### Categorical Ceilidh

This blog may have been a little neglectful lately of pure categorical adventures. Yesterday's seminar by Ross Street was on Kan extensions in group representation theory. This came about because Elango Panchadcharam has been writing a thesis here on Mackey functors and Ross Street is planning a basic text on category theory from a 2-categorical point of view.

Initially one allows G and H to be arbitrary small categories with an arrow i from H to G, and M a module category which is cocomplete. The restriction functor based on i between the functor (representation) categories has a left adjoint, which for those who know is where the left Kan extensions come in. One has categorical formulae for Lan(V) where V is a particular representation. In general, these use the comma category (i,A) for A an object of G, which is like a weak pullback of i and the arrow representing A in Cat. But in Mackey theory one really wants to consider two subgroups H,K in G. Then one ends up looking at the comma category (i,j) where j is now an inclusion for K.

Group theorists try to avoid the (i,j) because it's not a group, but a groupoid. But as Ross Street pointed out, the (i,j) contains a lot of important information. One gets the Mackey Decomposition theorem for instance, in terms of Res and Ind functors.

Dorette Pronk gave a wonderful talk about Conformal Field Theory and nuclear functors. In Segal's original definition of a CFT the source category of disjoint unions of oriented circles is not really a category. One adds in the familiar cylinders to get identities. But cylinders alter volumes, so one should be more careful about defining the source and target categories to get the conformal structure right. The categories used are always monoidal *-categories. The arrows are chosen according to the notion of nuclear ideal. This is a subset N(A,B) of each hom set which is closed under arbitrary composition, tensor, star and conjugation, and such that there exists a natural bijection from N(A,B) to Hom(I,A*oB). An example is Hilb with only the Hilbert-Schmidt maps.

In the CFT source category Pants (the name caused some amusement amongst the mathematicians) the nuclear ideal eliminates the cylinder problem. The target category is taken to be CLR, the category of correct linear relations, which I won't define here. Finally, a CFT is just a nuclear functor from Pants to CLR.

Initially one allows G and H to be arbitrary small categories with an arrow i from H to G, and M a module category which is cocomplete. The restriction functor based on i between the functor (representation) categories has a left adjoint, which for those who know is where the left Kan extensions come in. One has categorical formulae for Lan(V) where V is a particular representation. In general, these use the comma category (i,A) for A an object of G, which is like a weak pullback of i and the arrow representing A in Cat. But in Mackey theory one really wants to consider two subgroups H,K in G. Then one ends up looking at the comma category (i,j) where j is now an inclusion for K.

Group theorists try to avoid the (i,j) because it's not a group, but a groupoid. But as Ross Street pointed out, the (i,j) contains a lot of important information. One gets the Mackey Decomposition theorem for instance, in terms of Res and Ind functors.

Dorette Pronk gave a wonderful talk about Conformal Field Theory and nuclear functors. In Segal's original definition of a CFT the source category of disjoint unions of oriented circles is not really a category. One adds in the familiar cylinders to get identities. But cylinders alter volumes, so one should be more careful about defining the source and target categories to get the conformal structure right. The categories used are always monoidal *-categories. The arrows are chosen according to the notion of nuclear ideal. This is a subset N(A,B) of each hom set which is closed under arbitrary composition, tensor, star and conjugation, and such that there exists a natural bijection from N(A,B) to Hom(I,A*oB). An example is Hilb with only the Hilbert-Schmidt maps.

In the CFT source category Pants (the name caused some amusement amongst the mathematicians) the nuclear ideal eliminates the cylinder problem. The target category is taken to be CLR, the category of correct linear relations, which I won't define here. Finally, a CFT is just a nuclear functor from Pants to CLR.

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