom (U, CoDisc Γ 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 (Γ (U), Γ (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 $CartSp$ .

Smooth toposes

Proposition

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

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

Cohesive over-toposes

Proposition

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

A necessary conditions that the over topos $ℰ / X$ is a connected topos is that

$X$ is contractible $Π_{0} (X) ≃ *$ ;

Sufficient condition for $ℰ / X$ to be a local topos is that

$X$ is a tiny object.

Infinitesimal thickening and formal smoothness

See the section Formal smoothness at Q-category.

Counter-examples

Counter-Example

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

PSh (B G) ≃ G Set

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

(Π_{0} ⊣ Const ⊣ \lim_{\leftarrow}) : G Set \overset{\overset{\lim_{\to}}{\to}}{\overset{\overset{Const}{\leftarrow}}{\underset{\lim_{\leftarrow}}{\to}}} Set .

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

Π_{0} (G) ≃ *

but

Π_{0} (G \times G) ≃ \bar{n},

where $G$ denotes the fundamental representation of $G$ on itself. Therefore $Π_{0}$ does not preserve products in this case.

Related concepts

locally connected topos / locally ∞-connected (∞,1)-topos
local topos / local (∞,1)-topos
- Pi modality $⊣$ flat modality $⊣$ sharp modality
cohesive topos / cohesive (∞,1)-topos

and

References

The axioms for a cohesive topos originate in

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

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

This appears maybe first in

Bill Lawvere Cohesive toposes and Cantor’s “lauter Einsen” Philosophia Mathematica (3) Vol. 2 (1994), pp. 5-15. (pdf)

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

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

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

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

Bill Lawvere, Some Thoughts on the Future of Category Theory Como (1990)

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

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

Bill 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

Bill 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

Bill 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

Bill 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 (tac)

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)

Revised on December 6, 2012 03:35:09 by Urs Schreiber (82.169.65.155)

nLab cohesive topos

Context

Cohesive ∞-Toposes

Backround

Definition

Presentation over a site

Structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

Models

Discrete and concrete objects

Topos Theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

Definition

Definition

Remark

Remark

Definition

Definition

Definition

Remark

Definition

Remark

Properties

Adjoint quadruples

Proposition/Definition

Proposition

Proof

Relations between the axioms

Proposition

Proposition

Proposition

Subcategories of discrete and codiscrete objects

Proposition

Proof

Quasitoposes of concrete objects

Observation

Proof

Observation

Proof

Definition

Proposition

Proof

Remark

Proposition

Proof

Proposition

Proof

Objects with one point per cohesive piece

Theorem

Proof

Internal modal logic

Definition

Remark

Example

Examples

Over a discrete site

Families of sets – the Sierpinksi topos

Over a site with initial and terminal object

Observation

Proof

Corollary

Proposition

Reflexive directed graphs

Proposition

Definition

Observation

Simplicial sets

Proposition

Over a cohesive site

Theorem

Diffeological spaces

nLab
cohesive topos

Cohesive $∞$ -Toposes

Structures in a cohesive $(∞, 1)$ -topos