(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
An (∞,1)-sheaf – often called a ∞-stack – is the (∞,1)-categorical analog of a sheaf. Just like a category of sheaves is a topos, an (∞,1)-category of (∞,1)-sheaves is an (∞,1)-topos.
There is good motivation for sheaves, cohomology and higher stacks.
Here we recall basic definitions and then concentrate on 1-categorical models that present (∞,1)-categories of ∞-stacks.
What we describe is effectively the old theory of the model structure on simplicial presheaves seen in the new light of Higher Topos Theory.
We proceed as follow.
As a preparation,
we recall 1-categorical presheaves
and how a category of sheaves may equivalently be thought of as
a category geometrically embedded into that of presheaves
or equivalently as the localization of presheaves at local isomorphisms.
Then in the main part, after recalling
the definition of (∞,1)-category
we give
the general definition of (∞,1)-presheaves;
and finally
It is helpful to briefly recall the story that we want to tell in the category theory context, because in the full higher category theory context it will be literally the same with all notions such as adjoint functor, exact functor etc suitably regarded in the context of (∞,1)-functors.
Consider a category $C$ that we want to think as a category of “test spaces”. Classical choices would be $C =$ Top, the category of topological spaces, $C =$ Diff, the category of smooth manifolds or $C = Op(X)$, the category of open subsets of some topological space $X$.
Let Set be the category of sets. We write
for the category of presheaves on $C$. This is like a category of very general spaces modeled on $C$ as described at motivation for sheaves, cohomology and higher stacks.
In fact, this is a bit too general for most purposes: the objects of $PSh(C)$ may be very non-local in that they don’t respect the way test objects in $C$ are supposed to glue together. The full subcategory on those presheaves that do respect some kind of gluing of test objects is the category of sheaves.
A category of sheaves on $C$ is a category $Sh(C)$ equipped with a geometric embedding into $PSh(C)$
Recall that this means that
$Sh(C) \hookrightarrow PSh(C)$ is a full and faithful functor;
$\bar {(\cdot)} : PSh(C) \to Sh(C)$ is a left exact left adjoint.
in other words that
$Sh(C) \hookrightarrow PSh(C)$ is the inclusion of a reflective subcategory;
with the special property that the left adjoint to the inclusion is left exact (i.e. preserves finite limits).
In view of our models for $\infty$-sheaves it is of importance that this implies an equivalence characterization
The category $Sh(C)$ is equivalent to the full subcategory of $S$-local presheaves, where $S$ is the set of local isomorphisms.
Another useful kind of geometric embeddings is that of the point:
let ${*}$ be the category with a single morphism (the identity on a single object). Then $PSh({*}) \simeq Sh({*}) \simeq Set$. Geometric embeddings
are called points of $Sh(C)$. We say that $Sh(C)$ has enough points if isomorphisms of sheaves can be tested on points
This is the situation we shall concentrate on here.
The topos $Sh(Diff)$ has enough points, one for every $n \in \mathbb{N}$.
The topos $Sh(Op(X))$ has enough points: one for every ordinary point of $X$.
If $Sh(C)$ has enough points, we may characterize sheaves in yet another way, which is the one that directly suggests the local model structure on simplicial presheaves discussed below:
Let $S \subset Mor(PSh(C))$ be the set of stalkwise isomorphisms, i.e. those morphisms $f : A \to B$ of presheaves such that for all points $x$ the morphism $x^* f : x^* A \to x^* B$ is an isomorphism (of sets).
If $Sh(C)$ has enough points, then $Sh(C)$ is equivalent to the full subcategory of $S$-local presheaves.
The local model structure on simplicial presheaves that we are going to describe is obtained from this description of sheaves by
replacing sets by simplicial set
replacing stalkwise isomorphism of sets by stalkwise weak homotopy equivalences of simplicial sets.
So the model structures we shall encounter are plausible guesses. What is less trivial is that this plausible structure indeed presents the fully general notion of (∞,1)-sheaf/∞-stack.
This fully general notion we introduce now.
An ordinary locally small category is a category enriched over the category Set of sets.
An (∞,0)-category is an ∞-groupoid which we think of as modeled by a simplicial set that is a Kan complex.
Recall that there is a notion of nerve and realization
for SSet-enriched categories induced by a cosimplicial simplicially enriched category
where the nerve operation $N$ is called the homotopy coherent nerve of simplicially enriched categories.
An (∞,1)-category is a category enriched over $\infty$-groupoids, i.e. an SSet-enriched category all whose hom-objects happen to be Kan complexes.
Given two $(\infty,1)$-categories $\mathbf{C}$ and $\mathbf{D}$ the (∞,1)-functor $(\infty,1)$-category is
This is indeed itself an $(\infty,1)$-category (HTT, prop 1.2.7.3).
The (∞,1)-category of (∞,1)-categories $(\infty,1)Cat$ is that whose
objects are $(\infty,1)$-categories;
for $\mathbf{C}$ and $\mathbf{D}$ two $(\infty,1)$-categories the $\infty$-groupoid $(\infty,1)Cat(\mathbf{C}, \mathbf{D})$ is the maximal Kan complex inside the simplicial set of maps between the homotopy coherent nerves
Examples
Using the monoidal embedding $const : Set \hookrightarrow \infty Grpdf \subset SSet$ every ordinary category is an $(\infty,1)$-category.
