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i space will be a double category it turns out there is another way of seeing this double category by local descriptions of the gauge transformations in particular general gauge transformations in hgt are combinations of two special types described geometrically by valued functions or valued 1 forms where is the group of objects of and is the group of morphisms based at if we think of connections as functors from the fundamental 2 groupoid into these correspond to pseudonatural transformations between these functors the main point is that there are also two special types of these called strict and costrict the strict ones are just natural transformations where the naturality square commutes strictly the costrict ones also called icons for identity component oplax natural transformations see the paper by steve lack linked from the nlab page above for an explanation of costrictness they assign the identity morphism to each object but the naturality square commutes only up to a specified 2 cell any pseudonatural transformation factors into a strict and costrict part the point is that taking these two types of transformation to be the horizontal and vertical morphisms of a double category we get something that very naturally arises by the action of a big 2 group of symmetries on a category we also find something which doesn t happen in ordinary gauge theory that only the strict gauge transformations arise from this global symmetry the costrict ones must already be the morphisms in the category being acted on this category plays the role of the moduli space in the normal 1 group situation so moving to 2 groups reveals that in general we should distinguish between global rigid symmetries of the moduli space which are strict gauge transformations and costrict ones which do not arise from the global 2 group action and should be thought of as intrinsic to the moduli space string bundles david roberts gave a rather interesting talk called constructing explicit string bundles there are some notes for this talk here the point is simply to give an explicit construction of a particular 2 group bundle there is a lot of general abstract theory about 2 bundles around and a fair amount of work that manipulates physically motivated descriptions of things that can presumably be modelled with 2 bundles there has been less work on giving a mathematically rigorous description of specific concrete 2 bundles this one is of interest because it s based on the string 2 group details are behind that link but roughly the classifying space of a homotopy 2 type is fibred over the classifying space for a 1 type the exact map is determined by taking a pullback along a certain characteristic class which is a map out of saying the string 2 group is a bit of a misnomer by the way since such a 2 group exists for every simply connected compact lie group the group that s involved here is a the string 2 group associated to the universal cover of the rotation group this is the one that determines whether a given manifold can support a string structure a string structure on therefore is a lift of a spin structure which determines whether one can have a spin bundle over hence consistently talk about a spin connection which gives parallel transport for spinor fields on the string structure determines if one can consistently talk about a string bundle over and hence a 2 group connection giving parallel transport for strings in this particular example the idea was to find explicitly a string bundle over minkowski space or its conformal compactification in point of fact this particular one is for latek string 5 and is over 6 dimensional minkowski space whose compactification is this particular is convenient because it s possible to show abstractly that it has exactly one nontrivial class of string bundles so exhibiting one gives a complete classification the details of the construction are in the notes linked above the technical details rely on the fact that we can coordinatize nicely using the projective quaternionic plane but conceptually it relies on the fact that and because of how the lifting works this is also this quotient means there s a string bundle whose fibre is while this is only one string bundle and not a particularly general situation it s nice to see that there s a nice elegant presentation which gives such a bundle explicitly by constructing cocycles valued in the crossed module associated to the string 2 group which give its transition functions here endeth part i of this discussion of the workshop in edinburgh part ii will talk about urs schreiber s very nice introduction to higher geometric quantization this ends the first part of this update the next will describe the physics oriented talks and the third will describe urs schreiber s series on higher geometric quantization february 27 2014 tqft s in vienna part 2 posted by jeffrey morton under 2 groups category theory cohomology conferences gauge theory gerbes groupoids physics quantization sheaves spans stacks tqft leave a comment to continue from the previous post twisted differential cohomology ulrich bunke gave a talk introducing differential cohomology theories and thomas nikolaus gave one about a twisted version of such theories unfortunately perhaps in the wrong order the idea here is that cohomology can give a classification of field theories and if we don t want the theories to be purely topological we would need to refine this a cohomology theory is a contravariant functorial way of assigning to any space which we take to be a manifold a graded group that is a tower of groups of cocycles one group for each with some coboundary maps linking them in some cases the groups are also rings for example the group of differential forms graded by degree cohomology theories satisfy some axioms for example the mayer vietoris sequence has to apply whenever you cut a manifold into parts differential cohomology relaxes one axiom the requirement that cohomology be a homotopy invariant of given a differential cohomology theory one can impose equivalence relations on the differential cocycles to get a theory that does satisfy this axiom so we say the finer theory is a differential refinement of the coarser so in particular ordinary cohomology theories are classified by spectra this is related to the brown representability theorem whereas the differential ones are represented by sheaves of spectra where the constant sheaves represent the cohomology theories which happen to be homotopy invariants the twisting part of this story can be applied to either an ordinary cohomology theory or a differential refinement of one though this needs similarly refined twisting data the idea is that if is a cohomology theory it can be twisted over by a map into the picard group of this is the group of invertible modules where an module means a module for the cohomology ring assigned to essentially tensoring with these modules is what defines the twisting of a cohomology element an example of all this is twisted differential k theory here the groups are of isomorphism classes of certain vector bundles over and the twisting is particularly simple the picard group in the topological case is just the main result is that while topological twists are classified by appropriate gerbes on for k theory gerbes the differential ones are classified by gerbes with connection fusion categories scott morrison gave a talk about classifying fusion