This entry is about a refinement of the concept of cohesive topos. The definition here expresses an intuition not unrelated to that at cohesive (∞,1)-presheaf on E-∞ rings but the definitions are unrelated and apply in somewhat disjoint contexts.
Discrete and concrete objects
A cohesive -topos is a gros (∞,1)-topos that provides a context of generalized spaces in which higher geometry makes sense, in particular higher differential geometry. See also at motivation for cohesive toposes for a non-technical discussion.
Technically, it is an -topos whose global section (∞,1)-geometric morphism ∞Grpd admits a further left adjoint (∞,1)-functor and a further right adjoint :
with and both full and faithful (∞,1)-functors and such that moreover preserves finite (∞,1)-products. Here
the existence of induces a sub-(∞,1)-quasitopos of concrete objects that behave like ∞-groupoids equipped with extra cohesive structure , such as with continuous structure, smooth structure, etc.
the existence of induces a notion of geometric fundamental ∞-groupoid, hence under Top of geometric realization of objects in .
The functor itself may be thought of as sending a cohesive ∞-groupoid to its underlying bare -groupoid . This is with all cohesion forgotten (for instance with the continuous or the smooth structure forgotten).
Conversely, and send an -groupoid either to the discrete ∞-groupoid with discrete cohesive structure (for instance with discrete topology) or to the codiscrete ∞-groupoid with the codiscrete cohesive structure (for instance with codiscrete topology).
This kind of adjoint quadruple, directly analgous to the structure introduced by William Lawvere for cohesive toposes, induces two adjoint modalities which in turn are adjoint to each other, to yield the adjoint string shape modality flat modality sharp modality , which may be thought of as expressing “contiuum” and “quantity” in the sense of Georg Hegel’s Science of Logic (as explained in detail there.)
The existence of such an adjoint quadruple of adjoint -functors alone implies a rich internal higher geometry in that comes with its internal notion of Galois theory, Lie theory, differential cohomology, Chern-Weil theory.
Examples of cohesive -toposes include
the -topos Disc∞Grpd of discrete ∞-groupoids;
the -topos ETop∞Grpd of Euclidean-topological ∞-groupoids;
the -topos Smooth∞Grpd of smooth ∞-groupoids;
the -topos SynthDiff∞Grpd of synthetic differential ∞-groupoids;
the -topos Super∞Grpd of super ∞-groupoids;
the -topos SmoothSuper∞Grpd of smooth super ∞-groupoids.
In ETop∞Grpd and those contexts containing it, the internal notions of geometric realization, geometric homotopy and Galois theory subsume the usual ones (over well-behaved topological spaces). In Smooth∞Grpd also the notions of Lie theory, differential cohomology and Chern-Weil theory subsume the usual ones. In SynthDiff∞Grpd the internal notion of Lie algebra and Lie algebroid subsumes the traditional one — and generalizes them to higher smooth geometry.
We state the definition in several equivalent ways.
externally in the ambient context;
internally to the cohesive -topos itselfs;
internally and formulated in homotopy type theory
The definition is the immediate analog of the definition of a cohesive topos.
Sometimes it is desireable to add further axioms, such as the following.
Here the first morphismism is the image under of the -unit and the second is an inverse of the -counit (which is invertible because is full and faithful in a local (∞,1)-topos.)
We say discrete objects are concrete in if for all ∞Grpd the morphism
induces monomorphisms on all homotopy sheaves.
We reformulate the above axioms for a cohesive -topos without references to functors on it, and instead entirely in terms of structures in it.
A full sub-(∞,1)-category is
reflectively embedded precisely if for every object there is a morphism
(the unit) with , such that for all the value of the (∞,1)-categorical hom-space-functor
is an equivalence (of ∞-groupoids).
coreflectively embedded precisely if for every object there is a morphism
(the counit) with such that for all the value of the (∞,1)-categorical hom-space-functor
is an equivalence (of ∞-groupoids).
This is proven here.
