(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
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This is a sub-entry of homotopy groups in an (∞,1)-topos. It discusses the general notions of étale homotopy in the context of locally ∞-connected (∞,1)-toposes.
For the other notion of homotopy groups in an $(\infty,1)$-topos see categorical homotopy groups in an (∞,1)-topos.
An ordinary topos $E$ is a locally connected topos if the global sections geometric morphism $(LConst \dashv \Gamma) : E \stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}} Set$ is in fact an essential geometric morphism in that $LConst$ has also a left adjoint $(\Pi_0 \dashv LConst)$:
This left adjoint $\Pi_0$ sends each object $X$ of $A$ to its set $\Pi_0$ of connected components. In other words this left adjoint produces the degree 0-part of the homotopy groups of objects of $E$.
This has an obvious generalization of (∞,1)-toposes.
The obvious generalization of the notion of $\Pi_0$ for a locally connected topos is to say that for $n \in \mathbb{N}$ an (n,1)-topos $\mathbf{H}$ is a locally n-connected (n,1)-topos if again the terminal geometric morphism is an essential geometric morphism in that the constant n-stack functor $LConst$ has a left adjoint $\Pi_n$
Here we may take $n = \infty$ and say that an (∞,1)-topos is locally contractible if we have an essential geometric morphism
to ? Grpd?, with $\Pi$ the left adjoint (∞,1)-functor to the constant ∞-stack (∞,1)-functor $LConst$. For $X \in \mathbf{H}$ any object, the ∞-groupoid $\Pi(X)$ deserves to be called the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos of $X$ Its ordinary homotopy groups are the homotopy groups of $X$.
While an obvious slight generalization or refinement of what is considered in previous literature, it seems that the simple picture of a left adjoint (∞,1)-functor to the constant ∞-stack functor has not been made explicit in the existing literature (though possibly in the thesis by Richard Williamson).
However, up to some straightforward translations of concepts and notation, it turns out that essentially all aspects of this picture are present and well known – if somewhat implicitly – in existing literature. A detailed commented account of what is in the literature is in the following subsection and in particular in the section Examples below.
There are essentially three different methods concretely constructing the abstractly defined homotopy ∞-groupoid?-functor $\Pi(-)$.
by constructing the left adjoint $\Pi(-)$ as the functor that takes an object to its local contraction – this is described in the section In terms of local contractions;
by using monodromy/Galois theory of locally constant ∞-stacks to reproduce $\Pi()$ by Tannaka duality – this is described in the section In terms of monodromy and Galois theory;
by constructing $\Pi(-)$ explicitly as a path $\infty$-groupoid in terms of paths modeled on an interval object in $\mathbf{H}$ – this is described in the section In terms of concrete paths .
If the locally contractible (∞,1)-topos $\mathbf{H}$ has a site $C$ with $\mathbf{H} \simeq Sh_{(\infty,1)}(C)$ such that the objects of the site are geometrically contractible in that constant (∞,1)-presheaves already satisfy descent over objects in $C$, then the left adjoint $\Pi : \mathbf{H} \to \infty Grpd$ to $LConst$ may be constructed explicitly as follows.
Following the discussion at models for ∞-stack (∞,1)-toposes there is a model structure on simplicial presheaves $sPSh(C)_{proj}^{loc}$ wich presents $\mathbf{H}$.
Proposition The (∞,1)-adjunction $(\Pi \dashv LConst) : Sh_{(\infty,1)}(C) \stackrel{\leftarrow}{\to} \infty Grpd$ is presented by an SSet-enriched Quillen adjunction
where
for $S \in sSet$ the presheaf $LConst_S$ sends all $U \mapsto S$, for all $U$;
the functor $\Pi$ acts by $\Pi(X) = \int^{U \in C} X = \lim_\to X$.
The total left derived functor of $\Pi$ first takes an object $X$ to a simplicial presheaf that is degreewise a coproduct of representables and then contracts all these representables to the terminal object, regarding the resulting constant simplicial presheaf as a simplicial set:
Proof
This is discussed at fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos and cohesive (∞,1)-topos.
Essentially the construction of $\mathbb{L} \Pi$ as above is an old construction in terms of – somewhat implicitly – the structure of a category of fibrant objects on simplicial objects in a topos:
the discussion on page 18 of
which goes back to
goes as follows:
Let $E = Sh(C)$ be a locally connected topos
that here we think of as a petit over-topos over a given object $X$ in some ambient gros topos. Accordingly we write $X = *$ for the terminal object in $Sh(C)$.
Assume that $E$ has enough point. Then stalkwise Kan-fibrant simplicial objects in $E$, i.e. stalk-wise Kan-fibrant simplicial sheaves on $C$ form a category of fibrant objects. In particular a fibrant simplicial object $Y \in [\Delta^{op}, Sh(C)]$ equipped with an acyclic fibration $Y \to X$ to the terminal object $X = *$ is a hypercover of $X$.
