The n-Category Café

Question 6) homotopy hypothesis for generalized spaces

Forgot this one: this concerns figure 5.

In the context of the homotopy hypothesis we pick a notion of $\infty\mathbf{Groupoid}$ and pick a notion of $\mathbf{Spaces}$ such that each space has an fundamental $\infty$-groupoid and each $\infty$-groupoid has a geometric realization as a space – and ask if these two operations yield some kind of equivalence $\infty\mathbf{Groupoids} \simeq \mathbf{Spaces}$.

Consider this for the choice $$\mathbf{Spaces} := \mathrm{Sheaves}(\mathbf{Euclid})$$ where $\mathbf{Euclid}$ is the full subcategory of $\mathbf{Manifolds}$ on vector spaces and the choice $$\infty\mathbf{Groupoids} := \omega\mathbf{Groupoid}(\mathbf{Spaces})$$ ($\omega$-groupoids internal to $\mathbf{Spaces}$).

Let $$\Pi_\omega : \mathbf{Spaces} \to \omega\mathbf{Groupoids}(\mathbf{Spaces})$$ be the functor which sends each space $X$ to its fundamental $\omega$-groupoid $\Pi_\omega(X)$ whose $k$-morphisms are thin homotopy classes of smooth images of the standard $k$-disk in $X$, (constant in a neighbourhood of the boundary of the disk).

Let $$|-| : \omega\mathbf{Groupoids}(\mathbf{Spaces}) \to \mathbf{Spaces}$$ be the functor which sends each $\omega$-groupoid $C$ to the sheaf $$|C| : U \mapsto hom(\Pi_\omega(U),C) \,.$$

First sub-question: this $|-|$ is defined without mentioning the $\omega$-nerve of $C$. It should nevertheless yield the right notion of geometric realization in $\mathbf{Spaces}$. Is that right?

Second subquestion: we have Quillen model category structures on both sides. Do $|-|$ and $\Pi_\omega$ induce a Quillen model equivalence? If not, what else do we get? At least an adjunction one way or the other?

Hi Urs,

which notion of a $\w$-groupoid are you choosing? I suppose that in the strict case (factoring maps through thin homotopy) it may be not entirely trivial to shown that the compositions descend to the quotient.

More later,

João

Hi Urs,

I may be missing something, but it would appear to me that the premise of your question 1 is incorrect. That is, it is certainly not the case that the unique $\omega$-functor $!:\mathcal{O}_n\rightarrow \text{pt}$, from the $n^{\text{th}}$ oriental to the point, is a weak equivalence in the folkloric model category structure on $\omega-\text{Cat}$.

By definition, as you metioned, the weak equivalences in this structure are the $\omega$-functors $f:\mathbb{C}\rightarrow\mathbb{D}$ which are essentially surjective on all cells.

That is to say if $x$ and $y$ are parallel $(n-1)$-cells in $\mathbb{C}$ (which means that they have common $(n-2)$-sources and targets) and there is an $n$-cell $z$ in $\mathbb{D}$ with $(n-1)$-source $f(x)$ and $(n-1)$-target $f(y)$ then there exists an $n$-cell $w$ in $\mathbb{C}$ with $(n-1)$-source $x$ and $(n-1)$-target $y$ such that $f(w)$ is equivalent to $z$ in $\mathbb{D}$.

This condition, however, is certainly not satisfied by $!:\mathcal{O}_n\rightarrow\text{pt}$.

To see why this is the case, start by observing that $\mathcal{O}_n$ has a unique non-identity $n$-cell, which we might call $\alpha_n$, and that this cell has $(n-1)$-source $s_{n-1}(\alpha_n)$ and $(n-1)$-target $t_{n-1}(\alpha_n)$ which are distinct and parallel $(n-1)$-cells.

Of course, the point $\text{pt}$ as an $\omega$-category consists only of a single $0$-cell $\bullet$ along with its identity cell at each dimension. So it follows that $!$ maps both of $s_{n-1}(\alpha_n)$ and $t_{n-1}(\alpha_n)$ to the (unique) identity on $\bullet$ at dimension $(n-1)$. Furthermore, it also follows that the identity on $\bullet$ at dimension $n$ provides us with a n $n$-cell whose $(n-1)$-source and $(n-1)$-target are both equal to $!(s_{n-1}(\alpha_n))=!(t_{n-1}(\alpha_n))$.