A reflective embedding
and a coreflective embedding
fit into a single adjoint triple
(hence there is an equivalence that moreover makes the coreflector of coincide with the reflector of ) precisely if for the unit and counit given by lemma 1 we have that the morphisms on the left of
are (natural) equivalences for all objects , as indicated on the right.
It is clear that if we have an adjoint triple, then (1) and (2) are implied. We discuss now the converse.
First notice that the two embeddings always combine into an adjunction of the form
The natural equivalence (1) applied to a codiscrete object gives that of the counit of this composite adjunction is an equivalence
for all , and since is full and faithful, so is the composite counit
itself. Analogously, (2) implies that the unit of the composite adjunction is an equivalence. Therefore (1) and (2) together imply that the adjunction itself exhibits an equivalence .
Using this we then find for each a composite natural equivalence
where the first morphism uses the above equivalence on the codiscrete object and the second is a choice of natural inverse of (2).
Since is full and faithful, this mean that we have equivalences
natural in , hence that .
Using these lemmas we can now restate cohesiveness internally.
For a sub-(∞,1)-category, the inclusion extends to an adjoint quadruple of the form
precisely if there exists for each object
a morphism with ;
a morphism with ;
a morphism with
such that for all and the induced morphisms
are equivalences (the first three of ∞-groupoids the last two in ).
Moreover, if is a cartesian closed category, then preserves finite products precisely if the -inclusion is an exponential ideal.
The last statement follows from the -category analog of the discussion here.
In homotopy type theory
The axioms for cohesion, in the internal version, can be formulated in homotopy type theory, the internal language of an (∞,1)-topos.
The corresponging Coq-HoTT code is in (Shulman).
For more see cohesive homotopy type theory.
We discuss basic properties implied by the axioms for cohesive -toposes in
Then we discuss presentations over special sites in
As a point-like space
Over an -cohesive site
We discuss a presentation of classes of cohesive (∞,1)-toposes by a model structure on simplicial presheaves over a suitable site.
The detailed discussion is at ∞-cohesive site.
For a cohesive -topos over an ∞-cohesive site, the functor preserves (∞,1)-pullbacks over discrete objects.
We first consider a lemma. Notice that for the (∞,1)-Grothendieck construction gives an equivalence of (∞,1)-categories
from the over (∞,1)-category of over to the (∞,1)-functor (∞,1)-category from to .
We establish this via a presentation of by a model structure on simplicial presheaves.
Let be an ∞-cohesive site of definition for . Then by the discussion there we have
Moreover, picking a Kan complex presentation for , which we shall denote by the same symbol, we have that the constant simplicial presheaf is fibrant. Therefore by this proposition the induced model structure on an overcategory on presents the given over (∞,1)-category
Now observe that we have an ordinary equivalence of categories
under which the model structure becomes that of the local projective model structure on functors with values in the model structure that presents .
Let then denote the model structure for left fibrations. By the discussion there, this also presents . Hence by this proposition we have an equivalence of (∞,1)-categories
This allows now to apply this presentation of the (∞,1)-Grothendieck construction to find
where is the simplicially enriched category corresponding to (as discussed at relation between quasi-categories and simplicial categories ) and is the global model structure on sSet-enriched presheaves.
Then using the cartesian closure of the category of simplicial presheaves (which is a topos) inside the we have
Finally this implies the claim using this proposition.
With this lemma we can now give the proof of prop. 5.
By the discussion at adjoint (∞,1)-functors on slices we have that induces an adjoint pair
Under the equivalence from lemma 3 the functor maps to
Since products of -functor -categories are computed objectwise, and since preserves finite products by the axioms of cohesion, also preserves finite products, and hence so does . But products in the slice over are (∞,1)-pullbacks over . So this proves the claim.
Base of discrete and codiscrete objects
In the internal definition the base of discrete/codiscrete objects is not explicitly axiomatized to be an (∞,1)-topos itself (the base (∞,1)-topos), but this is implied by the axioms. We deduce that and related properties in stages.
In the following, let be an (∞,1)-topos equipped with an adjoint quadruple of functors to an (∞,1)-category – the base of cohesion, where and are full and faithful.
