2010 Category Theory Octoberfest

Steve Awodey (CMU): “Sketch of the homotopy interpretation of intensional type theory” Abstract: As a tutorial of sorts, I will outline the homotopy interpretation of intensional type theory and survey some of the recent results by various people. As a tutorial of sorts, I will outline the homotopy interpretation of intensional type theory and survey some of the recent results by various people. Nathan Bowler (Cambridge): “Unwirings and exponentiability for multicategories” Abstract: I’ll introduce the concept of an unwiring (roughly, a way of pulling things apart over the arities given by a monad, just as the structure map of an algebra gives you a way of pasting them together over those arities), and explain how it arises in the study of categories of games. I’ll explain the close links of this concept to exponentiability, including a characterisation of exponentiable multicategories in well-behaved contexts. Robin Cockett (Calgary): “Integral categories” Abstract: An integral category is a cartesian left additive category with an integral operator: A⊗A // B; (a, v) 7→ f(a) · v A⊗A // B; (a, v) 7→ [∫ v0 a0 f(a0).v0 ] (a, v) An integral category is a cartesian left additive category with an integral operator: A⊗A // B; (a, v) 7→ f(a) · v A⊗A // B; (a, v) 7→ [∫ v0 a0 f(a0).v0 ] (a, v) which takes a map which is additive in the second argument, a form, and produces a function of the same type (but which is, in general, no longer additive in the second argument). Integral categories provide an abstraction of integration on forms, rather than measure theory. Integration in these categories is treated a binding operator on forms which is first explored independently of differentiation. When an integral category is also a (cartesian) differential category it is reasonable to link the two structures by the two fundamental theorems of calculus. One then obtains a setting in which the relation between integration on forms and differentiation is formally the same as in calculus. Geoff Cruttwell (Calgary): “Differential and tangent structure for restriction categories” Abstract: Differential restriction categories capture partial settings in which one Differential restriction categories capture partial settings in which one