The definition of the ∞-groupoid $\Pi(X)$ as defined in the above references (notice that only its homotopy groups are written down explicitly there, but it’s immediate to equivalently write it as we do now) is
where
the colimit is taken over the category of acyclic fibrations/hypercovers $Y \to X$;
the connected components functor $\Pi_0 : Sh(C) \to Set$ is applied degreewise to the simplicial sheaf $Y = (Y_\bullet)$ to produce a simplicial set.
In Artin-Mazur it is discussed that this prescription does produce the right homotopy groups for $X$ a topological space if one assumes that this space is locally contractible space.
If we therefore interpret this as saying that for the above prescription to yield the correct result we generally ought to assume that $Sh_{(\infty,1)}(C)$ is a locally contractible (∞,1)-topos, then this prescription can be seen to model implicitly the left Quillen functor $\Pi(-)$ that we described above:
In terms of the full model category structure on $sPSh(C)_{proj}^{loc}$ among all these hypercovers is one that is the cofibrant object
mentioned above, consisting degreewise of coproducts of representables with $\Pi_0(U_i) = *$. For instance if $X$ admits a good open cover, we can take $Y$ to be the Cech nerve of that good cover. (For more on this see ∞-Lie groupoid.) Due to the lifting property of cofibrant objects, any colimit over all hypercovers can be computed by evaluating just at that hypercover.
There the Artin-Mazur-Moerdijk-prescription yields
This is indeed the action of the left Quillen functor from above.
It is the nerve theorem that asserts that for $Y$ the Cech nerve of a good open cover, this simplicial set is homotopy equivalent to the original paracompact space.
A closely related, implicitly slightly more general statement is in on p. 25 of
which describes this construction for the case $\mathbf{H} = Sh_{(\infty,1)}(Diff)$ (the gros topos of $\infty$-stacks on Diff).
With even more general sites allowed, but working only at the level of homotopy categories the left adjoint $\Pi$ and its construction is described in Proposition 2.18 of
See also the discussion at locally contractible (∞,1)-topos.
Given an (∞,1)-topos $\mathbf{H} = Sh_{(\infty,1)}(C)$ we define the ∞-groupoid of locally constant ∞-stacks on an object $X \in \mathbf{H}$ to be
where $LConst_{Core(\infty Grpd)}$ is the constant ∞-stack on the core ∞-groupoid of ? Grpd?.
If $\mathbf{H}$ is a locally contractible (∞,1)-topos in that $LConst$ has the left adjoint (∞,1)-functor $\Pi(-)$, then by definition of adjunction we have the equivalence
with locally constant ∞-stacks/$\infty$-covering spaces on the one hand and (∞,1)-functors from $\Pi(X)$ to ∞Grpd on the other.
Concrete realizations of this equivalence are discussed in the Examples-section below. Here we describe how one may reconstruct in terms Tannaka duality $\Pi(X)$ from just knowing $\infty CovBund(X)$ in terms of the automorphism ∞-group of a fiber functor
from $\infty$-coverin bundles/locally constant ∞-stacks over $X$ to ∞-groupoid.
– these automorphism are called the monodromy of $X$.
We want to show that these automorphism ∞-groups are the loop space objects of $\Pi(X)$, hence the geometric homotopy $\infty$-groups.
This is the reconstruction of the geometric homotopy ∞-groups of an ∞-stack $X$ from its monodromy or Galois theory.
Proof
The underlying mechanism is just $(\infty,1)$-Tannaka duality, i.e. essentially the (∞,1)-Yoneda lemma applied a few times in a row:
suppose we knew $\Pi(X)$, so that by adjunction we have
Then for each point $x \in \Pi(X)$ given by a morphism $i : {*} \to \Pi(X)$ we get a fiber functor
which takes a local system $\rho : \Pi(X) \to \infty Grpd$ and evaluates it on $x$. By the (∞,1)-Yoneda lemma this means that $F_x$ is given by homming out of the local system $Y_{\Pi(X)^{op}} x$ represented by $x$:
But this in turn means that $\infty Func(i,\infty Grpd) : \infty Func(\Pi(X),\infty Grod) \to \infty Grpd$ is itself a representable functor, in the (∞,1)-category of (∞,1)-presheaves $PSh_{(\infty,1)}(PSh_{(\infty,1)}(\Pi(X)^{op})^{op})$:
This way we find, by applying the (∞,1)-Yoneda lemma two more times, that the automorphism ∞-group of the fiber functor is
Now, the same is of course true even if we don’t have $\Pi(X)$ in hands yet, but only know the ∞-groupoid $CovBund(X)$ of covering $\infty$-bundles / locally constant ∞-stacks in $X$: in terms of this we may reconstruct the automorphism ∞-groups of $\Pi(X)$ as
The idea that geometric homotopy groups of generalized spaces given by sheaves, stacks, ∞-stacks is detected and definable by the behaviour of locally constant sheaves, stacks, $\infty$-stacks on these objects goes back to Grothendieck's Galois theory and the notion of fundamental group of a topos. The state of the art treatment of the Galois theory of coverings in a topos is in
In Pursuing Stacks Grothendieck talked about how this 1-categorical situation generalizes to ∞-stacks.
