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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 26th 2013
• (edited Apr 26th 2013)

I have seen (e.g. here) and been involved in various discussions of the universal characterization of, ultimately, the (infinity,n)-category of spans in, ultimately, some ambient $\infty$-category with finte $\infty$-limits.

But I don’t seem to have seen a definite answer.

Guess. For $\mathbf{H}$ an $\infty$-category with finite $\infty$-limits, $Span_n(\mathbf{H})$ is the symmetric monoidal $(\infty,n)$-category with fully-self-dual objects, free on $\mathbf{H}$.

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeApr 26th 2013

So is $Bord_n = Span_n(\mathbf{H})$, for some $\mathbf{H}$?

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 26th 2013
• (edited Apr 26th 2013)

No, crucially not. $Bord_n$ is free for fully dualizable objects, but the objects in $Span_n(\mathbf{H})$ are fully self-dual.

Accordingly.

$Span_n(\ast) \simeq \ast \,,$

clearly, while of course the free etc. with duals on $\ast$ is $Bord_n(\ast) = Bord_n$, where the whole structure comes precisely from the fact that a dual to $\ast$ is thrown in which is not equivalent to the original $\ast$.

So if $Span_n(-)$ has a similarly universal description it must be different from that of $Bord_n(-)$. At the same time it is kind of similar. I am wondering how to express this, precisely.

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeApr 27th 2013

So what might $Span_{n}(\mathbf{H})$ be used for?

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeApr 27th 2013
• (edited Apr 27th 2013)

Ah, I see. I have now written a note explaining this here:

This certainly can do with more polishing and expansion, but this is how far I got right now. Let me know what you think.

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeApr 27th 2013

So the dualizable $\ast$ gets sent by Fields to an object in the span, which as such is self-dual, and so the dual of $\ast$ gets sent to the same object?

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeApr 27th 2013
• (edited Apr 27th 2013)

Yes, that’s right.

So a choice of fields for an $n$-dimensional topological local prequantum field theory, which is a map

$\mathbf{Fields} \colon Bord_n^\otimes \to Span_n(\mathbf{H})^\otimes \,,$

assigns one and the same moduli stack of fields to any point

$\mathbf{Fields}(\ast^+) \simeq \mathbf{Fields}(\ast^-) \,.$

But when we add a localized action functional, hence lift this to a map $S$ in

$\array{ && Span_n(\mathbf{H}, \flat \mathbf{B}^n U(1)) \\ & {}^{\mathllap{S}}\nearrow & \downarrow \\ Bord_n &\underset{\mathbf{Fields}}{\to}& Span_n(\mathbf{H}) }$

then one point is sent to the localized action functional

$\left[ \array{ \mathbf{Fields} \\ \downarrow^{\mathrlap{S}} \\ \flat \mathbf{B}^n U(1) } \right]$

while the other is sent to minus that action functional

$\left[ \array{ \mathbf{Fields} \\ \downarrow^{\mathrlap{-S}} \\ \flat \mathbf{B}^n U(1) } \right]$

(At prequantum field theory this is currently discussed here, but will be expanded…)

• CommentRowNumber8.
• CommentAuthorDavid_Corfield
• CommentTimeApr 28th 2013

Is this then a special way to find a codomain for an extended TQFT? Rather than just looking for a symmetric monoidal $(\infty, n)$-category containing a fully dualizable object, you first find one with fully self-dual objects, and then take mappings into $\flat \mathbf{B}^n U(1)$. So the latter is a good choice because of its symmetry?

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeApr 29th 2013
• (edited Apr 29th 2013)

This is maybe part of the question about what makes $Span_n(\cdots)$ special on general abstract grounds.

Heuristically, $Span_n(\cdots)$ is a good codomain for pre-quantum field theories because it nicely captures the old concept of trajectories/histories of fields, in a localized way. And so classes of examples show that it is a good codomain for capturing pre-quantum field theory.

But in general the question as to what kind of codomain to choose for which general reason when describing topological field theories as $n$-functors out of $Bord_n$ is open. For instance in the last months Mike Hopkins has been giving talks on the question of how to universally obtain codomains for the full quantum theory from their value on codimension 1.

