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July 1, 2014

The Linearity of Traces

Posted by Mike Shulman

At long last, the following two papers are up:

I’m super excited about these, and not just because I like the results. Firstly, these papers are sort of a culmination of a project that began around 2006 and formed a large part of my thesis. Secondly, this project is an excellent “success story” for a methodology of “applied category theory”: taking seriously the structure that we see in another branch of mathematics, but studying it using honest category-theoretic tools and principles.

For these reasons, I want to tell you about these papers by way of their history. (I’ve mentioned some of their ingredients before when I blogged about previous papers in this series, but I won’t assume here you know any of it.)

To begin with, recall that an object XX of a symmetric monoidal category is dualizable if, when regarded as a 1-cell in the associated one-object bicategory, it has an adjoint DXD X. Then any endomorphism f:XXf:X\to X has a trace defined by

IηXDXf1XDXDXXϵI. I \xrightarrow{\eta} X \otimes D X \xrightarrow{f\otimes 1} X\otimes D X \xrightarrow{\cong} D X \otimes X \xrightarrow{\epsilon} I.

In VectVect, the dualizable objects are the finite-dimensional ones, traces reproduce the usual trace of a matrix (incarnated as a 1×11\times 1 matrix), and in particular tr(1 X)=dim(X)tr(1_X) = dim(X). In the stable homotopy category, this is Spanier-Whitehead duality, traces produce the Lefschetz number L(f)L(f) (incarnated as the degree of a self-map of a sphere), and we have L(1 X)=χ(X)L(1_X) = \chi(X), the Euler characteristic. The Lefschetz fixed point theorem follows by abstract nonsense.

The recent part of the story began in 2001, when Peter May wrote “The additivity of traces in triangulated categories”. The Euler characteristic (and Lefschetz number) are additive: if XX is a cell complex and AXA\subseteq X a subcomplex, then χ(X)=χ(A)+χ(X/A)\chi(X) = \chi(A) + \chi(X/A) and L(f)=L(f| A)+L(f/A)L(f) = L(f|_A) + L(f/A). Peter showed an abstract version of this: if a symmetric monoidal category is compatibly triangulated, then for any distinguished triangle XYZΣXX\to Y\to Z\to \Sigma X, we have χ(Y)=χ(X)+χ(Z)\chi(Y) = \chi(X) + \chi(Z).

A few years later, Peter and Johann Sigurdsson realized that “Costenoble-Waner duality” for parametrized spaces was naturally about adjunctions in a bicategory whose objects were topological spaces and whose 1-cells from AA to BB are spectraparametrized over A×BA\times B”. (The 2-cells are fiberwise stable maps; note the conspicuous absence of continuous maps of base spaces.) Peter thus wondered whether additivity generalized to bicategories. In the book that he and Johann wrote, they generalized some of his axioms for triangulated monoidal categories to “locally triangulated” bicategories, but the final axiom (TC5) used the symmetry, which doesn’t make sense in a bicategory. It was also not clear how to generalize “traces”, since the definition of trace also uses symmetry.

Enter Kate Ponto, who was studying topological fixed-point theory. This subject “begins” with the Lefschetz fixed point theorem, but continues with more refined invariants such as the Reidemeister trace, which supports a converse to this theorem (under suitable hypotheses). One definition of the Reidemeister trace uses the Hattori-Stallings trace, which is a sort of trace for matrices over a noncommutative ring: you’d like to sum along the diagonal, but the result is basis-dependent until you map it from RR into the quotient abelian group R=R/(rssr)\langle\langle R \rangle\rangle = R / (r s \sim s r). Kate realized that the Hattori-Stallings trace, and hence also the Reidemeister trace, was a sort of “bicategorical trace” that she was able to define for endo-2-cells of dualizable 1-cells in any bicategory equipped with some extra structure that she named a shadow.

Pleasingly to fans of the microcosm principle, a shadow on a bicategory B\mathbf{B} is a “categorified trace”, consisting of functors :B(A,A)T\langle\langle-\rangle\rangle:\mathbf{B}(A,A) \to \mathbf{T} that are “cyclic up to isomorphism”: XYYX\langle\langle X \odot Y \rangle\rangle \cong \langle\langle Y \odot X \rangle\rangle plus some coherence axioms. Given this, if X:ABX:A\to B has an adjoint DXD X and f:XXf:X\to X, Kate defined its trace tr(f)\tr(f) to be U AηXDXf1XDXDXXϵU B \langle\langle U_A \rangle\rangle \xrightarrow{\eta} \langle\langle X \odot D X\rangle \xrightarrow{f\odot 1} \langle\langle X\odot D X \rangle\rangle \xrightarrow{\cong} \langle\langle D X \odot X \rangle\rangle \xrightarrow{\epsilon} \langle\langle U_B \rangle\rangle where U AU_A, U BU_B are the unit 1-cells. I blogged about this here. So Kate had solved half of the problem of generalizing additivity to bicategories.

