generalisations of the Seifert-van Kampen Theorem?

Question

I have been reading Jacob Lurie's book "Higher Algebra", version May 8, 2011. One is grateful to him for covering such a lot of ground and for making it all so readily available.

My attention was drawn to the Section A.3 on "The Seifert-van Kampen Theorem" p. 845.

It starts by stating the classical theorem determining the fundamental group of pointed space which is a union of two open sets with path-connected intersection. (The most general theorem of this type is for the fundamental groupoid on a set $A$ of base points for a space $X$ which is the union of a family $\mathcal U$ of open sets and such that $A$ meets each path-component of all 1-,2-,3-fold intersections of the sets of $\mathcal U$).

It states: " In this section, we will prove a generalization of the Seifert-van Kampen theorem, which describes the entire weak homotopy type of $X$ in terms of any sufficiently nice covering of X by open sets: Theorem A.3.1." However this Theorem makes no mention of groups or connectivity conditions.

So my question is: How does one deduce the SvKT as stated there, or its more general version, from Theorem A.3.1?

Theorem A.3.1 itself seems closely related to classical theorems on excision, showing the singular complex is chain homotopy equivalent to the singular complex of $\mathcal U$-small simplices. (I don't have the earliest reference for this, but I like the proof by R. Sch\"on from Proc. AMS 59 (1976).)

A particular point for the deduction of the most general version of the SvKT is: why the number 3? One explanation is that it has to do with the Lebesgue dimension of $\mathbb R^2$.

Answer 1

I don't actually see how to deduce the version of the classical SvKT on a set of base points $A$ directly from Lurie's version. It seems that to apply Lurie's theorem, we would need the stronger hypothesis that $A$ meets every path component of every finite intersection of open sets in $\mathcal{U}$ (which is reasonable, since the conclusion of Lurie's theorem is a stronger statement about fundamental $\infty$-groupoids rather than just fundamental 1-groupoids). But I think we can adapt his proof to derive the classical version.

Here's the first thing I'm going to try to prove: let $\chi:C\to \mathcal{O}(X)$ be a functor, with $C$ a small category and $\mathcal{O}(X)$ the poset of opens in $X$. For each $x\in X$, define $C_x$, as Lurie does, to be the full subcategory of $C$ spanned by those objects $c$ with $x\in\chi(c)$. Assume that for every $x$, the nerve of $C_x$ is simply connected. Then we have $\Pi_1(X) \cong \mathrm{colim}_{c\in C}\; \Pi_1(\chi(c))$, the colimit being a weak 2-colimit of groupoids.

Mimicking Lurie's argument in the 1-truncated case, we have a (pseudo 2-)functor $F:\mathrm{Gpd}^{\mathcal{O}(X)^{\mathrm{op}}} \to \mathrm{Gpd}$ defined by Kan extension from the functor $\Pi_1 : \mathcal{O}(X) \to \mathrm{Gpd}$. The (2,1)-topos $\mathrm{Sh}_{(2,1)}(X)$ is the (bicategorical) localization of $\mathrm{Gpd}^{\mathcal{O}(X)^{\mathrm{op}}}$ at the covering sieves, and Lurie's A.3.2 shows that $F$ inverts these covering sieves and hence factors through $\mathrm{Sh}_{(2,1)}(X)$. In particular, this induced functor $F:\mathrm{Sh}_{(2,1)}(X) \to \mathrm{Gpd}$ preserves (bicategorical) colimits.

Thus, it suffices to show that our functor $\chi:C\to \mathcal{O}(X)$ has colimit $X$ (the terminal object) when composed with the Yoneda embedding into $\mathrm{Sh}_{(2,1)}(X)$. And since $\mathrm{Sh}_{(2,1)}(X)$ has enough points (being sheaves on a topological space), it suffices to check this on all stalks. (At a finite categorical dimension, there is no hyper-incompleteness to worry about.) But at the stalk over $x\in X$, the $C$-diagram is trivial at those $c\in C_x$ and empty at the others, so its colimit is simply the groupoid reflection of $C_x$, which was assumed to be terminal (this is equivalent to the nerve of $C_x$ being simply connected).

