nForum - Search Results Feed (Tag: cohesive-homotopy-theory) 2021-12-08T09:51:50-05:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher axiomatic tangent structure of étale homotopy types https://nforum.ncatlab.org/discussion/6288/ 2014-10-25T23:18:39-04:00 2014-12-02T17:35:36-05:00 Urs https://nforum.ncatlab.org/account/4/ For XX a manifold of dimension nn, there is a canonical map &tau; X&colon;X&longrightarrow;BGL(n)\tau_X \colon X \longrightarrow \mathbf{B}\mathrm{GL}(n) to the moduli stack of the smooth ...

For $X$ a manifold of dimension $n$, there is a canonical map $\tau_X \colon X \longrightarrow \mathbf{B}\mathrm{GL}(n)$ to the moduli stack of the smooth general linear group, modulating the bundle to which the tangent bundle is associated.

I would like to formulate this construction axiomatically in differential cohesion.

It is clear to me how the formalization ought to proceed, but there are some gaps where I know what to do “informally”, but not yet how to do it fully axiomatically.

Here is how I imagine to start:

First, we fix some types that are to play the role of the local model spaces. One option is to ask for a type $\mathbb{A}^1$ that exhibits the cohesion in that the shape modality is $\mathbb{A}^1$-localization. Then take the $n$-dimensional model space to be $\mathbb{A}^n \coloneqq (\mathbb{A}^1)^{\times_n}$.

Then an $n$-dimensional manifold is a type $X$ such that there exists a morphism (the atlas)

$\coprod_i \mathbb{A}^n \longrightarrow X$

which is 1) a 1-epimorphism and 2) is formally étale in that it is modal for the infinitsimal shape modality $\Pi_{inf}$.

Next we need to axiomatize the group $\mathrm{GL}(n)$. In the standard model of differential cohesion this group is the following:

First, the formal disk $D^n$ around the origin in $\mathbb{A}^n$ is the homotopy fiber of the unit $\mathbb{A}^n \longrightarrow \Pi_{inf}(\mathbb{A}^n)$ of the infinitesimal shape modality

$D^n \coloneqq \mathrm{fib}(\mathbb{A}^n \longrightarrow \mathbb{A}^n)$

Then, $\mathrm{GL}(n)$ is just the automorphism group of that:

$\mathrm{GL}(n) \coloneqq \mathbf{Aut}(D^n) \,.$

To produce the canonical morphism

$X \longrightarrow \mathbf{B} \mathrm{GL}(n)$

we should proceed by producing a canonical morphism from the Cech nerve of $\coprod\mathbb{A}^n\to X$ to the Cech nerve of $\ast \to \mathbf{B}\mathrm{GL}(n)$, hence a map of simplicial types that starts out looking like this:

$\array{ \vdots && \vdots \\ \downarrow \downarrow \downarrow && \downarrow \downarrow \downarrow \\ (\coprod_i \mathbb{A}^n) \underset{X}{\times} (\coprod_i \mathbb{A}^n) &\stackrel{(\tau_X)_1}{\longrightarrow}& \mathrm{GL}(n) \\ \downarrow\downarrow && \downarrow\downarrow \\ \coprod_i \mathbb{A}^n &\longrightarrow& \ast }$

This is where I am presently stuck. I know how to proceed rigorously, but not how to proceed fully formally, if you see what I mean.

Rigorously, what I am supposed to say is that $(\tau_X)_1$ is the map which to a point

$\hat x \;\colon\; \ast \longrightarrow (\coprod_i \mathbb{A}^n) \underset{X}{\times} (\coprod_i \mathbb{A}^n)$

assigns the element in $\mathrm{GL}(n)$ obtained by first forming the formal disk $D^n_{\hat X}$ around that point, as above, and then using the induced diagram

$\array{ && D^n_{\hat x} \\ & {}^{\mathllap{i_1}}\swarrow && \searrow^{\mathrlap{i_2}} \\ \mathbb{A}^n && && \mathbb{A}^n \\ & \searrow && \swarrow \\ && X }$

together with the fact that all maps here are formally étale to form something that one would denote

$(i_1^{-1}) \circ i_2 \in \mathbf{Aut}(D^n_{\hat x}) \simeq \mathrm{GL}(n) \,.$

where “$i_1^{-1}$” means the inverse of the corestriction of $p_1$ onto its image, and where some kind of rigid translation $\mathbb{A}^n \stackrel{\simeq}{\longrightarrow} \mathbb{A}^n$ making the base points coincide is implicit.

But here I am a little stuck with the formalization. How would one formalize this “obvious” construction of $(\tau_X)_\bullet$?

]]>
higher Penrose-Ward transform in cohesive homotopy theory https://nforum.ncatlab.org/discussion/6331/ 2014-11-19T11:49:16-05:00 2014-11-28T20:16:29-05:00 Urs https://nforum.ncatlab.org/account/4/ I’d like to formalize in cohesive homotopy theory the Penrose-Ward transform for higher bundles as it appears for 3-bundles in section 4 of Christian Saemann, Martin Wolf, Six-Dimensional ...

I’d like to formalize in cohesive homotopy theory the Penrose-Ward transform for higher bundles as it appears for 3-bundles in section 4 of

• Christian Saemann, Martin Wolf, Six-Dimensional Superconformal Field Theories from Principal 3-Bundles over Twistor Space (arXiv:1305.4870)

I think I know in principle how it goes. But at one step I need a fiberwise $\flat$-modality. This is something that Mike has kept asking about recently and I didn’t give an answer. I still don’t have the answer, but at least now I have a good motivating example.