At about the same time, I was intrigued by a different aspect of Peter and Johann’s bicategory. Parametrized spaces and spectra can be “pulled back” and “pushed forward” along maps of base spaces. Moreover, pushforward and “copushforward” generalize homology and cohomology, hence should preserve duality. But how can we show this abstractly, since the maps between base spaces are missing from the bicategory of parametrized spectra? Peter and Johann solved this with “base change objects”: for any continuous map f:ABf:A\to B they defined spectra S fS_f and fS{}_f S over B×AB\times A and A×BA\times B such that composing with them was equivalent to pulling back and pushing forward. Moreover, S fS_f and fS{}_f S are dual; thus, since adjunctions compose, if XX is Costenoble-Waner dualizable, so is its pushforward to a point (π A) !X(\pi_A)_! X. This clean and easy argument, when they noticed it, replaced a long and messy calculation.

I, however, was unsatisfied with the fact that the maps of base spaces were not actually present in the bicategory, leading me to invent framed bicategories, which are actually double categories with extra properties. The horizontal arrows give it an underlying bicategory, while the vertical arrows supply the missing morphisms, and the additional 2-cells let us characterize the base change objects with a universal property. Soon, I realized that a “framing” on a bicategory was equivalent to giving pseudofunctorial “base change objects” with adjoints, a structure which had been defined by Richard Wood under the name proarrow equipment. However, the double-categorical viewpoint has certain advantages: e.g. it looks a little less ad hoc, it makes it easier to define functors and transformations between such structures (this had already been observed by Dominic Verity), and it generalizes to situations where the horizontal 1-cells can’t be composed.

Another thing that bothered me about Peter and Johann’s bicategory was that, to be honest, they hadn’t finished constructing it. They defined the composition and units and constructed associativity and unit isomorphisms, but didn’t prove the coherence axioms. In order to remedy this cleanly and abstractly, I isolated the properties of parametrized spectra that were necessary for the construction, leading to the notion of monoidal fibration or indexed monoidal category: a pseudofunctor C:S opMonCat\mathbf{C}:S^{op} \to MonCat. The only assumptions needed beyond this are that SS is cartesian monoidal and that the “pullback” functors f *:C(B)C(A)f^\ast:\mathbf{C}(B) \to \mathbf{C}(A) have “pushforward” Hopf left adjoints f !f_! satisfying the Beck-Chevalley condition for pullback squares (or homotopy pullback squares). Thus, from any such C\mathbf{C} we can construct a (framed) bicategory Fr(C)Fr(\mathbf{C}), whose objects are those of SS and with Fr(C)(A,B)=C(A×B)Fr(\mathbf{C})(A,B)=\mathbf{C}(A\times B). This was the main result of Framed bicategories and monoidal fibrations.

Now since Kate and I were both graduate students of Peter’s at the time, it was natural to put our work together. The mass of material that we produced eventually got sorted into three papers:

  1. Traces in symmetric monoidal categories, an expository paper containing the background we wanted to assume in the other papers, plus some fun unusual examples.

  2. Shadows and traces in bicategories. Kate originally defined shadows and traces in her thesis, but here we took a more systematic category-theoretic perspective. We described a string diagram calculus for shadows, and generalized the basic properties of symmetric monoidal trace that had been axiomatized by Joyal, Street, and Verity in Traced monoidal categories. For example, we showed that if XX and YY are right dualizable and f:XXf:X\to X and g:YYg:Y\to Y, then tr(fg)=tr(g)tr(f)\tr(f\odot g) = \tr(g) \circ \tr(f). You might say the intent was to make bicategorical traces “category-theoretically respectable”.

  3. Duality and traces for indexed monoidal categories, in which we finally combined our theses. Using another string diagram calculus, we showed that Fr(C)Fr(\mathbf{C}) has a shadow and related its bicategorical traces to symmetric monoidal traces in the C(A)\mathbf{C}(A)s.

To elaborate on this last one, any XC(A)X\in \mathbf{C}(A) can be regarded as a 1-cell in Fr(C)Fr(\mathbf{C}) in two ways: from AA to 11 or from 11 to AA. We denote these by X^\hat{X} and Xˇ\check{X} respectively. Then X^\hat{X} has a (right) adjoint just when XX is dualizable in C(A)\mathbf{C}(A). If we think of XX as an “AA-indexed family(X a) aA(X_a)_{a\in A}, or as a map XAX\to A with fiber X aX_a over aAa\in A, then this generally means just that each X aX_a is dualizable. However, the trace of tr(f^)tr(\hat{f}) contains more information than tr(f)tr(f), and sometimes strictly more. The former has domain U A\langle\langle U_A \rangle\rangle, which is generally like the free loop space of AA, and tr(f^)tr(\hat{f}) maps a loop α\alpha to the trace of f aX αf_a\circ X_\alpha, where X αX_\alpha is the monodromy around α\alpha and f af_a is the action of ff over some point aαa\in \alpha. By contrast, the trace of ff in C(A)\mathbf{C}(A) only knows about these traces for constant loops.