The main point to notice maybe is that while it is nice that the cobordism theorem is so general, for the purposes of physics it is too general. For most choices of $\mathcal{C}^\otimes$ maps $Bord_n^\otimes \to \mathcal{C}^\otimes$ are nothing like what in physics one would recognize as a “field theory”, quantum or not. And there is not yet a good idea of how to characterize those $\mathcal{C}^\otimes$ for which it is, and why, I think. There is just a bunch of hints, maybe.

I imagine $Span_n(\mathbf{H}, \flat \mathbf{B}^n U(1))$ for $\mathbf{H}$ a cohesive $\infty$-topos has a good chance of being not only good in examples for pre-quantum field theory, but also having a general abstract reason of existence. But I am not 100% sure yet.

Then next I imagine that the correct codomain for genuine quantum theory is the image of $Span_n(\mathbf{H}, \flat \mathbf{B}^n U(1))$ under a kind of stabilization operation, and that postcomposition with the stabilization map is quantization. But this is just a vague hunch at the moment.

• CommentRowNumber10.
• CommentAuthorDavidRoberts
• CommentTimeApr 29th 2013

Yes, Hopkins was saying (in Singapore in January) that in practice we tend to get objects (field theories etc) where the top two layers of homotopy information are rolled up into a $U(1)$, rather than being a $\mathbf{B}\mathbb{Z}$, and was asking if there is a categorification of this: a universal object where the top three layers are rolled up into some nice $\mathbb{C}$-linear category such as $dgVect$.

• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeApr 29th 2013

I see you, David R. and Urs, were discussing Hopkins’ talk here. I’m not sure I see which features to look for in whatever takes the place of $\mathbf{B}^n U(1)$.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeApr 29th 2013
• (edited Apr 29th 2013)

Maybe the following is clear, but it’s the fundamental basic idea:

For 1-dimensional field theory (quantum mechanics) such as the 1d Dijkgraaf-Witten theory (see there for more details) the (local) action functional has coefficients in $\mathbf{B}U(1)$ – reflecting the fact that it is trajectories between two field configurations that are assigned an action in $U(1)$ – and this is canonically “linearized” by postcomposing with the conononical functor

$\mathbf{B}U(1)\to Vect_{\mathbb{C}}$

that exhibits the caonical representation of $U(1)$ on $\mathbb{C}$.

Whatever happens in higher dimension is supposed to follow this pattern somehow.

Certainly we can for instance consider

$\mathbf{B}^n U(1)\to n Vect_{\mathbb{C}}$

where on the right we have n-modules as discussed there. This seems to be all right at least for higher Dijkraaf-Witten type theories.

• CommentRowNumber13.
• CommentAuthorDavid_Corfield
• CommentTimeApr 29th 2013

Is there anything to say about why $U(1)$? Is this just given by the physical theories we happen to have, or is there a universal characterisation that explains its presence?

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeApr 29th 2013
• (edited Apr 29th 2013)

1. when starting with a continuum line object $\mathbb{A}$ (such as the real line $\mathbb{R}$) then cocycles that are locally $\mathbb{R}$-valued will descend to globally defined cocycle only with coefficients in some quotient $\mathbb{R}/\Gamma$. Under mild conditions, that’s $\simeq \mathbb{R}/\mathbb{Z} \simeq U(1)$. But more general quotients $\mathbb{R}/\Gamma$ can be considered, too.

The Lie integration of $L_\infty$-cocycles to higher local Chern-Simons type prequantum field theories which we discuss in Cech cocycles for differential characteristic classes all arise with coefficients $\mathbb{R}/\Gamma$ in this way. And this covers quite a range of field theories (if we include their holographic duals), as surveyed for instance at the end of Higher geometric prequantum theory.

In summary this means that if one believes that higher Chern-Simons-type theories are the fundamental field theories (everything else being induced as boundary/defect theories of these) then $\mathbb{A}/\Gamma$-coefficients are “god given” and $\mathbb{A}/\mathbb{Z}$ is something like the non-pathological generic subcase.

