structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
typical contexts
Could not include topos theory - contents
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 $\Gamma$
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
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 $\Gamma$ 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 $E$ that we may think of as point sets equipped with extra structure is the quasitopos $Conc_\Gamma(E)$ of the concrete sheaves inside $E$
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
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:
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.
A topos $\mathcal{E}$ over some base topos $\mathcal{S}$, i.e. equipped with a geometric morphism
is cohesive if it is a
In detail this means that it has the following properties:
it is a locally connected topos: there exists a further left adjoint $(f_! \dashv f^*)$ 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 $(f_* \dashv f^!)$ (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 a quadruple of adjoint functors
such that $f_!$ preserves finite products.
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
The above adjoint quadruple canonically induces an adjoint triple of endofunctors on $\mathcal{E}$
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
Notice the existence of the following canonical natural transformations induced from the structure of a cohesive topos (a special case of the construction at unity of opposites).
Given a cohesive topos $\mathcal{E}$ with ($ʃ \dashv \flat$) its (shape modality $\dashv$ flat modality)-adjunction of def. 2, then the natural transformation
(given by the composition of the $\flat$-counit followed by the $ʃ$-unit) may be called the transformation from points to their pieces or the points-to-pieces transformation, for short.
The $(f^\ast \dashv f_\ast)$-adjunct of the transformation from pieces to points, def. 3,
is (by the rule of forming right adjuncts by first applying the right adjoint functor and then precomposing with the unit and by the fact that the adjunct of a unit is the identity) the map
Observe that going backwards by applying $f^\ast$ to this and postcomposing with the $(f^\ast \dashv f_\ast)$-counit is equivalent to just applying $f^\ast$, since by idempotency of $\flat$ the counit is an isomorphism on the discrete object $f^\ast f_! X$. Therefore the points-to-pieces transformation and its adjunct are related by
Observe then finally that since $f^\ast$ is a full and faithful left and right adjoint, the points-to-pieces transform is an epimorphism/isomorphism/monomorphism precisely if its adjunct $f_\ast X \longrightarrow f_! X$ is, respectively.
Below in Further axioms we discuss further axioms that one may want to impose on the points-to-pieces transform.
Below in Properties – Adjoint quadruples we discuss further properties of the points-to-pieces transform
In infinitesimal cohesion the points-to-pieces transform, def. 3, is required to be an equivalence.
In tangent cohesion the points-to-pieces transform, def. 3, is part of the canonical differential cohomology diagram.
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.
For $f : \mathcal{E} \to \mathcal{S}$ a cohesive topos, we say that pieces have points in $\mathcal{E}$ (or that the cohesion “verifies the Nullstellensatz”) if the points-to-pieces transformation from def. 3
is an epimorphism for all $X \in \mathcal{E}$.
This is equivalent to the following condition (see the proposition below):
We say that discrete objects are concrete in $\mathcal{E}$ if the transformation
is a monomorphism for all $S \in \mathcal{S}$ .
We say pieces of powers are powers of pieces if for all $S \in \mathcal{S}$ and $X \in E$ the natural morphism
is an isomorphism.
This morphism is the internal hom-adjunct of
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).
For $f : \mathcal{E} \to \mathcal{S}$ a cohesive topos, we say that its subobject classifier is contractible if for the subobject classifier $\Omega \in \mathcal{E}$ we have
This implies that for all $X \in \mathcal{E}$ also $f_! \Omega^X \simeq *$.
This appears as axiom 2 in (Lawvere, Categories of spaces).
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):
Say that a cohesive topos, def. 1, has Aufhebung of becoming if the sharp modality, def. 2, preserves the initial object
The Aufhebungs axiom def. 8 is satisfied by all cohesive toposes with a cohesive site of definition, see at Aufhebung – Over cohesive sites.
In the Aufhebungs-axiom of def. 8, the extra exactness condition on the shape modality in def. 1, which says (in particular) that shape preserves the terminal object
finds a dual companion. It might make sense to consider the variant of the axioms of cohesion which say
there is an adjoint triple of idempotent (co-)monads $ʃ \dashv \flat \dashv \sharp$;
such that $\sharp$ satisfies Aufhebung and $ʃ$ satisfies co-Aufhebung.
We discuss properties of cohesive toposes. We start in
with some generalities on situations where a sequence of four adjoint functors is given. Then in
we comment on the interdependency of the collection of axioms on a cohesive topos. In
we discuss the induced notion of concrete objects that comes with every cohesive topos and in
the induced subcategory of objects with one point per piece.
Some of these phenomena have a natural
For a long list of further structures that are canonically present in a cohesive context see
For more structure available with a few more axioms see at
Let $(p_! \dashv p^* \dashv p_*\dashv p^!) : \mathcal{E} \to \mathcal{S}$ 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, in particular concerning the induced points-to-pieces transforms, def. 3.
We write
etc. for units and
etc. for counits.
We have commuting diagrams, natural in $X \in \mathcal{E}$, $S \in \mathcal{S}$
and
where the diagonal morphisms
(the points-to-pieces transform, def. 3) and
are defined to be the equal composites of the sides of these diagrams.
This appears as (Johnstone, lemma 2.1, corollary 2.2).
The following conditions are equivalent:
for all $X \in \mathcal{E}$ the morphism $\theta_X : p_*X \to p_! X$ is an epimorphism;
for all $S \in \mathcal{S}$,, the morphism $\phi_S : p^*S \to p^! S$ is a monomorphism;
$p_*$ is faithful on morphisms of the form $A \to p^* S$.