(Right) dualizability of Xˇ\check{X} is a stronger condition; in parametrized spectra, for I Aˇ\check{I_A} (with I AI_A the unit of C(A)\mathbf{C}(A)) it is Costenoble-Waner duality. The composing-adjunctions argument mentioned above shows that if Xˇ\check{X} is right dualizable, then (π A) !X(\pi_A)_! X is dualizable in C(1)\mathbf{C}(1). In particular, a Costenoble-Waner dualizable space is also Spanier-Whitehead dualizable. Now Kate and I showed that tr(fˇ)tr(\check{f}) also contains more information than tr((π A) !(f))tr((\pi_A)_!(f)): the latter is the composite I 1tr(fˇ)AI 1I_1 \xrightarrow{tr(\check{f})} \langle\langle A \rangle\rangle \to I_1. This also follows completely formally, from the basic property of bicategorical traces that I mentioned above: if you compose two dualizable 1-cells, then the trace of an induced endomorphism is the composite of the original two traces. (The map AI 1\langle\langle A \rangle\rangle \to I_1 is the trace of the identity map of the base change object for π A\pi_A.) In particular, this explains how the Reidemeister trace refines the Lefschetz number.

As we worked on these papers, Kate and I were also trying to generalize additivity to bicategories. This was harder than we expected, mainly because triangulated categories are no good. Since their axioms are about nonunique existence, when you add more axioms like Peter’s, you get “there exists an X as in axiom A, and also a Y as in axiom B, together satisfying axiom C, and also …”. Peter’s axioms were manageable, but the bicategorical generalization was too much for us. If we had believed triangulated categories were a “correct thing”, we might have pushed through; but clearly the “correct thing” is a stable (∞,1)-category. However, we weren’t really enthusiastic about using those either. This led us to derivators; which may not really be a “correct thing” either, but their structure is categorically sensible and characterizes objects by universal properties, so they are much nicer to work with than triangulated categories.

The obvious place to start was to prove that symmetric monoidal derivators satisfy Peter’s axioms. In May 2011 I visited Kate in Kentucky, and we spent an intense week filling blackboards with string diagrams and checking that squares were homotopy exact. I even wrote a little computer program to do the latter for us. Eventually we joined forces with Moritz Groth, who contributed (among other things) the right definition of “closed monoidal derivator”. But we stayed stuck on things like Peter’s axiom (TC3).

Then in November 2011 we discovered a totally different approach to additivity. Consider the bicategory Prof(V)Prof(\mathbf{V}) of categories and profunctors enriched in a symmetric monoidal V\mathbf{V}. We have embeddings like ()^\hat{(-)} and ()ˇ\check{(-)}, but with variance: a functor X:AVX:A\to \mathbf{V} becomes a profunctor X^:A1\hat{X}:A ⇸ 1, while a functor Φ:A opV\Phi:A^{op}\to \mathbf{V} becomes a profunctor Φˇ:1A\check{\Phi}:1 ⇸ A. As before, X^\hat{X} is right dualizable when each X aX_a is dualizable, and tr(f^)tr(\hat{f}) records the traces of f aX αf_a\circ X_\alpha as α\alpha ranges over endomorphisms in AA. And right dualizability of Φˇ\check{\Phi} says that Φ\Phi is a weight for absolute colimits in V\mathbf{V}; thus the composing-adjoints argument implies

Theorem: If X:AVX:A\to \mathbf{V} is such that each X aX_a is dualizable, while Φ\Phi is a weight for absolute colimits, then the weighted colimit colim Φ(X)\colim^{\Phi}(X) is dualizable.

I would be surprised if no one had noticed this before, but I don’t recall seeing it written down. Even more interestingly, however, the “composition of traces” property now implies:

Theorem: In the above situation, given f:XXf:X\to X, the trace of colim Φ(f)\colim^\Phi(f) is the composite Itr(1 Φ)U Atr(f^)II \xrightarrow{\tr(1_\Phi)} \langle\langle U_A \rangle\rangle \xrightarrow{\tr(\hat{f})} I .

If V\mathbf{V} is additive and AA is finite, U A\langle\langle U_A \rangle\rangle is a direct sum of copies of II over “conjugacy classes” of endomorphisms in AA. Thus, tr(colim Φ(f))\tr(\colim^\Phi(f)) is a linear combination of the traces of f aX αf_a\circ X_\alpha, with coefficients determined by Φ\Phi. So for completely formal reasons, we have a very general “linearity formula” (hence the paper titles) for traces of absolute colimits. We obtain Peter’s original additivity theorem by generalizing V\mathbf{V} to be a symmetric monoidal derivator, with Φ\Phi the weight for cofibers. Absoluteness of this weight is equivalent to stability of V\mathbf{V}, and its coefficients are 11 and 1-1, yielding the original formula in a rewritten form:

L(f/A)=L(f)L(f| A).L(f/A) = L(f) - L(f|_A).

Finally, this argument can be entirely straightforwardly generalized to bicategories, since we know how to define “categories and profunctors enriched in a bicategory”.