This completes the proof of the 1-groupoidal version of Lurie's theorem. Now let's deduce a more classical statement. Let $X$ be our space and $\mathcal{U}$ an open cover of it. Define $C$ to be the category of 1-, 2-, or 3-fold intersections of open sets in $\mathcal{U}$, whose morphisms are the canonical inclusions from an $(n+k)$-fold intersection to an $n$-fold intersection, and let $\chi$ be the obvious functor.

For any $x\in X$, the category $C_x$ is obviously nonempty (because $\mathcal{U}$ covers $X$) and connected (because if $x\in U$ and $x\in V$, then $x\in U\cap V$). It is not much harder to see that it is simply connected: any two parallel zigzags of inclusions can be made equal by passing through at most triple intersections. Thus, we have

$$\Pi_1(X) \cong \mathrm{colim}_{c\in C} \; \Pi_1(\chi(c)).$$

Now let $A\subseteq X$ be a subset which meets all path components of all 1-, 2-, and 3-fold intersections of open sets in $\mathcal{U}$. Then $\Pi_1(\chi(c))$ is equivalent to its full sub-groupoid $\Pi_1(\chi(c),A)$ spanned by objects that are points of $A$, as is $\Pi_1(X)$. Since 2-dimensional colimits are invariant under equivalence of groupoids, the above statement passes to these groupoids as well.

Now the generalized SvKT of Ronnie and his coauthors amounts to asking that $\Pi_1(X)$ be the strict colimit of the functor $c\mapsto \Pi_1(\chi(c),A)$, in the 1-category of groupoids, so it basically suffices to show that this strict 1-colimit is also a (weak) 2-colimit. Now $C$ is a direct category, and $\mathrm{Gpd}$ is a model category with the canonical model structure (weak equivalences are equivalences, cofibrations are injective on objects), so $\mathrm{Gpd}^C$ inherits a Reedy model structure for which the adjunction

$$ \mathrm{colim} : \mathrm{Gpd}^C \;\rightleftarrows\; \mathrm{Gpd} : \Delta $$

is Quillen. It follows that the 1-colimit of a Reedy cofibrant diagram is also a 2-colimit. Unfortunately, $c\mapsto \Pi_1(\chi(c),A)$ is not Reedy cofibrant, but it is "partly" so. For instance, for any $U\in \mathcal{U}$, consider the latching object

$$ L_U = \mathrm{coeq}\left( \coprod_{V,W} \Pi_1(U\cap V\cap W,A) \;\rightrightarrows\; \coprod_{V}\Pi_1(U\cap V,A) \right) $$

Then the map $L_U \to \Pi_1(U,A)$ is injective on objects; thus our functor is at least Reedy cofibrant "at the top level". It will suffice to show that if $G\in \mathrm{Gpd}^C$ is "sufficiently Reedy cofibrant" in senses like this, then $\mathrm{colim}(G)$ can be calculated in a homotopy-invariant way.

Let $G\in \mathrm{Gpd}^C$, and enumerate the elements of $\mathcal{U}$ (perhaps transfinitely) as $(U_\alpha)_{\alpha<\lambda}$. We will define a transfinite sequence of groupoids $$ H_0 \to H_1 \to H_2 \to \cdots $$ such that $H_\alpha$ is the colimit of $G$ restricted to the subcategory of $C$ determined by the $U_\beta$ with $\beta<\alpha$, and their pairwise and triple intersections. Of course we can take $H_0 = 0$. Now given $H_\alpha$, define $H_{\alpha+1}$ to be the pushout of $H_\alpha$ and $G(U_{\alpha})$ along $$ K_\alpha = \mathrm{coeq}\left( \coprod_{\beta,\gamma < \alpha} G(U_\alpha \cap U_\beta \cap U_\gamma) \;\rightrightarrows\; \coprod_{\beta<\alpha} G(U_\alpha \cap U_\beta) \right). $$ For limit $\alpha$, we of course define $H_\alpha$ to be the colimit. It is easy to verify that each $H_\alpha$ is the colimit as asserted, and thus $\mathrm{colim}_{\alpha<\lambda} \; H_\alpha = \mathrm{colim} \; G$.