This appears as (Johnstone, lemma 2.3).
By the above definition, $\phi_S$ is a monomorphism precisely if $\iota_{p^* S} : p^* S \to p^! p_* p^* S$ is. This in turn is so (see monomorphism) precisely if the first function in
and hence the composite is a monomorphism in Set.
By definition of adjunct and using the $(p_* \dashv p^!)$-zig-zag identity, this is equal to the action of $p_*$ on morphisms
Similarly, by the above definition the morphism $\theta_X$ is an epimorphism precisely if $p_!(\eta_X) : p_! p^* p_* X \to p_! X$ is so, which is the case precisely if the top morphism in
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 $(p_! \dashv p^*)$-zig-zag identity.
We record some relations between the various axioms characterizing cohesive toposes.
The axioms pieces have points and discrete objects are concrete are equivalent.
This is just a reformulation of the above 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.
For a sheaf topos the condition that it
is connected;
satisfies pieces have points
is equivalent to the condition that it
is hyperconnected;
is local.
This is (Johnstone, theorem 3.4).
The reflective subcategories of discrete objects and of codiscrete objects are both exponential ideals.
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.
A cohesive topos comes canonically with various subcategories, sub-quasi-toposes and subtoposes of interest. We discuss some of these.
For $(\Pi \dashv \Disc \dashv \Gamma \dashv coDisc) : \mathcal{E} \to Set$ a cohesive topos, $(\Gamma \dashv coDisc)$ exhibits $Set$ as a subtopos
By general properties of local toposes. See there.
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
is an isomorphism.
By general properties of reflective subcategories. See there for details.
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
for the full subcategory on concrete objects.
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.
Observe that the composite morphism
is given (see adjunct) by postcomposition with the $(\Gamma \dashv coDisc)$-unit $\iota_Y : Y \to coDisc \Gamma Y$
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.
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 .
The category $Conc(\mathcal{E})$ is a quasitopos and a reflective subcategory of $\mathcal{E}$.
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 $C$
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.
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.
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 $Conc(\mathcal{E})$.
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
$\mathcal{L}$ is a reflective subcategory and a coreflective subcategory
$s_*$ preserves coproducts.
the components of the reflector $X \to s_! s^* X$ are epimorphisms.
This is theorem 2 in (Lawvere).
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
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…)
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).
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
is an isomorphism, where $A \to # A$ is the $(\Gamma\dashv coDisc)$-unit on $A$.
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
is an isomorphism, then $\phi$ is discretely true.
Because if so, then
is an isomorphism and hence
is for all $X$. Therefore in this case $\Gamma \phi \to \Gamma A$ is an isomorphism and hence so is $# \phi \to # A$.
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.
We discuss some cohesive toposes over sites $C$ with trivial coverage/topology, so that the category of sheaves is the category of presheaves
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
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
since a presheaf $X$ on $C$ is given by a morphism
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
is
Plugging in the above this is just
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.
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
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
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
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
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.
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
If $C$ has both an initial object $\emptyset$ as well as a terminal object $*$ then there is a quadruple of adjoint functors
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
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
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.
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 prehseaf $X : C^{op} \to Set$ 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
sends each edge to its source and the projection
sends each edge to its target. The identities
and
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 $[1]$
In fact this is the Cauchy completion of $\mathbf{B}End(\Delta[1])$, obtained by splitting the idempotents.
In summary this shows that
We have an equivalence of categories
To see that this presheaf topos is cohesive, notice that the terminal geometric morphism
$Disc S$ 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
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 .
Reflexive directed graphs are equivalently skeleta of simplicial sets.
The category sSet of simplicial sets is a cohesive topos in which cohesive pieces have points .
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.
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
A class of examples is obtained from toposes over a cohesive site:
Let $C$ be a cohesive site. The sheaf topos $Sh(C)$ is cohesive.
See cohesive site for examples.
The site CartSp with the standard open cover coverage is a cohesive site and even an (∞,1)-cohesive site.
The quasitopos $Conc(Sh(CartSp))$ of concrete objects in the cohesive topos over $CartSp$ is the category of diffeological spaces.
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 $(\Gamma \dashv CoDisc)$)
is a monomorphism. Monomorphisms of sheaves are tested objectwise, so that this is equivalent to
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
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 $CartSp$.
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.
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
Sufficient condition for $\mathcal{E}/X$ to be a local topos is that
See at differential cohesion. Examples include the Cahiers topos
Consider a full subcategory inclusion
which has a left adjoint $\Pi_0^{\mathcal{P}}$ and a right adjoint $\Gamma^{\mathcal{P}}$ that coincide with each other
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 $\mathcal{E}$ 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.
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
is the category of permutation representations of $G$. It comes with a triple of adjoint functors
The colimit over a representation $(V, \rho) : \mathbf{B}G \to Set$ is quotient set $V/\rho(G)$. So we have
but
where $G$ denotes the fundamental representation of $G$ on itself. Therefore $\Pi_0$ does not preserve products in this case.
local topos / local (∞,1)-topos
cohesive topos / cohesive (∞,1)-topos
and
The axioms for a cohesive topos originate in
where however the term “cohesive topos” was not yet used.
This appears maybe first in
Under the name categories of cohesion a formal axiomatization is given in
(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.
In
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
and specifically the term “cohesion” probably from
In
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
on page 14.
The precise term cohesive topos apparently first appeared publically in the lecture
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
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
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)
A good deal of the structure of cohesive toposes is also considered in
under the name Q-categories .
The internal logic of local toposes is discussed in