Before going on, I want to emphasize why I consider this a success story for applied category theory. We started out by looking at something that arose naturally in another branch of mathematics; in this case, the Reidemeister trace in topological fixed-point theory. Its definition looked somewhat ad hoc, but it was a generalization of something that did have a nice category-theoretic description (the Lefschetz number), so we (and here I mean Kate) trusted that it probably had one too. So we (i.e. Kate) wrote down a categorical description of the structure being used, and then abstracted away the particulars to arrive at a general definition: shadows and bicategorical traces.

This general definition might have looked a bit peculiar to a category theorist, but we took it seriously and went on to study it using category-theoretic tools. We proved a coherence theorem (the string diagram calculus), ensuring that the definition was not missing any axioms. We investigated its abstract properties, not because we had any particular reason to need them at the moment, but because past experience suggested that they would eventually be necessary to know, and useful to have collected in one place.

It then turned out that one of these abstract properties — the composition theorem for traces — enabled a clean and essentially completely formal proof, and generalization, of a result (additivity) that used to require long calculations and lots of commutative diagrams. It took us a while to notice this. But I dare say it would have taken much longer if we hadn’t previously written down the composition theorem. That’s why I say it was a success story for applying category theory seriously.

In fact, there are a couple more similar success stories hiding inside this larger story. The first involves shadows on framed bicategories, which were slated for inclusion in Shadows and traces in bicategories but got omitted out of consideration for the intended readership. Such a shadow is easiest to define using the double-categorical perspective: it’s a single functor whose domain is the category whose objects are all the endo-horizontal-1-cells and whose morphisms are the squares with equal horizontal sources and targets:

A X A f f B Y B. \array{ A & \xrightarrow{X} & A \\ ^f\downarrow & \Downarrow & \downarrow^f \\ B & \xrightarrow{Y} & B. }

Such a shadow can be defined on any double category, but in the framed case, a shadow on the horizontal bicategory extends uniquely to one on the framed bicategory — by the construction of twisted traces! When we first noticed it, this seemed like just a cute bit of trivia. But in the linearity paper, it turned out to be crucial in identifying the components of tr(f^)\tr(\hat{f}), which we did by applying the composition theorem again using a base change object, whose trace we identified using this characterization of framed shadows. I’ll omit the details; you can find them in the paper. The point is that just as before, having previously found and studied abstractly the correct categorical structure gave us the tools we needed later on for a concrete result.

The second additional success story has to do with derivator bicategories: bicategories enriched over the monoidal bicategory of derivators. We needed these to get linearity for the Reidemeister trace, which is a bicategorical trace and also requries “stable” additivity. In particular, we needed to extend Peter and Johann’s bicategory to a derivator bicategory. This might have been a lot of work, except that in Framed bicategories and monoidal fibrations I had already shown that FrFr was 2-functorial. My motivation for this was pure category-theoretic principle: every construction should be a functor. But now, since a monoidal derivator is a 2-functor Cat opMONCATCat^{op}\to MONCAT (with extra properties), we can essentially just apply the 2-functor FrFr to an “indexed monoidal derivator” to obtain a derivator bicategory. And the indexed monoidal derivator is essentially right there in Peter and Johann’s book. (When we shared these papers with Peter, he remarkede “so that is what we were doing way back then!”)

I’ll finish this long post by mentioning a story that has yet to be told, relating to the construction of Prof(V)Prof(\mathbf{V}) for a derivator V\mathbf{V}. Kate and I needed this bicategory for the linearity story, so we joined forces with Moritz Groth (who had the first idea of how to construct it) to do it in a separate paper. However, the three of us then discovered that Prof(V)Prof(\mathbf{V}) would also solve the original problem of proving that Peter’s axioms hold in a stable monoidal derivator. This seemed a good way to make the bicategory paper stand on its own, so we retitled it The additivity of traces in monoidal derivators (and eventually split it in two as well).

(We still don’t know whether Peter’s proof generalizes directly to bicategorical trace. Even using derivators, there seems to be another roadblock or two. I’d be happy to elaborate if anyone is interested; it’s possible they could be circumvented with a little thought.)

Now unfortunately, the objects of Prof(V)Prof(\mathbf{V}) are not actually categories enriched in V\mathbf{V}, but ordinary unenriched categories. (No one knows how to define “categories (coherently) enriched in a monoidal derivator”; it may be impossible with the current definition of derivator.) Now given a monoidal derivator V:Cat opMONCAT\mathbf{V}:Cat^{op}\to MONCAT, the hom-category Prof(V)(A,B)Prof(\mathbf{V})(A,B) should be V(A×B op)\mathbf{V}(A\times B^{op}). This should look familiar! Indeed, a monoidal derivator is a CatCat-indexed monoidal category, and the construction of Prof(V)Prof(\mathbf{V}) is very similar to that of Fr(C)Fr(\mathbf{C}) (recall Fr(C)(A,B)=C(A×B)Fr(\mathbf{C})(A,B) = \mathbf{C}(A\times B)). However, the pushforward functors in a derivator don’t satisfy the Beck-Chevalley condition for (homotopy) pullback squares, which we required for FrFr; instead, they satisfy it for comma squares, or more generally homotopy exact squares.