2. when taking $\mathbb{A} = \mathbb{C}$ as the basic line object and then forming the corresponding multiplicative group this is $\mathbb{G}_m = \mathbb{C}^\times = U(1) \times contractible$. So in this sense $U(1)$ is the “canonical group” induced by our ambient lined topos — if it is lined by the complex numbers.

• CommentRowNumber15.
• CommentTimeJan 21st 2015

Is there now a universal property recorded somewhere?

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeJan 21st 2015
• (edited Jan 21st 2015)

As far as I am aware (will be very happy to be corrected), the only discussion in this direction is still Dawson-Paré-Pronk 04

• CommentRowNumber17.
• CommentTimeJan 21st 2015

Thanks, I’ll take a look at that.

• CommentRowNumber18.
• CommentAuthorFosco
• CommentTimeJan 21st 2015

The Fourier-Mukai transform is a pull-push operation through correspondences equipped with objects in a cocycle given by an object in a derived category of quasi-coherent sheaves.

Can anybody expand on this point?

• CommentRowNumber19.
• CommentAuthorJon Beardsley
• CommentTimeJan 22nd 2015

This is potentially a red herring, but are you guys familiar with Rune Haugseng’s work on categories of spans? He records a universal property on page 19 of http://arxiv.org/pdf/1409.0837.pdf, which ultimately references his paper with David Gepner: http://arxiv.org/pdf/1312.3178.pdf.

Anyway, just a thought. For $(\infty,1)$-categories, Haugseng’s construction reduces to Clark Barwick’s construction here: http://arxiv.org/pdf/1301.4725v2.pdf.

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeJan 22nd 2015
• (edited Jan 22nd 2015)

Hi Jon, yes, I am familiar with Haugseng 14. While it’s true that he speaks of a “universal property” on that page, I’d think where we ask here for “the universal property” we have something else in mind. Of course that may depend, and in any case should be made more explicit, so let me try to do so:

What it says on that p. 19 is that the $(\infty,n)$-category $Corr_n$ of $n$-fold spans is characterized by the category $\Sigma^n$ of def. 3.1. But, as example 3.3 highlights, this $\Sigma^n$ is just the category of abstract $n$-fold spans, kind of like what the simplex category $\Delta$ is to a simplicial set. So the statement on p. 19 is technically neat, but is not a conceptual explanation of “why spans”, because it just says that $n$-fold spans are characterized by abstract $n$-fold spans. If you see what I mean.

I’d think what we are rather after in the discussion here – but clearly we should say so more explicitly – is a characterization of $Corr_n(\mathcal{C})$ on more abstract universal grounds. The thing is that for any $\mathcal{C}$ with products, then $Corr_n(\mathcal{C})$ is symmetric monoidal with duals and receives a monoidal map $\mathcal{C} \to Corr_n(\mathcal{C})$. That makes one hope that $\mathrm{Corr}_n(\mathcal{C})$ is in some way the universal completion of $\mathcal{C}$ to a symmetric monoidal $(\infty,n)$-category with duals, such that $\mathcal{C} \to Corr_n(\mathcal{C})$ is monoidal, and maybe subject to some further constraints (such as mentioned in #1).

I think that’s what we are after when we ask for “the universal property” of $Corr_n$. But of course now Adeel should say whether he maybe had something else in mind!

• CommentRowNumber21.
• CommentTimeJan 22nd 2015
• (edited Jan 22nd 2015)

Fosco: well, the Fourier-Mukai transform defines a functor on a category of what might be called “derived correspondences”, where the objects are varieties and the morphisms are complexes on the product. At the moment I am not completely sure how this fits in with the categorical definition of span/correspondence, but maybe Urs had something in mind.