So we may regard the $n$-identity on $\bullet$ as being an $n$-cell with $n$-source $!(t_{n-1}(\alpha_n))$ and $n$-target $!(s_{n-1}(\alpha_n))$ (note the orientation here, identities are reversible). Now were $!$ a weak equivalence, we could appeal to essential surjectivity and find an $n$-cell in $\mathcal{O}_n$ whose $n$-source is $t_{n-1}(\alpha_n)$ and whose $n$-target is $s_{n-1}(\alpha_n)$. In other words, it would allow us to find an $n$-cell in $\mathcal{O}_n$ whose $n$-orientation is reversed relative to that of $\alpha_n$. So since $s_{n-1}(\alpha_n)$ and $t_{n-1}(\alpha_n)$ are distinct it follows that this $n$-cell cannot be equal to $\alpha_n$ and cannot be an identity, thus contradicting the fact that $\alpha_n$ is the unique non-identity $n$-cell of $\mathcal{O}_n$.

Indeed, if we work just a little harder more can be said. For instance, although I haven’t checked the details, it appears clear that we should be able to prove that any $\omega$-category which is weakly equivalent to an $\omega$-groupoid is itself an $\omega$-groupoid. However, in a precise sense $\mathcal{O}_n$ is about as far from being an $\omega$-groupoid as you can get. Indeed, this observation is pretty much a direct restatement of Street’s (very strong) loop freeness results for the parity complex which generates $\mathcal{O}_n$.

While I am thinking of such things, rather than painting ceilings which is my foreground task for this weekend, let me comment on questions 2 and 3.

First some notation, we take $\otimes$ to denote the (lax) Gray tensor on $\omega$-Cat. This is a non-symmetric tensor which has left and right closures, which we will denote by $\text{lax}_l(\mathbb{C},\mathbb{D})$ and $\text{lax}_r(\mathbb{C},\mathbb{D})$ respectively. We adopt this notation because these are precisely the appropriate (left and right handed) generalisations of the 2-categories of $2$-functors, lax-natural transformations and modifications of the 2-dimensional theory.

Now, although $\otimes$ is not symmetric (or braided) as a monoidal structure on $\omega$-Cat, we may still do enriched category theory relative to it - so long as we take a little care with our definitions. In general, we can follow much of the elementary presentation of enriched category theory given in the first few chapters of Max Kelly’s book on this topic. Alternatively, we could think of $\omega$-Cat as a 1-object bicategory and use Street’s presentation of enrichment in a bicategory to provide us with the appropriate background.

In particular we may use the right closure $\text{lax}_r$ to enrich $\omega$-Cat over itself. Notice here that were we to use $\text{lax}_l$ for this we would not obtain a category enriched wrt $\otimes$, but would instead obtain a category enriched wrt the reverse tensor $X\otimes^\circ Y = Y\otimes X$.

Furthermore, $\Delta$ can also be made into a category enriched wrt $\otimes$, simply by regarding each of its homsets as a discrete $\omega$-category.

Now we are able to follow Kelly and enrich the functor category $[\Delta,\omega-\text{Cat}]$, using the right closure $\lax_r$ and certain kinds of limits in $\omega$-Cat called ends. Explicitly, if $F$ and $G$ are two (enriched) functors from $\Delta$ to $\omega$-Cat then the hom $\omega$-category between them is given by the formula

wherein we use the integral notation (as introduced by MacLane) for ends.

Observe now that if $\mathbb{C}$ is any $\omega$-category, then by definition an $\omega$-functor $\mathbb{C}\rightarrow [\Delta,\omega-\text{Cat}](F,G)$ corresponds to a family of $\omega$-functors $\mathbb{C}\rightarrow\text{lax}_r(F(n),G(n))$ which is extra-ordinarily natural in $n\in\Delta$ (by the universal property of ends). Such a family, in turn, corresponds to a family of $\omega$-functors $\mathbb{C}\otimes F(n)\rightarrow G(n)$ which is natural in $n\in\Delta$ (using the adjunction $-\otimes F(n)\dashv \text{lax}_r(F(n),-)$). Consequently, taking $\mathbb{C}$ to be the 1-glob $\mathbb{G}^1=\mathcal{O}(G^1)$, we find that the 1-cells of $[\Delta,\omega-\text{Cat}](F,G)$ correspond to $\omega$-functors $\mathbb{G}^1\rightarrow[\Delta,\omega-\text{Cat}](F,G)$ which in turn correspond precisely to the cells pictured in Urs’ figure 2. Taking $\mathbb{C}$ to be the $n$-glob $\mathbb{G}^{n-1}=\mathcal{O}(G^n)$ we recover analogous descriptions of the $n$-cells of this hom-$\omega$-category.

