Alan Cigoli (Università di Torino) will give a talk on
Cartesian and additive opindexed categories (joint work with S. Mantovani and G. Metere)
Abstract:
We give a characterization of cartesian objects in the cartesian 2-category OpICat of opindexed categories. They are given by pseudofunctors F: B —> Cat, where B has finite products and the canonical oplax monoidal structure L on F admits a right adjoint R (in a suitable sense), which makes F a lax monoidal pseudofunctor. As a special case, if we restrict our attention to functors F: B —> Set, the cartesian ones are just finite-product preserving functors. When moreover B is additive, such F factorizes through the category Ab of abelian groups, and the corestriction is an additive functor.
Then we consider opindexed groupoids, i.e. pseudofunctors F: B —> Gpd. The cartesian objects here are pseudofunctors preserving finite products up to equivalences. When moreover B is additive, we find that such F factorizes through the 2-category Sym2Gp of symmetric 2-groups. In fact, we characterize the latter as 2-additive pseudofunctors (in the sense of Dupont).