categories the point of which was just to collect together a bunch of results constructing particular examples the talk opens with a quote by rutherford all science is either physics or stamp collecting that is either about systematizing data and finding simple principles which explain it or about collecting lots of data this talk was unabashed stamp collecting on the grounds that we just don t have a lot of data to systematically understand yet and for that very reason i won t try to summarize all the results but the slides are well worth a look over the point is that fusion categories are very useful in constructing tqft s and there are several different constructions that begin given a fusion category and yet there aren t all that many examples and very few large ones known scott also makes the analogy that fusion categories are noncommutative finite groups which is a little confusing since not all finite groups are commutative anyway but the idea is that the symmetric fusion categories are exactly the representation categories of finite groups so general fusion categories are a non symmetric generalization of such groups since classifying finite groups turned out to be difficult and involve a laundry list of sporadic groups it shouldn t be too surprising that understanding fusion categories which for the symmetric case include the representation categories of all these examples should be correspondingly tricky since as he points out we don t have very many non symmetric examples beyond rank 12 analogous to knowing only finite groups with at most 12 elements it s likely that we don t have a very good understanding of these categories in general yet there were a couple of talks one during the workshop by sonia natale and one the previous week by sebastian burciu whom i also had the chance to talk with that week about equivariantization of fusion categories and some fairly detailed descriptions of what results the two of them have a paper on this which gives more details which i won t summarize but i will say a bit about the construction an equivariantization of a category acted on by a group is supposed to be a generalization of the notion of the set of fixed points for a group acting on a set the category has objects which consist of an object which is fixed by the action of together with an isomorphism for each satisfying a bunch of unsurprising conditions like being compatible with the group operation the morphisms are maps in between the objects which form commuting squares for each their paper and the talks described how this works when is a fusion category namely is also a fusion category and one can work out its fusion rules i e monoidal structure in some cases it s a group theoretical fusion category it looks like for some group or a weakened version of such a thing it s morita equivalent to a nice special case of this is if the group action happens to be trivial so that every object of is a fixed point in this case is just the category of objects of equipped with a action and the intertwining maps between these for example if then in particular a group theoretical fusion category what s more this construction is functorial in itself given a subgroup we get an adjoint pair of functors between and which in our special case are just the induced representation and restricted representation functors for that subgroup inclusion that is we have a mackey functor here these generalize however to any fusion category and to nontrivial actions of on the point of their paper then is to give a good characterization of the categories that come out of these constructions quantizing with higher categories the last talk i d like to describe was by urs schreiber called linear homotopy type theory for quantization urs has been giving evolving talks on this topic for some time and it s quite a big subject see the long version of the notes above if there s any doubt however i always try to get a handle on these talks because it seems to be describing the most general framework that fits the general approach i use in my own work this particular one borrows a lot from the language of logic the linear in the title alludes to linear logic basically urs motivation is to describe a good mathematical setting in which to construct field theories using ingredients familiar to the physics approach to field theory namely fields see the description of kevin walker s talk also lagrangian functionals that is the notion of a physical action constructing tqft from modular tensor categories for instance is great but the fields and the action seem to be hiding in this picture there are many conceptual problems with field theories like the mathematical meaning of path integrals for instance part of the approach here is to find a good setting in which to locate the moduli spaces of fields and the spaces in which path integrals are done then one has to come up with a notion of quantization that makes sense in that context the first claim is that the category of such spaces should form a differentially cohesive infinity topos which we ll call the infinity part means we allow morphisms between field configurations of all orders 2 morphisms 3 morphisms etc the topos part means that all sorts of reasonable constructions can be done for example pullbacks the differentially cohesive part captures the sort of structure that ensures we can really treat these as spaces of the suitable kind cohesive means that we have a notion of connected components around it s implemented by having a bunch of adjoint functors between spaces and points the differential part is meant to allow for the sort of structures discussed above under differential cohomology really that we can capture geometric structure as in gauge theories and not just topological structure in this case we take to have objects which are spectral valued infinity stacks on manifolds this may be unfamiliar but the main point is that it s a kind of generalization of a space now the sort of situation where quantization makes sense is we have a space i e object of field configurations to start then a space of paths this is where path integrals are defined and a space of field configurations in the final system where we observe the result there are maps from the space of paths to identify starting and ending points that is we have a span now in fact these may all lie over some manifold such as the classifying space for gerbes that is we don t just have these spaces but these spaces equipped with one of those pieces of cohomological twisting data discussed up above that enters the quantization like an action it s what you integrate in a path integral aside to continue the parallel quantization is playing the role of a cohomology theory and the action is the twist i really need to come back and complete an old post about motives because there s a close analogy here if quantization is a cohomology theory it should come by factoring through a universal one in the world of motives where space now means something like scheme the target of this universal cohomology theory is a mild variation on just the category of spans i just alluded to then all others come from some functor out of it then the issue is what quantization looks like on this sort of scenario the atiyah singer viewpoint on tqft isn t completely lost here quantization should be a functor into some monoidal category this target needs properties which allow it to capture the basic quantum phenomena of superposition i e som...