After Pursuing Stacks, apparently the first to publish a detailed formalization and proof of how the homotopy groups of a topological space $X$ may be recovered from the behaviour of locally constant ∞-stacks on $X$ was
This has a followup construction in
Program Stacks, Intersection theory and Non-abelian Hodge Theory, MSRI, Berkeley, January-May 2002 (arxiv:math/0212330).
Very similar constructions and statement then appeared in
Pietro Polesello and Ingo Waschkies, Higher monodromy , Homology, Homotopy and Applications, Vol. 7(2005), No. 1, pp. 109-150 (pdf)
Mike Shulman, Parametrized spaces model locally constant homotopy sheaves (arXiv:0706.2874)
and, building on that, in example 1.8 of
Notably the article by Pietro Polesello and Ingo Waschkies makes fully explicit the observation that locally constant $n$-stacks are precisely the sections of the constant $(n+1)$-stack on the $(n+1)$-groupoid $n Grpd$. This is a key observation for bringing the full power of the adjunction $(\Pi \dashv LConst)$ into the picture, as we do here.
It was pointed out to Urs Schreiber by Richard Williamson that these constructions should generalize from topological spaces to objects in any (∞,1)-topos, maybe along the lines outlined above, and that this way suitable $(\infty,1)$-toposes $\mathbf{H}$ comes canonically equipped with a notion of homotopy ∞-groupoid? $\Pi(X)$ of every object $X \in \mathbf{H}$.
…
The following references discuss fundamental groupoids of an entire topos constructed from concrete interval objects. In the context of the above discussion these toposes are to be thought of as petit over-toposes over a given object in an ambient gros topos, and as such are concerned with the fundamental groupoid of that object, in our sense.
The construction of the fundamental groupoid of a topos from interval objects is in
The comparison of this construction with the one by monodromy/Galois theory is in
Here we discuss the 0-th geometric homotopy group $\Pi_0 : Sh(X) \to Set$ of objects in a sheaf topos in terms of a left adjoint $\Pi_0$ of the constant sheaf functor. This is a special case of the more general situation discussed in Pi0 of a general object in a locally connected topos below.
Let $X$ be a sufficiently nice topological space.
There is a triple of adjoint functors
where
$(LConst \dashv \Gamma)$ is the usual global section geometric morphism with $LConst_S$ the constant sheaf of locally constant functions with values in $S \in Set$ and
$\Pi : Sh(X) \to Set$ is left adjoint to $LConst$ and sends each sheaf $A$ to the set of connected components of the corresponding etale space $p_A : Et(A) \to X$:
The etale space of $LConst_S$ is $E(LConst_S) = X \times S$. By the relation of sheaves on $X$ with etale spaces over $X$ we have
For $\gamma : I \to E(A)$ any continuous path in $E(A)$, and for $f : E(A) \to X \times S$ a morphism in $Et/X$, the image of $\gamma$ in $X \times I$ is fixed by, say, the image $f(\gamma(0)) = (p_A(\gamma_0),s)$ to be $f(\gamma) : t \mapsto (p_A(\gamma(t)),s)$. This means that the value of $f$ on any path component of $E(A)$ is uniquely fixed by its value on any point in that path component.
Choosing a basepoint in each path component therefore induces bijection
More generally, if $E$ is a locally connected topos then the global sections geometric morphism $(LConst \dashv \Gamma) : E \stackrel{\leftarrow}{\to} Set$ has also a left adjoint $\Pi_0$ to $LConst$:
For instance page 17 of
The general idea is that of
A discussion of of how this produces first homotopy groups of a 1-topos is at
The general construction of the first geometric homotopy group of objects in a Grothendieck topos is for instance in section 8.4 of
This case is discussed in
We indicate briefly how the results stated in this article fit into the general abstract picture as indicated above:
The authors consider locally constant 1-stacks and 2-stacks on sites of open subsets of sufficiently nice topological spaces.
Prop. 1.1.9 gives the adjunction
between forming constant stacks and taking global sections.