The unit and composition in Fr(C)Fr(\mathbf{C}) and Prof(V)Prof(\mathbf{V}) also look very similar. For instance, in Fr(C)Fr(\mathbf{C}) we compose XC(A×B)X\in \mathbf{C}(A\times B) and YC(B×C)Y\in \mathbf{C}(B\times C) by pulling them both back to A×B×B×CA\times B\times B\times C and tensoring them there, pulling back again along the diagonal to A×B×CA\times B\times C, then pushing forward to A×CA\times C. In Prof(V)Prof(\mathbf{V}), we compose XV(A×B op)X\in \mathbf{V}(A\times B^{op}) and YV(B×C op)Y\in \mathbf{V}(B\times C^{op}) by pulling them both back to A×B op×B×C opA\times B^{op}\times B\times C^{op} and tensoring them there, pulling back again along the projection of the twisted arrow category to A×tw(B) op×C opA\times tw(B)^{op}\times C^{op}, then pushing forward to A×C opA\times C^{op}. Note that if AA, BB, and CC are groupoids, then B opBB^{op} \cong B and tw(B)Btw(B)\simeq B, and the two constructions do agree. This leads to a natural

Question: Is there an abstract construction producing a (framed) bicategory from some input data, which reduces in one case to FrFr and in another case to ProfProf?

If such a thing existed, maybe we could apply it to “derivators” with CatCat replaced by something else, such as the 2-category of internal categories in a topos, or a 2-category of (∞,1)-categories. The latter would include in particular the (∞,0)-categories, i.e. spaces; thus when V=Spectra\mathbf{V}=Spectra it should reproduce Peter and Johann’s bicategory (c.f. also Ando-Blumberg-Gepner).

In fact, Kate and I had already used a version of the linearity story with Peter and Johann’s bicategory replacing Prof(V)Prof(\mathbf{V}) to prove the multiplicativity of the Lefschetz number and Reidemeister trace. Roughly, multiplicativity means that given a fibration EBE\to B with fiber FF, and compatible endomorphisms f:EEf:E\to E and f¯:BB\bar{f}:B\to B, we have L(f)=L(f| F)L(f¯)L(f) = L(f|_F) \cdot L(\bar{f}). However, if BB is not simply-connected, then L(f| F)L(f|_F) can differ between fibers; thus instead of a simple product we need a sum of fiberwise traces over loops in BB — whose coefficients turn out to be none other than the Reidemeister trace of f¯\bar{f}. In other words, it is another linearity formula, with BB acting like the weight Φ\Phi and the Reidemeister trace acting like its coefficient vector. And we proved it in the same way: composing the Costenoble-Waner dualizable I Bˇ:1B\check{I_B}:1⇸ B with the fiberwise dualizable E^:B1\hat{E}:B⇸1 yields the ordinary space E:11E:1⇸1, and we can apply the composition-of-traces theorem.

Now a fibration EBE\to B is equivalently an (∞,1)-functor BGpdB \to \infty Gpd, and the total space EE is its (homotopy) colimit. Thus, additivity and multiplicativity are really two special cases of a single theorem about absolute colimits of (∞,1)-diagrams; the only thing missing is a construction of the appropriate bicategory.

Posted at July 1, 2014 4:35 AM UTC

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Re: The Linearity of Traces

Bravo! And thanks for explaining the story behind how all this came to be; it’s not something that goes in a paper, but certainly helps others to understand the flow of ideas.

Posted by: David Roberts on July 2, 2014 4:16 PM | Permalink | Reply to this

Re: The Linearity of Traces

About the question you ask: have you considered the extension of derivators to (,1)(\infty,1)-categories? I don’t know it this reaches the level of generality you are after, but the construction of Prof(V)Prof(\mathbf{V}) for a derivator defined on (,1)(\infty,1)-categories can be restricted to \infty-groupoids and thus gives a theory of parametrized objects in V\mathbf{V}, generalizing May and Sigurdsson’s parametrized spectra. Moreover, any derivator in the usual sense gives rise to such a thing. Indeed, given a derivator V\mathbf{V}, one can extend it to (,1)(\infty,1)-categories in two (equivalent) ways. One can associate to a small category CC together with a subcategory WCW\subset C, the full subcategory V(C,W)\mathbf{V}(C,W) of V(C)\mathbf{V}(C) which consists of objects whose corresponding functor CV(𝟙)C\to\mathbf{V}(\mathbb{1}) sends arrows of WW to isomorphisms. This category is equivalent to the category of cocontinuous morphisms of derivators from P(C,W)\mathbf{P}(C,W) to V\mathbf{V}, where P(C,W)\mathbf{P}(C,W) is the derivator associated to the left Bousfield localization of the projective model structure on the category of simplicial presheaves on CC by WW; in particular, it only depends on the (,1)(\infty,1)-category obtained by inverting the maps of WW in CC. Equivalently, one can also define directly, for any simplicial set XX, V(X)\mathbf{V}(X) as the category of cocontinuous functors from P(X)\mathbf{P}(X) to V\mathbf{V}, where P(X)\mathbf{P}(X) is the derivator associated to the contravariant model structure on SSet/XSSet/X. The assignment XV(X)X\mapsto\mathbf{V}(X) is a 22-functor from the (opposite of the) 22-category of (,1)(\infty,1)-categories (as considered by Riehl And Verity for instance) to the 22-category of categories (because XP(X)X\mapsto\mathbf{P}(X) has this property). Furthermore, one can see that this extension of V\mathbf{V} satifies all the axioms of the theory of derivators, replacing small categories by simplicial sets and pullbacks of small categories by homotopy pullbacks in the Joyal model structure. In this sense, there is no difference between the theory of derivators defined on CatCat or on (,1)-Cat(\infty,1)\text{-}Cat. Finally, let me add that you can consider the restriction of (the extension of) Prof(V)Prof(\mathbf{V}) to \infty-groupoids, and that, if we stick to the language of pairs (C,W)(C,W) as above, but with C=WC=W, this essentially is the content of my paper “Locally constant functors”, the last section of which being closely related to the sructure of a framed bicategory constructed functorially from any derivator, whose objects are \infty-groupoids.