So we find that the hom-$\omega$-category defined in the end formula above is precisely the $\omega$-category described in Urs’ pictures.

However, we have now also succeeded in showing that the descent $\omega$-category of Street and Urs’ suggestion are indeed one in the same, as conjectured in question 3. What Urs’ calls $\text{hom}_{\omega\text{CoSimp}}(\mathcal{O}(\Delta^{(-)}),\mathcal{E})$ is simply the enriched homset I would denote by $[\Delta,\omega-\text{Cat}]_r(\mathcal{O}(\Delta^{(-)}), \mathcal{E})$, as in the end formula above. Then as we have seen, $n$-cells of that hom-$\omega$-category correspond to families of $\omega$-functors $\mathcal{O}(G^n\otimes \Delta^m)\cong\mathbb{G}^n\otimes\mathcal{O}(\Delta^m)\rightarrow \mathcal{E}^n$ which are natural in $n\in\Delta$. And these in turn are simply the elements of the (un-enriched) homset $[\Delta,\omega-\text{Cat}](\mathcal{O}(G^n\otimes \Delta^\bullet),\mathcal{E})$ given by Street in his definition of $\text{Desc}(\mathcal{E})$.

BTW On the topic of the Gray tensor product of strict $\omega$-categories, Street’s nerve construction allows us to give a purely simplicial construction of this tensor. For those who are simplicially minded this approach provides a simpler and more direct presentation of the Gray tensor, which can be used to circumvent problems that arise from its traditional presentation in terms of colimits of free $\omega$-categories on products of globs. More details on this topic can be found in my recently published Memoir of the AMS on the topic of Complicial Sets.

Here is what I am imagining to say:

Quickly recall the basic setup and notation I am using:

$\mathbf{Euclid}$: the full subcategory of $\mathbf{Manifolds}$ on vector spaces.

$\mathbf{Spaces} := Sh(\mathbf{Euclid})$: sheaves on $\mathbf{Euclid}$ with respect to the standard notion of covers.

$\omega\mathrm{Cat}(\mathbf{Spaces})$: $\omega$-categories internal to $\mathbf{Spaces}$.

$C \stackrel{\simeq_w}{\to} D$: a weak equivalence of $\omega$-categories, i.e. an $\omega$-functor essentially $k$-surjective for all $k$.

$\omega\mathrm{Grpd}(\mathbf{Spaces})$: $\omega$-categories with specified strict inverses for all $k$-morphisms for all $k$ with inverse-assigning maps being morphisms in $\mathbf{Spaces}$.

$\omega\mathrm{Grp}(\mathbf{Spaces})$ : $\omega$-groupoids of the form $G = C(a,a)$ for $C \in \omega\mathrm{Grpd}(\mathbf{Spaces})$ and $a \in Obj(C)$.

$\mathbf{B}G$: the unique $\omega$-groupoid with a single object $\bullet$ and $C(\bullet,\bullet) = G$.

Then:

Definition [$\omega$-covers] For $\mathbf{X} \in \omega\mathrm{Cat}(\mathbf{Spaces})$, a cover of $X$ is an epimorphic weak equivalence $$\pi : \mathbf{Y} \stackrel{\simeq_w}{\to \gt} \mathbf{X} \,.$$ A refinement of covers is an epimorphism of covers $$\array{ \mathbf{Y} &\to \gt& \mathbf{Y}' \\ & {}_\pi\searrow \swarrow_{\pi'} \\ & \mathbf{X} } \,.$$ Denote the category formed by covers of $\mathbf{X}$ by $\mathrm{Covers}(\mathbf{X})$.

Definition [$\omega$-cohomology] For $\mathbf{X},C \in \omega\mathrm{Cat}(\mathbf{Spaces})$, the cohomology of $\mathbf{X}$ with coefficients in $C$ is $$H(\mathbf{X},C) := \int^{\mathbf{Y} \in \mathrm{Covers}(\mathbf{X})} \mathrm{hom}(\mathbf{Y},C) \,.$$

cocycles - objects of $H(\mathbf{X},C)$

coboundaries - 1-morphisms in $H(\mathbf{X},C)$; if $C = \mathbf{B}^{n-1}U(1)$ then these have the familiar form $a + d\lambda = b$.

cohomology classes - equivalence classes in $H(\mathbf{X},C)$

Definition [various kinds of cohomologoes]

Nonabelian cohomology: $G$ an $\omega$-group and $C := \mathbf{B}G$.