Urs: That seems like what I had in mind. I am trying to understand the construction of $(\infty,1)$-categories of motivic complexes, which goes roughly as follows. One takes the $(\infty,1)$-category $Corr_1(Stk;S)$ where $Stk$ is the $(\infty,1)$-category of (derived) stacks, and $S$ is a base scheme. Consider the subcategory of this whose objects are smooth $S$-schemes and whose 1-morphisms induce closed integral subschemes of the product. Then let $Sm_S^{cor}$ be the “linearization” of this, formed by taking the free simplicial abelian groups of the mapping spaces. Finally one takes sheaves of chain complexes on the associated $\infty$-topos. This is just a direct translation to $\infty$-categories of what is in the literature, but I would like a more conceptual description of $Sm_S^{cor}$. If anyone happens to have any thoughts on this, I would be happy to hear them…

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeJan 22nd 2015
• (edited Jan 22nd 2015)

Sorry, what’s the question regarding the Fourier-Mukai transform? We have a correspondence of varieties – which happens to be of the simple product form $X \leftarrow X \times Y \rightarrow Y$, and the transform is a pull-tensor-push integral transform through this correspondence, with a twist $E$ on $X \times Y$. That’s the archetype of all these transforms.

• CommentRowNumber23.
• CommentTimeJan 22nd 2015
• (edited Jan 22nd 2015)

I was hoping that it could be realized as a functor on the plain category of correspondences in the categorical sense, but it seems one should rather consider the fibered product of $Corr(Var)$ and $\sqcup_Z D(Z)$ over $Var$, i.e. the category of correspondences $X \leftarrow Z \to Y$ equipped with a complex $E \in D(Z)$, and then the Fourier-Mukai transform can be described as a functor from this category to the category of stable infinity-categories.

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeJan 22nd 2015
• (edited Jan 22nd 2015)

So that’s why I like to speak about “phased” correspondences. If the kind of twisting (co)cycle on the correspondence space to be considered has a moduli stack $Tw$, then one may consider correspondences in the slice over $\mathbf{B}Tw$ (delooping of the tensor operation). These are given by diagrams of the form

$\array{ && Z \\ & \swarrow && \searrow \\ X && \swArrow_E && Y \\ & {}_{\mathllap{\alpha}}\searrow && \swarrow_{\mathrlap{\beta}} \\ && \mathbf{B}Tw }$

(this is a plain correspondence in the slice $\mathcal{C}_{/\mathbf{B}Tw}$). If here $\alpha$ and $\beta$ are trivial, then $E$ is equivalently an element in $Tw(Z)$.

(More generally, $E$ is a cocycle in $(\alpha,\beta)$-bitwisted bivariant $Tw$-cohomology).

• CommentRowNumber25.
• CommentTimeJan 22nd 2015
• (edited Jan 22nd 2015)

Ah, very nice. So you’re saying one should look at the category that Haugseng would write as $\Span_1(Stk; \mathcal{M})$, where $Stk$ is the infinity-topos of derived stacks, and $\mathcal{M}$ is the derived moduli stack of objects in a dg-category in the dg-category of perfect complexes. I think this is exactly the kind of thing he does in the last section of the paper Jon mentioned, where he constructs infinity-categories of spans of Lagrangian correspondences.

• CommentRowNumber26.
• CommentTimeJan 22nd 2015

I think this should answer my question in 21 as well: it seems like I can just look at $Span_1(Stk; Cyc)$ where $Cyc$ is a moduli stack of algebraic cycles. That should formalize the idea of a motivic correspondence being a categorical span equipped with an algebraic cycle. Thanks for pointing this out!

• CommentRowNumber27.
• CommentAuthorUrs
• CommentTimeJan 22nd 2015
• (edited Jan 22nd 2015)

Yes, exactly. I’d dare say that the observation that Lagrangian correspondences are naturally expressed this way is due to section 1.2.10.4 in dcct, that pre-geometric quantum field theory (“classical” field theory) is naturally expressed this way is discussed in section 3.9.14, and a survey of how this really does reproduce the traditional concept of quantization via a kind of motivic linearization is in the last section there.

• CommentRowNumber28.
• CommentTimeJan 22nd 2015

Ah, sorry for the misattribution! I was slow to make the connection with a discussion we had on this a while ago.

• CommentRowNumber29.
• CommentAuthorUrs
• CommentTimeJan 22nd 2015

Ah, no problem at all. I just thought I’d be allowed to mention it. :-) Rune cites a bunch of my notes at the end of his introduction.

• CommentRowNumber30.