Then prop 1.2.5, 1.2.6, culminating in theorem 1.2.9, p. 121 gives (somewhat implicitly) the other adjunction
with the right adjoint to $LConst$ being the fundamental groupoid functor on representables. (Where we change a bit the perspective on the results as presented there, to amplify the pattern indicated above. For instance where the authors write $\Gamma(X,C_X)$ we think of this here equivalently as $Sh_{(2,1)}(X)(X,LConst(C))$, so that the theorem then gives the adjunction equivalence $\cdots \simeq Grpd(\Pi_1(X),C)$).
Then in essentially verbatim analogy, these results are lifted from stacks to 2-stacks in section 2, where now prop 2.2.2, 2.2.3, culminating in theorem 2.2.5, p. 132 gives (somewhat implicitly) the adjunction
now with the path 2-groupoid operation (locally) left adjoint to forming constant 2-stacks.
Let $X$ be a sufficiently nice (I think this should be locally (relatively) contractible. -DR) (paracompact) topological space. The canonical map $X \to {*}$ induces the geometric morphism
where the right adjoint $\Gamma$ is taking global sections and the left adjoint is forming the constant ∞-stack on an $\infty$-groupoid $K$. If $K = Core (\infty Grpd)$ then $LConst_K$ is the constant ∞-stack of locally constant ∞-stacks and we write
for the $\infty$-groupoid of locally constant $\infty$-stacks on $X$.
Write $\Pi(X) := Sing X$ for the fundamental ∞-groupoid of $X$.
There is an equivalence of $\infty$-groupoids
Urs Schreiber: I think this is proven in the literature, if maybe slightly implicitly so. I’ll now go through the available references to discuss this.
After old ideas by Alexander Grothendieck from Pursuing Stacks, it seems that the first explicit formalization and proof of this statement is given in
In theorem 2.13, p. 25 the author proves an equivalence of (∞,1)-categories (modeled there as Segal categories)
of locally constant ∞-stacks on $X$ and Kan fibrations over the fundamental ∞-groupoid $\Pi(X) = Sing(X)$.
But Kan fibrations over a Kan complex such as $\Pi(X)$ are equivalently left fibrations (as discussed there) and by by the (∞,1)-Grothendieck construction these are equivalent to (∞,1)-functors $\Pi(X) \to \infty Grpd$. So under the (∞,1)-Grothendieck construction Toën’s result does actually produce an equivalence
In
Program Stacks, Intersection theory and Non-abelian Hodge Theory, MSRI, Berkeley, January-May 2002 (arxiv:math/0212330).
this is discussed in the context of Segal-toposes.
Very similar statements are discussed in
and, building on that, in example 1.8 of
A variant of this statement – more general in one respect, less general in another – appears in
as theorem 7.1.0.1.
There it is shown that for any $K \in \infty Grpd$ there is a bijection of homotopy sets of morphisms
where $(p^* \dashv p_*) : Sh_{(\infty,1)}(X) \to \infty Grpd$ is the geometric morphism we denoted $(LConst \dashv \Gamma)$ above.
If we also rewrite the left using the equivalence of $Top$ with $sSet$, this reads
For $K = Core(\infty Grpd)$ this is the $\pi_0$-decategorification of the above statement.
The geometric $\Pi_\infty$ of the terminal object in a locally ∞-connected (∞,1)-topos can be called the fundamental ∞-groupoid of the topos. It represents the shape of the topos.
On page 18-19 of
is described the construction of $\Pi(X) \in \infty Grpd$ for $X$ the terminal object in $Sh_{(\infty,1)}(C)$ on an ordinary site $C$ with $\Pi(X)$ as described above in Geometric fundamental oo-groupoid.
This reviews in particular (slightly implicitly)
Let $X$ be a topological space that has a basis of contractible open subsets. Write $X$ also for $X$ regarded as the terminal object in $Sh_{(\infty,1)}(X)$. Then the image of $X$ under $\Pi : Sh_{(\infty,1)}(X) \to \infty Grpd$ has the same homtopy groups as $X$ regarded as an object in Top:
This is a slight reformulation of the statement in
M. Artin, B. Mazur, Etale homotopy , Springer lecture notes in mathematics 100, Berlin 1969
Notice the local contractibility assumption. This is necessary in general for $\Pi(X)$ to make sense.
Let $C =$ Diff and consider in $Sh_{(\infty,1)}(Diff)$ the two objects
$S^1$, the $\infty$-stack represented by the standard circle in $Diff$;
$\mathbf{B}\mathbb{Z}$ – the $\infty$-stack constant on the delooping groupoid of the additive group $\mathbb{Z}$.
Then
the categorical homotopy groups of $S^1$ are all trivial
the geometric homotopy groups of $S^1$ are the usual ones obtained from regarding $S^1$ as an object in Top:
etc.
For $\mathbf{B}\mathbb{Z}$ it is the other way round:
the categorical homotopy groups of $\mathbf{B}\mathbb{Z}$ are
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