Equivariant cohomology: $\mathbf{X}$ not a discrete $\omega$-category

group cohomology: equivariant cohomology of a point, for instance for $G$ a group and $A$ an abelian group $H^n(G,A) = H(\mathbf{B}G, \mathbf{B}^{n}A)$.

differential cohomology : $\mathbf{X} = \Pi_\omega(X)$ (flat) or $\mathbf{X} = P_n(X)$ (fake-flat) – see next definition [there is also the non-fake flat case, but that requires more discussion]

Definition [fundamental $\omega$-groupoids]

For $X \in \mathbf{Spaces}$, let $\Pi_\omega(X)$ be the $\omega$-groupoid whose $k$-morphisms are maps of the standard $k$-disk in $X$, constant in a neighbourhood of the boundary of the disk and modulo thin-homotopy.

Denote the truncation of $\Pi_\omega(X)$ at $n$-morphisms by $P_n(X)$ and the result of identifying in $P_n(X)$ $n$-morphisms equivalence in $\Pi_\omega(X)$ by $\Pi_n(X)$.

In particular $P_0(X)$ is the discrete $\omega$-category over $X$ and $\Pi_1(X)$ the fundamental 1-groupoid of $X$. (I had the notation for $P_0(X)$ consistently wrong in the above comments. Sorry.)

Definition [geometric realization]

Let $$|-| : \mathbf{\omega}\mathrm{Cat}(\mathbf{Spaces}) \to \mathbf{Spaces}$$ be defined such that for $C \in \omega\mathrm{Cat}(\mathbf{Spaces})$ the realization $|C|$ is the sheaf $$|C| : U \mapsto hom(\Pi_\omega(U),C) \,.$$

[“smooth homotopy hypothesis”]

The fundamental $\omega$-groupoid construction and realization fit together nicely (ahem) $$\Pi_\omega : \mathbf{Spaces} \leftrightarrow \omega\mathrm{Cat}(\mathbf{Spaces}) : |-|$$

Definition [differential forms on $\mathbf{Spaces}$]

Write $\Omega^\bullet \in \mathbf{Spaces}$ for the deRham sheaf $$\Omega^\bullet : U \mapsto \Omega^\bullet(U) \,.$$ For $X \in \mathbf{Spaces}$ write $$\Omega^\bullet(X) := hom(X,\Omega^\bullet)$$ for the space of differential forms on $X$. This naturally carries the structure of a differential non-negatively graded-commutative algebra (“DGCA”) and yields a contravariant functor $$\Omega^\bullet : \mathbf{Spaces} \to DGCAs \,.$$

Definition [classifying space for algebra-valued differential forms]

For $A$ any DGCA, denote by $S(A)$ the space given by the sheaf $$S(A) : U \mapsto hom(A,\Omega^\bullet(X)) \,.$$ This yields a contravariant functor $$S : DGCAs \to \mathbf{Spaces} \,.$$

[“differential homotopy hypothesis”] The contravariant functors $$\Omega^\bullet : \mathbf{Spaces} \leftrightarrow DGCAs : S$$ form a contravariant adjunction with unit $$Id_{DGCAs} \Rightarrow \Omega^\bullet(S(A))$$ given by the inclusion $$A \hookrightarrow \Omega^\bullet(S(A))$$ that acts as $$(a \in A) \mapsto \forall U : ( (f \in hom(A,\Omega^\bullet(U))) \mapsto (f(a) \in \Omega^\bullet(U))) \,.$$

Definition [$L_\infty$-algebroid]

A (finite rank) $L_\infty$-algebroid is a manifold $X$ and a cochain complex $g$ concentrated in non-positive degree of finite rank $(A := C^\infty(X))$-modules together with a linear (over the ground field )degree +1 algebra derivation $$d : \wedge^\bullet_A g^* \to \wedge^\bullet_A g^*$$ (where $g^*$ is the dual complex (over $A$), hence non-negatively graded) such that $$d^2 = 0 \,.$$ The DGCA thus defined $$CE(g) := (\wedge^\bullet_A g^*, d)$$ is the Chevalley-Eilenberg DGCAof $g$. We identify the category of $L_\infty$-algebroids with the image $$CE : L_\infty \to DGCAs \,.$$

If $g$ is concentrated in the first $n$ degrees this is a Lie $n$-algebroid. If $X = pt$ this is a (finite dimensional) $L_\infty$-algebra or a Lie $n$-algebra if $g$ is concentrated in the first $n$-degrees. If $X = pt$ and $d : g^* \to g^* \wedge g^*$ this is a dg-Lie algebra.