Posted by: Denis-Charles Cisinski on July 3, 2014 1:38 PM | Permalink | Reply to this

Re: The Linearity of Traces

Indeed, I have thought about that, and about the first of the ways you mention of extending an ordinary derivator to (,1)(\infty,1)-categories. However, I have not yet managed to verify the axioms for the extension, or to extend the construction of Prof(V)Prof(V) to “(,1)(\infty,1)-derivators”. Are you saying you’ve done both of those? Are they written down anywhere?

Posted by: Mike Shulman on July 3, 2014 4:57 PM | Permalink | Reply to this

Re: The Linearity of Traces

The case of the extension to \infty-groupoids is essentially done in the paper “Locally constant functors” (which you may find on my web page). I don’t know a reference where the fact that the extension to (,1)(\infty,1)-categories satisfies all the axioms of derivators is written down explicitely, but this follows right away from results which can be found in Lurie’s `Higher topos theory’. If we define V(X)\mathbf{V}(X) as the category of cocontinuous functors from P(X)\mathbf{P}(X) to V\mathbf{V}, everything follows from known properties of the functorialities on the P(X)\mathbf{P}(X)’s. Indeed, for any map of simplicial sets u:XYu:X\to Y, we have an adjunction in the 22-category of derivators (where 11-cells are cocontinuous morphisms of derivators)

u !:P(Y)P(X):u *.u_{!}:\mathbf{P}(Y)\rightleftarrows\mathbf{P}(X): u^{*}.

Moreover, the inverse image functor u *:V(Y)V(X)u^*:\mathbf{V}(Y)\to\mathbf{V}(X) is obtained by composition with u !u_!. In other words, if you already know that presheaves of \infty-groupoids on (,1)(\infty,1)-categories satisfy the axioms of derivators (which is already in Lurie’s book), we get the axioms of derivator over (,1)(\infty,1)-categories for an arbitrary V\mathbf{V} by pure 22-functoriality (note that we can play the game of replacing V\mathbf{V} by V op\mathbf{V}^{op} whenever we want). I did not check that one gets framed bicategories or proarrow equipements from there, but I am confident that the proofs you already know in the standard case of derivators over ordinary categories extend to this `higher level of generality’.

Posted by: Denis-Charles Cisinski on July 6, 2014 10:36 PM | Permalink | Reply to this

Re: The Linearity of Traces

Can you point to where these facts are in HTT? I expect they are there, but I remember looking for them and failing to find them, specifically (Der4).

Assuming they are there, this is the second way that you suggested to extend a derivator to (,1)(\infty,1)-categories. Does that mean you haven’t found a way to make the first one work directly?

There is then, as I mentioned, the additional problem of generalizing the construction of a bicategory from a derivator to have (,1)(\infty,1)-categories as objects. Do you know how to do that?

Posted by: Mike Shulman on July 7, 2014 5:56 PM | Permalink | Reply to this

Re: The Linearity of Traces

To check Der4 for presheaves of \infty-groupoids (in the language of u *u^* and f !f_!), you just reproduce the proof of Prop. 1.5.9 in these notes of Maltsiniotis), then apply Prop. in HTT to the smooth maps of the form x\XXx\backslash X\to X (here, for readability with respect to HTT, I refer to Lurie’s notion of smoothness, which corresponds to what Joyal and Grothendieck (in the case of ordinary categories) call properness). This will give you Der4 (in the laguage of u *u^* and f !f_!) for any derivator V\mathbf{V}, and thus the dual version as well (replacing V\mathbf{V} by V op\mathbf{V}^{op}).

I have never tried to check the axioms of derivators using the first approach directly (the main difficulty is then to get a good notion of comma categories (or of `Grothendieck fibration with \infty-groupoidal fibers’) at hand in order to formulate Der4 in a nice way).