This is a general property of a reflectively and coreflectively embedded subcategory. The limits are computed by computing them in and then applying and the colimits are computed by computing them in and then applying . For any diagram we have
We have also the following stronger statement.
By one of the equivalent characterizations of presentable (∞,1)-categories these are reflective sub-(∞,1)-categories of (∞,1)-categories of (∞,1)-presheaves where the embedding is by an accessible (∞,1)-functor.
Since is itself accessibly and reflectively embedded into the presheaves on a (∞,1)-site of definition, we have a composite reflective inclusions
Since even preserves all (∞,1)-colimits, it is in particular an accessible (∞,1)-functor, hence so is the above composite.
Finally, since preserves all (∞,1)-limits, hence in particular the finite limits, is a geometric embedding that exhibits an sub-(∞,1)-topos.
Notice that the reflection does not in general constitute a geometric embedding, since is only required to preserve finite products (and in interesting examples rarely preserves more limits than that).
The following statement and its proof about cohesive 1-toposes should hold verbatim also for cohesive -toposes.
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 preserves even all limits and for the discrete objects the reflector does so by assumotion of strong connectedness.
Factorization systems associated to
We discuss orthogonal factorization systems in a cohesive -topos that characterize or are characterized by the reflective subcategory of dicrete objects, with reflector .
For a morphism in , write for the (∞,1)-pullback in
where the bottom morphism is the -unit. We say that is the -closure of , and that is -closed if .
If has an ∞-cohesive site of definition, then every morphism in factors as
such that is a -equivalence in that it is inverted by .
The factorization is given by the naturality of and the universal property of the -pullback in def. 4.
Then by prop. 5 the functor preserves the -pullback over the discrete object and since is an equivalence, it follows that is an equivalence.
The pair of classes
is an orthogonal factorization system in .
This follows by the general reasoning discussed at reflective factorization system:
By prop. 9 we have the required factorization. It remains to check the orthogonality.
be a square diagram in where the left morphism is a -equivalence and the right morphism is -closed. Then by assumption there is a pullback square on the right in
By naturality of the adjunction unit, the total rectangle is equivalent to
Here by assumption the middle morphism is an equivalence. Therefore there is an essentially unique lift in the square on the right and hence a lift in the total square. Again by the universality of the adjunction, any such lift factors through and hence also this lift is essentially unique.
Finally by universality of the pullback, this induces an essentially unique lift in
For a -closed morphism and a global element, the homotopy fiber is a discrete object.
By the def. 4 and the pasting law we have that is equivalently the -pullback in
Since the terminal object is discrete, and since the right adjoint preserves -pullbacks, this exhibits as the image under of an -pullback of -groupoids.
Structures in a cohesive -topos
A cohesive -topos is a general context for higher geometry with good cohomology and homotopy properties. We list fundamental structures and constructions that exist in every cohesvive -topos.
This section is at
Types of cohesion
We discuss extra structure on a cohesive -topos that encodes a refinement of the corresponding notion of cohesion to infinitesimal cohesion . More precisely, we consider inclusions of cohesive -toposes that exhibit the objects of as infinitesimal cohesive neighbourhoods of objects in .
This section is at
Locally ringed cohesion
Every cohesive -topos equipped with differential cohesion comes canonically equipped with a notion of formally étale morphisms (as discussed there). Combined with the canonical interpretation of as the classifying topos of a theory of local T-algebras, this caninically induces a notion of locally algebra-ed (∞,1)-toposes with cohesive structure, generalizing the notion of locally ringed spaces and locally ringed toposes.
This section is at
Examples of cohesive -toposes
We list examples of cohesive -toposes, both specific ones as well as classes of examples constructed in a certan way.
Cohesive diagram -toposes
Cohesive diagrams in a cohesive -topos
Let be an cohesive -topos.
Let be a small category (diagram) with initial object and terminal object , or else a presentable (∞,1)-category. Write
for the triple of adjoint (∞,1)-functors given by including and .
Then the (∞,1)-functor category is again a cohesive -topos, exhibited by the adjoint quadruple which is the composite
where the adjoint quadruple on the left is that induced under (∞,1)-Kan extension from .