Definition [$\infty$-Lie integration and differentiation]

Combining the smooth and the differential “homotopy hypotheses” with this definition we obtain the situation $$\omega\mathrm{Cat}(\mathbf{Spaces}) \leftrightarrow \mathbf{Spaces} \leftrightarrow DGCAs \leftarrow L_\infty \,.$$ We say that going from right to left through this is $\infty$-Lie integration. Going from left to right is $\infty$-Lie differentiation.

Definition [$L_\infty$-algebraic cocycle]

For $g$ a Lie $n$-algebra, $X$ a space and $Y \to\gt X$ a regular epimorphism, a $g$-cocycle on $X$ is a DGCA morphism

$$\Omega^\bullet_{vert}(Y) \leftarrow CE(g) : A_{vert} \,.$$

Let $G$ be the simply connected $n$-group integrating $g$, i.e. $$\mathbf{B}G := \Pi_n \circ S \circ CE (g) \,.$$ Notice that the integration of $\Omega^\bullet_{vert}(Y)$ is $$\Pi_n^{vert}(Y) := \Pi_n \circ S (\Omega^\bullet_{vert}) \,,$$ the sub $n$-groupoid of $\Pi_n(Y)$ of all those $n$-paths that project down to a point in $X$.

Definition [Lie integration of $L_\infty$-algebraic cocycles]

Let $\mathbf{Y}$ be a quotient of $\Pi_n^{vert}(Y)$ which is weakly equivalent to $P_0(X)$. Let $G'$ be a quotient of $G$ such that there is a commuting diagram

$$\array{ \Pi_n^{verty}(Y) &\stackrel{\Pi_n\circ S \circ CE (A_{vert})}{\to}& \mathbf{B}G \\ \downarrow && \downarrow \\ \mathbf{Y} &\stackrel{g}{\to}& \mathbf{G}' \\ \downarrow^{\simeq_w} \\ P_0(X) } \,.$$ The existence of at least one such diagram is the integrability condition on $A_{vert}$.

Then this nonabelian cocycle $$g : \mathbf{Y} \to \mathbf{B}G'$$ is a nonabelian cocycle integrating $A_{\mathrm{vert}}$.

Proposition. This integration of $L_\infty$-algebraic cocylces to nonabelian cocycles yields in particular a refinement (of the method and of the result) in nonabelian cohomology of the Chern-Simons cocycles constructed by Brylinski&MacLaughlin:

where B&M obtain Chern-Simons $n$-cocycles obstructing lifts of $G$-cocycles through shifted central extensions

$$\mathbf{B}^{n-1}U(1) \to \hat G \to G$$

this procedure constructs the corresponding twisted $\hat G$-cocycles which generalize the twisted bundles that appear in twisted K-theory.

Definition [shifted central extension of $\omega$-groups]

The $\omega$-functors $$\mathbf{B}^{n-1}U(1) \to \hat G \to G$$ are a shifted central extension is the first one is monic, the second epic and if there is a weak equivalence from the weak quotient to $G$: $$\mathbf{B}(U(1) \to \hat G) \stackrel{\simeq_w}{\to} \mathbf{B}G \,.$$

Given a $G$-cocycle $$\mathbf{Y} \stackrel{g}{\to} \mathbf{B}G$$ we ask for the obstruction to lifting this through $$\array{ && \mathbf{B}^n U(1) \\ && \downarrow \\ && \mathbf{B}\hat G \\ && \downarrow \\ \mathbf{Y} &\stackrel{g}{\to}& \mathbf{B}G } \,.$$ With $\mathbf{Y}$ chosen fine enough we can always lift to a twisted $\hat G$-cocycle, namely a $\mathbf{B}(U(1) \to \hat G)$-cocycle as in figure 4.

In [SSS] this is described for $L_\infty$-algebraic cocycles. Hit everything in there with $\Pi_n \circ S : DGCAs \to \omega\mathrm{Cat}(\mathbf{Spaces})$ to get the corresponding statements in nonabelian cohomology.