Once we work with quasi-categories with the second construction, I don’t see any particular difficulty to get nice bicategories. Note that we have (again by 22-functoriality) the analogue of Prop. in HTT for any derivator, as well as nice (co)finality properties (obtained from Prop. in HTT). We have a bicategory QcatQcat whose objects are quasi-categories (or even simplicial sets!) and whose category of morhisms from XX to YY is given by the full subcategory of simplicial sets MX×YM\to X\times Y such that the projection p:MXp:M\to X is a coCartesian fibration while q:MYq:M\to Y is a Cartesian fibration. If DerDer denotes the 22-category of derivators (with cocontinuous functors as 11-cells), we get a bifunctor QcatDerQcat\to Der defined by XP(X)X\mapsto\mathbf{P}(X) on objects and by the functors Mq!p *M\mapsto q!p^* (here, as coCartesian fibrations are smooth (see Prop. in HTT), we have the adequate Beck-Chevalley properties to see the compatibility conditions for composition of such things in DerDer). This bifunctor is symmetric monoidal in a suitable sense with respect to the cartesian product of simplicial sets, which follows from the fact that, given simplicial sets X 1,,X nX_1,\ldots,X_n and a derivator V\mathbf{V}, there is a canonical equivalence of categories between morphisms of prederivators P(X 1)××P(X n)V\mathbf{P}(X_1)\times\dots\times\mathbf{P}(X_n)\to\mathbf{V} which preserve colimits in each variables and cocontinuous morphisms of derivators P(X 1××X n)V\mathbf{P}(X_1\times\dots\times X_n)\to\mathbf{V} (to see this, we may assume that each X iX_i is the localisation of an ordinary category, and reduce the problem to the case where eaxh X iX_i is (the nerve of) a small 11-category). Note that the functors Qcat(X,Y)Der(P(X),P(Y))Qcat(X,Y)\to Der(\mathbf{P}(X),\mathbf{P}(Y)) send Joyal weak equivalences over X×YX\times Y to isomorphisms, and thus induce functors Ho(Qcat(X,Y))Der(P(X),P(Y))=P(X op×Y)(pt)Ho(Qcat(X,Y))\to Der(\mathbf{P}(X),\mathbf{P}(Y))=\mathbf{P}(X^{op}\times Y)(pt) These functors have fully faithful right adjoints whose essential images can be described in terms of the objects (p,q):MX×Y(p,q):M\to X\times Y with pp a left fibration and qq a right fibration.

From there, it looks straightforward to me that, if V\mathbf{V} is a monoidal derivator, then composing the functor Der(,V)Der(-,\mathbf{V}) with the monoidal bifunctor above gives a bicategory whose objects are quasi-categories with categories of morphisms given by V(X×Y op)\mathbf{V}(X\times Y^{op}) (although writing down all this with care might require some time!).

Posted by: Denis-Charles Cisinski on July 8, 2014 1:06 AM | Permalink | Reply to this

Re: The Linearity of Traces

With a good dependent linear type theory in place, would it be possible to couch some/many of your ideas in its terms?

Posted by: David Corfield on July 7, 2014 11:30 AM | Permalink | Reply to this

Re: The Linearity of Traces

Thanks for asking that! The theory of indexed monoidal traces, and hence the multiplicativity theorem, can indeed be described in dependent linear type theory. For additivity, however, it seems that we would need an answer to the question I asked at the end (which is another motivation for asking it).

Ordinary traces in symmetric monoidal categories can be described in ordinary linear type theory. Recall (for bystanders) that in linear type theory every variable present in a context must be used exactly once. Just as ordinary “nonlinear” type theory corresponds to categories with (among other things) finite products, linear type theory corresponds to (usually symmetric) monoidal categories, in which there are no diagonals XX×XX\to X\times X or projections X1X\to 1; thus we cannot “duplicate” or “discard” information.

In particular, instead of a cartesian product type X×YX\times Y, we have a “tensor product type” XYX\otimes Y. It has the same introduction rule: given x:Xx:X and y:Yy:Y we can form (x,y):XY(x,y):X\otimes Y. But the linearity restriction means that the same variable can’t occur in both xx and yy, so that in particular we don’t have (x,x):XX(x,x):X\otimes X. The elimination rule is best phrased in terms of pattern-matching rather than projections: given p:XYp:X\otimes Y and a term z:Zz:Z involving two variables x:Xx:X and y:Yy:Y (exactly once each!) we have a term

(let(x,y)pinz):Z. (let\;(x,y)\coloneqq p\;in\; z) : Z.

Note that pp appears only once in this expression, as required. Technically, xx and yy appear twice, once in (x,y)(x,y) and once in zz, but the first is a binding occurrence which cancels out the other, so overall the term does not contain them as free variables.

Similarly, the “unit type” II has the usual introduction rule tt:Itt:I, and an elimination rule that essentially says any variable of type II can be ignored: given a variable u:Iu:I and a term z:Zz:Z not involving uu, we have a term

(discarduinz):Z (discard\;u\;in\;z) : Z

which does involve uu.