By the discussion at (∞,1)-Kan extension each of the original three functors induces [adjoint triples etc, as indicated. In particular is a right adjoint and therefore preserves finite products (and all small (∞,1)-limits, even).]
By the original adjunctions one finds that and , which implies the adjoint quadruple as indicated above by essential uniqueness of adjoints.
Finally it is clear that , which implies that is a full and faithful (∞,1)-functor (and hence so is ).
In particular we have
For a cohesive -topos, also its arrow (∞,1)-category is cohesive.
For ∞Grpd (“discrete cohesion”, see below) the corresponding cohesive -topos is known as the Sierpinski (∞,1)-topos.
Simplicial objects in a cohesive -topos
For a cohesive -topos its (∞,1)-category of simplicial objects is cohesive over
sends a simplicial object to the homotopy colimit over its components, hence to its “geometric realization” as seen in .
evaluates on the 0-simplex;
sends an object in to the simplicial object which is simplicially constant on .
Hence cohesion of relative to expresses the existence of a discrete and directed notion of path.
Notice that there is an inclusion
of the groupoid objects internal to and of the category objects internal to inside .
Here is also the classifying topos for linear intervals. Its homotopy type theory internal language is equipped with an interval type.
For more see at simplicial object in an (∞,1)-category.
Bundles of cohesive spectra
The tangent (∞,1)-category to a cohesive -topos is itself cohesive, (by the discussion at tangent ∞-category – Examples – Of an ∞-topos), the tangent cohesive (∞,1)-topos.
This the -topos of parameterized spectra in , hence is context for cohesive stable homotopy theory.
Global equivariant homotopy theory
Rezk-global equivariant homotopy theory:
From -Cohesive sites of definition
From this one obtains the following list of examples of cohesive -toposes.
Complex analytic -groupoids
Synthetic differential -groupoids
Smooth Super -groupoids
Smooth -groupoids over algebraic -stacks
One can consider the tangent (∞,1)-topos of the cohesive (∞,1)-topos
of ∞-stacks on the site of smooth manifolds with values in turn in ∞-stack over a site of arithmetic schemes, hence by smooth ∞-groupoids but over a base (∞,1)-topos of algebraic ∞-stacks.
This leads to differential algebraic K-theory. See there for details.
Smooth cohesion over arbitrary base -toposes
The above examples relativize to arbitrary bases.
For instance smooth cohesion over an (∞,1)-topos over a suitable site of schemes is the natural context for differential algebraic K-theory. See there for more.
As a context for geometric spaces and paths in geometric spaces, cohesive -toposes are a natural context in which to formulate fundamental fundamental physics. See higher category theory and physics for more on this.
unrelated is the notion of cohesive ∞-prestack
The category-theoretic definition of cohesive topos was proposed by Bill Lawvere. See the references at cohesive topos.
The observation that the further left adjoint in a locally ∞-connected (∞,1)-topos defines an intrinsic notion of paths and geometric homotopy groups in an (∞,1)-topos was suggested by Richard Williamson.
The observation that the further right adjoint in a local (∞,1)-topos serves to characterize concrete (∞,1)-sheaves was amplified by David Carchedi.
Some aspects of the discussion here are, more or less explicitly, in
For instance something similar to the notion of ∞-connected site and the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos is the content of section 2.16. The infinitesimal path ∞-groupoid adjunction is essentially discussed in section 3. The notion of geometric realization (see structures in a cohesive (∞,1)-topos – geometric realization), is touched on around remark 2.22, referring to
But, more or less explicitly, the presentation of geometric realization of simplicial presheaves is much older, going back to Artin-Mazur. See geometric homotopy groups in an (∞,1)-topos for a detailed commented list of literature.
A characterization of infinitesimal extensions and formal smoothness by adjoint functors (discussed at infinitesimal cohesion) is considered in
in the context of Q-categories .
On cohesive -toposes proper
The material presented here is also in section 3 of
In homotopy type theory
Expositions and discussion of the formalization of cohesion in homotopy type theory is in
The corresponding Coq-code is in
Exposition is at