Below is a summary of the talk given at the Séminaire Itinérant de Catégories (8 October 2021), prepared before the talk. The talk is mainly based on Zurab Janelidze's joint work in progress with Francois van Niekerk, as well as his earlier work on forms with former collaborators.
The talk assumes that the listener is familiar with basic ideas and concepts of category theory found in Categories for the Working Mathematician by Saunders Mac Lane (in particular, Chapters I, VII and VIII), as well as with the notions of factorization system and Grothendieck fibration.
1. Biproducts, products, sums and monoidal categories
The goal of this talk is to explain the following diagram:
- A Grandis exact category is a category equipped with an ideal of null morphisms in the sense of C. Ehresmann, such that every morphism admits a decomposition into a cokernel followed by a kernel, relative to the ideal. In the pointed case, this becomes the notion of a Puppe-Mitchell exact category: a pointed category where every morphism decomposes as a cokernel followed by a kernel (both in the usual sense of a pointed category).
- An Isbell bicategory is a category equipped with a proper factorization system. The corresponding "form of quotients" is the fibration of quotients and the "form of subobjects" is the opfibration of subobjects.
- A monoidal structure is an internal monoid in the 2-category of categories. A form is not an internal poset in the 2-category of categories.
- Products and sums, once they exist, are unique (up to isomorphism). A category may have several non-isomorphic proper factorization systems.
- Isomorphism between the monoidal structure of product and the monoidal structure of sum forces the category to be pointed. Exact categories need not be pointed categories.