The $(\infty,1)$-category $\infty Grpd$ (∞Grpd) is the full [[SSet]-subcategory of SSet on Kan complexes.
The simplicial connected components functor
is strong monoidal and hence induces a functor
The image $H(\mathbf{C})$ of an $(\infty,1)$-category $\mathbf{C}$ with $H(\mathbf{C})(x,y) = \pi_0(\mathbf{C}(x,y))$ is the homotopy category of an (∞,1)-category.
Two $(\infty,1)$-categories $\mathbf{C}$ and $\mathbf{D}$ are equivalent if they are isomorphic in $H((\infty,1)Cat)$
It is often convenient to present $(\infty,1)$-categories by 1-categorical models.
For $\mathbf{A}$ a combinatorial simplicial model category, the $(\infty,1)$-category presented by it is the full subcategory $\mathbf{A}^\circ \subset \mathbf{A}$ on objects that are both cofibrant and fibrant.
Remark The axioms of a simplicial model category ensure that the hom-simplicial sets of $\mathbf{A}^\circ$ are indeed Kan complexes. (for instance HTT, remark 3.1.8).
Let $\mathbf{A}$ and $\mathbf{B}$ be combinatorial simplicial model categories. Then the corresponding $(\infty,1)$-categories $\mathbf{A}^\circ$ and $\mathbf{B}^\circ$ are equivalent precisely if there is a sequence of SSet-enriched Quillen equivalences
There is now an obvious definition of $(\infty,1)$-categories of $(\infty,1)$-presheaves and of $(\infty,1)$-sheaves by interpreting the 1-categorical story in the $(\infty,1)$-categorical context.
Now we generalize the above from sheaves to (∞,1)-sheaves also known as ∞-stacks.
The $(\infty,1)$-category of (∞,1)-presheaves on $C$ is
see also HTT, prop. 5.1.1.1)
The $(\infty,1)$-category presented by the global model structure on simplicial presheaves $SPSh(C)_{proj}$ on $C$ (either the projective or the injective one) is equivalent to that of $(\infty,1)$-presheaves on $C$:
There are $(\infty,1)$-category analogs of all the familiar notions from category theory, in particular
exact functor (preserving finite limits).
Using this we obtain a definition of geometric embedding of $(\infty,1)$-toposes , i.e. left exaxt reflective (∞,1)-subcategories by literally copying the 1-categorical definition.
An (∞,1)-category of (∞,1)-sheaves is a geometric embedding into an (∞,1)-category of (∞,1)-presheaves
Let the combinatorial simplicial model category $\mathbf{B}$ be a left Bousfield localization of the combinatorial simplicial model category $\mathbf{A}$ then
is the inclusion of a reflective (∞,1)-subcategory.
By HTT, prop A.3.7.4 every combinatorial simplicial left Bousfield localization is given by a set $S$ of cofibrations such that
the fibrant objects of $\mathbf{B}$ are precisely the fibrant objects in $\mathbf{A}$ that are $S$-local object;
the weak equivalences of $\mathbf{B}$ are the $S$-local morphisms in $\mathbf{A}$.
Accordingly $\mathbf{B}^\circ$ is the full $\infty Grpd$-enriched subcategory of $\mathbf{A}^\circ$ on $S$-local objects. (see also HTT, prop 6.5.2.14).
By HTT, prop. 5.5.4.15 this means that $\mathbf{B}$ is a reflective (∞,1)-subcategory of $\mathbf{A}$.
Remark Notice that this does not yet say that the localization is left exact .
But this makes at least plausible that the local model structure on simplicial presheaves is a presentation for an (∞,1)-category of (∞,1)-sheaves.
That this is indeed the case is
The local model structure on simplicial presheaves $SSh(C)^{l loc}_{proj}$ presents the hypercompleted version of the (∞,1)-category of (∞,1)-sheaves $Sh^{hc}(C)$ on $C$.
Remark See the discussion at Čech cohomology for the role of hypercompletion.
The nerve operation of the Dold-Kan correspondence
embeds sheaves with values in non-negatively graded chain complexes of abelian groups into simplicial sheaves as those simplicial sheaves with values in Kan complexes that carry a struict abelian group structure. This way homological algebra and abelian sheaf cohomology are realized as special cases of models for $\infty$-stacks: a complex of abelian sheaves presents a stably abelian $\infty$-stack.
Under the Dold-Kan correspondence abelian sheaf cohomology identifies with the hom-set of the homotopy category corresponding infinity-stack (infinity,1)-topos.
More precisely, let
the underlying site be the category of open subsets $C = Op(X)$ of a topological space $X$,
let $A \in Sh(X)$ be a sheaf with values in abelian groups on $X$;
let $\mathbf{B}^n A \in Sh(X,SSet)$ be the image of the complex of sheaves $A[-n]$ concentrated in degree $n$ under the Dold-Kan nerve;
write $X \in Sh(X)$ for the terminal object sheaf in $Sh(X)$ (the sheaf constant on the singleton set).
Then degree $n$ abelian sheaf cohomology of $X$ with coefficients in $A$ is homotopy classes of maps from $X$ to $\mathbf{B}^n A$:
The original proof was given in BrownAHT in terms of the category of fibrant objects structure on locally Kan simplicial sheaves.
The analogous arguments in terms of the full injective model structure were given by Jardine. See section 6 of his lecture notes.