So we consider a correspondence

$\array{ && Z \\ & {}^{\mathllap{p_1}}\swarrow && \searrow^{\mathrlap{p_2}} \\ X_1 && && X_2 }$

Let $G$ be an $\infty$-group and consider a $G$-principal $\infty$-bundle $P \to X_1$ equipped with a trivialization of its pullback along $p_1$.

The transform in question is supposed to do the following: from the chosen trivialization on $Z$ we are to produce a $p_1$-fiberwise $G$-valued function on $Z$. Associated with this is a $p_1$-fiberwise Maurer-Cartan form on $Z$ which glues to a global form on $Z$ and this we are to push down along $p_2$. (This is an abstract rephrasing of what appears in components in section 4 of the above article.)

Here I’ll ignore the push-forward for the moment and consider only the first step of producing that fiberwise Maurer-Cartan form.

On a single fiber $x \colon \ast \to X$ it works like this: let $g$ be the map that modulates $P \to X_1$. Then the assumed trivialization of the pullback of this $G$-bundle to $Z$ gives the diagram

$\array{ p_1^{-1}(x) &\longrightarrow& Z &\longrightarrow& \ast \\ \downarrow && \downarrow && \downarrow \\ \ast &\stackrel{x}{\longrightarrow}& X_1 &\stackrel{g}{\longrightarrow}& \mathbf{B}G }$

and, by looping, this induces a map $p_1^{-1}(x) \longrightarrow G$. This is the $G$-valued function on the fiber. Its fiber MC form is given by the composite

$p_1^{-1}(x) \longrightarrow G \longrightarrow \flat_{dR} \mathbf{B}G \,.$

Now we need to generalize this to a construction that does this for all fibers at once.

]]>
axiomatic theta-functions https://nforum.ncatlab.org/discussion/6214/ 2014-09-02T21:22:55-04:00 2014-09-29T16:20:53-04:00 Urs https://nforum.ncatlab.org/account/4/ This here is to vent some thoughts on how to formalize theta functions and, if possible, bundles of conformal blocks, in cohesive homotopy theory. It is related to the note Local prequantum field ...

This here is to vent some thoughts on how to formalize theta functions and, if possible, bundles of conformal blocks, in cohesive homotopy theory. It is related to the note Local prequantum field theory (schreiber) and, just as that note, is based on discussion with Domenico Fiorenza.

One basic idea here is that

1. theta functions are transgressions of Chern-Simons-type functionals to codimension 1.

2. along the lines of example 3.2.17 in Local prequantum field theory (schreiber) that transgression is universally provided by the cobordism hypothesis for coefficients being the $n$-category of $n$-fold correspondences in the slice of the given cohesive $\infty$-topos over the “$n$-group of phases”.

More concretely, let $\mathbf{Fields} \in \mathbf{H}$ be any cohesive homotopy type, let $\mathbb{G}$ a cohesive abelian $\infty$-group object, then a “Chern-Simons $n$-bundle” is a map

$\exp(\tfrac{i}{\hbar} S_{CS}) \colon \mathbf{Fields}\longrightarrow \mathbf{B}^n \mathbb{G} \,.$

Regard this as an object in the slice

$\exp(\tfrac{i}{\hbar} S_{CS}) \in \mathbf{H}_{/\mathbf{B}^n \mathbb{G}} \,.$

Consider the $(\infty,n)$-category of $n$-fold correspondences in this slice

$\mathcal{C}\coloneqq Corr_n(\mathbf{H}_{/\mathbf{B}^n \mathbb{G}}) \,.$

Every object here is supposed to be fully dualizable, hence $\exp(\tfrac{i}{\hbar} S_{CS})$ defines the local action functional of a local prequantum field theory of dimension $n$

$\exp(\tfrac{i}{\hbar} S_{CS}) \colon Bord_n^{fr} \longrightarrow Corr_n(\mathbf{H}_{/\mathbf{B}^n \mathbb{G}}) \,.$

The claim is that to a closed $n$-framed $(n-1)$-manifold $\Sigma_{n-1}$ this monoidal $n$-functor assigns a map

$[\Pi(\Sigma), \mathbf{Fields}] \longrightarrow \mathbf{B}\mathbb{G}$

regarded as an $(n-1)$-fold homotopy between trivial homotopies between the 0-map $\ast \to \mathbf{B}^n \mathbb{G}$.

This should be the theta bundle. Here $[\Pi(\Sigma),\mathbf{Fields}]$ is the mapping stack from the fundamental $\infty$-groupoid $\Pi(\Sigma)$ to $\mathbf{Fields}$, hence is the moduli stack of flat $\mathbf{Fields}$-valued $\infty$-connections on $\Sigma$, hence is the covariant phase space of the Chern-Simons theory. The “theta bundle” is equivalently the prequantum line bundle of the CS-theory on $\Sigma$.

One question to be thought about is this:

to turn $\exp(\tfrac{i}{\hbar} S_{CS})$ into a local prequantum field theory on cobordisms which are not framed but are equipped with $(G\to O(n))$-structure it needs to be equipped with the structure of a $G$-homotopy fixed point in $Core(\mathcal{C})$. What is this structure more explicitly?

]]>
arithmetic cohesion -- table https://nforum.ncatlab.org/discussion/6162/ 2014-08-15T15:04:15-04:00 2014-08-15T15:04:15-04:00 Urs https://nforum.ncatlab.org/account/4/ started a table-for-inclusion arithmetic cohesion – table and included it into relevant entries. In the course of this I started a minimum at adic residual.

started a table-for-inclusion arithmetic cohesion – table and included it into relevant entries.

In the course of this I started a minimum at adic residual.

]]>