Now suppose $G$ has the property that the map $K_\alpha \to G(U_\alpha)$ is injective on objects (a cofibration) for every $\alpha$. Then the pushout defining $H_{\alpha+1}$ is a homotopy pushout and thus homotopy-invariant. Moreover, the map $H_\alpha \to H_{\alpha+1}$ is again a cofibration, so the colimits at limit stages are also homotopy colimits and thus homotopy-invariant. Therefore, for $G$ with this property, strict colimits are 2-colimits. But our functor $c\mapsto \Pi_1(\chi(c),A)$ does have this property (it is a slightly stronger version of being "Reedy cofibrant at the top level").

Thus, its strict colimit is also a homotopy colimit, so $\Pi_1(X,A)$ is equivalent to this strict colimit. The theorem of Ronnie and coauthors asserts that it is in fact isomorphic to this strict colimit, but it is easy to check that it has the same set of objects as the strict colimit (namely $A$), and an equivalence of groupoids which is bijective on objects is an isomorphism.

Answer 2

My impression (which is that of a total non-expert) is the following:

Spaces, according to the "homotopy hypothesis," are the same thing as $(\infty, 0)$-categories. This comes out of Lurie's (and Joyal's) work if one accepts that quasi-categories are an appropriate model for $(\infty, 1)$-categories: it's a result that a quasi-category with every 1-morphism invertible is the same thing as a Kan complex.

Let $X$ be a space, and let $U, V$ be open subsets which cover $X$. Then we have a homotopy pushout diagram expressing $X$ as the homotopy pushout $U \sqcup_{U \cap V} V$. In other words, if we think in terms of higher groupoids, $X$ is the homotopy pushout of the $\infty$-groupoids corresponding to $U, V, U \cap V$.

Describing the homotopy type of a homotopy pushout completely (i.e., in terms of homotopy groups and such) is really hard: otherwise we would know all the homotopy groups of spheres! What I take from this is that explicit models for higher category theory that one can easily compute with are likely to be very complicated, or otherwise homotopy theory would become easy.

However, there might be more luck if we restrict to special cases. For instance, there is an equivalence between 1-truncated spaces (spaces with no higher homotopy groups than $\pi_1$) and groupoids, given by taking the fundamental groupoid. As it happens, we can compute with groupoids: there is, for instance, a nice model category presentation of groupoids, and we can work out what the homotopy pushout of groupoids is.

Since truncation below $n$ is a left adjoint, this then amounts to saying that taking the fundamental groupoid sends homotopy pushout squares in spaces to homotopy pushout squares in groupoids. This is precisely the classical van Kampen theorem (stated for groupoids rather than groups, though.) More generally, we can say that the "fundamental $n$-groupoid of a space" (by which I mean the truncation below $n$) commutes with homotopy push-outs.

(Example: if we want to show that $\pi_1(S^1) = \mathbb{Z}$, we observe that $S^1 $ is a homotopy pushout $\ast_{\sqcup S^0} \ast$, so we have to compute the "suspension" of the discrete groupoid on two elements. Taking the homotopy pushout amounts to adding two isomorphisms identifying the two points, which gives a groupoid equivalent to $\mathbb{Z}$.)

I understand that you have done work generalizing the classical van Kampen theorem to higher homotopy groups. My guess that, in Lurie's language, some of that would translate into the construction of explicit algebraic model for 2-and-higher-groupoids (as opposed to 2-truncated spaces) and a means of computing homotopy pushouts (but I'm only speculating here), i.e., it seems to me that it would be quite different from what "Higher Algebra" does.