In linear type theory, we can say that a type XX is dualizable if there is a type DXD X, a term η:XDX\eta : X\otimes D X (the domain II can be left out), and a term (p:DXX)ϵ(p):I(p:D X\otimes X) \vdash \epsilon(p):I, such that

(let(x,ξ)ηin(discardϵ(ξ,x)inx))=x (let\;(x',\xi)\coloneqq \eta\;in\;(discard\;\epsilon(\xi,x)\;in\;x')) = x

for any x:Xx:X, and

(let(x,ξ)ηin(discardϵ(ξ,x)inξ))=ξ (let\;(x,\xi')\coloneqq \eta\;in\;(discard\;\epsilon(\xi,x)\;in\;\xi')) = \xi

for any ξ:DX\xi:D X. Then given f:XXf:X\to X, its trace is

let(x,ξ)ηinϵ(ξ,f(x)). let\;(x,\xi)\coloneqq\eta\;in\;\epsilon(\xi,f(x)).

Note that the order of xx and ξ\xi gets reversed between their binding and their use. This corresponds to the use of symmetry in the categorical definition of trace.

One way to think of this is that when we break η\eta into xx and ξ\xi, we can consider ξ\xi to be a sort of “equals-xx predicate”, which is applied by ϵ\epsilon. Thus, the trace measures the extent to which f(x)f(x) equals xx.

The linear type theory I just described doesn’t have any dependent types. It’s tricky to say what it would mean for one linear type to be dependent on another, but if we allow both “linear” and “nonlinear” types, we can easily have the linear types dependent on the nonlinear ones (and the nonlinear ones dependent on each other). This is the sort of theory that David is asking about, and it corresponds to an indexed monoidal category, where the base category is cartesian monoidal (the nonlinear types) but the fibers are only symmetric monoidal (the linear types).

One of the type constructors we can have in this type theory is the dependent sum of a family of linear types dependent on a nonlinear one. That is, if we have A:NLTypeA:NLType and (a:A)X(a):LType(a:A)\vdash X(a)\;:LType, then we have (a:A)X(a):LType\sum_{(a:A)} X(a) : LType, with the usual pairing constructor (a,x): (a:A)X(a)(a,x):\sum_{(a':A)} X(a') and a matching eliminator like the above “letlet”, except that the first variable matched is nonlinear and can be used multiple times (or none).

In particular, we can “linearize” a nonlinear type AA by summing up the unit linear type, writing ΣA\Sigma A for (a:A)I\sum_{(a:A)} I. In Peter and Johann’s world, the nonlinear types are spaces, the linear ones are spectra, and ΣA\Sigma A is the suspension spectrum of a space.

Now suppose we have (a:A),(b:B)X(a,b):LType(a:A),(b:B)\vdash X(a,b)\;:LType and (b:B),(c:C)Y(b,c):LType(b:B),(c:C)\vdash Y(b,c)\;:LType. Their bicategorical composite is defined by

(XY)(a,c)= (b:B)X(a,b)Y(b,c). (X\odot Y)(a,c) = \sum_{(b:B)} X(a,b)\otimes Y(b,c).

And the bicategorical unit of a nonlinear type AA is

(U A)(a,a)=Σ(a=a) (U_A)(a,a') = \Sigma(a=a')

(assuming the nonlinear type theory has an identity type). Finally, a nonlinear type family (a:A)X(a):LType(a:A)\vdash X(a):LType is Costenoble-Waner dualizable if there is a family of linear types (a:A)DA(a):LType(a:A)\vdash D A(a) : LType, a term η: (a:A)A(a)DA(a)\eta:\sum_{(a:A)} A(a) \otimes D A(a), and a term (a,a:A),(ξ:DA(a)),(x:A)ϵ(a,a,ξ,x):Σ(a=a)(a,a':A),(\xi:D A(a)),(x:A) \vdash \epsilon(a,a',\xi,x):\Sigma(a=a'), such that the triangle identities hold in an appropriate way.

However, to deal with linearity, we need a bicategory whose objects are categories. To formulate that in a similar way to the above, we’d need the “nonlinear types” to be categories, so that their type theory would be some sort of directed type theory. But without a general context in which to perform the derivator-to-bicategory construction, it’s not obvious to me exactly what this directed type theory should look like, or how it should interact with the dependent linear types.

Posted by: Mike Shulman on July 7, 2014 11:49 PM | Permalink | Reply to this

Re: The Linearity of Traces

In the first paper,

…derivators, which were inverted by Grothendieck…

should be ‘invented’.

Posted by: David Corfield on July 8, 2014 10:14 AM | Permalink | Reply to this

Re: The Linearity of Traces

Thanks, will fix.

Posted by: Mike Shulman on July 8, 2014 3:51 PM | Permalink | Reply to this

Re: The Linearity of Traces

So that example in 4.16 of the first paper is what is called the Frobenius formula here.

Can you approach things like the Selberg trace formula with your resources?

Posted by: David Corfield on July 12, 2014 10:58 AM | Permalink | Reply to this

Re: The Linearity of Traces

I don’t have any ideas about how to approach the Selberg trace formula, but I suppose it might be possible.

Posted by: Mike Shulman on July 12, 2014 8:36 PM | Permalink | Reply to this

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