“On the mystery of generic object”
Thursday, June, 25th, 16:00 - Zoom - Ingo Blechschmidt (University of Augsburg)
Let G be a group. Which group does this utterance refer to? “None in particular” is a straightforward answer, but we argue that the “generic group” is a more intriguing response.
The generic group is a reification of all the individual groups into a single coherent entity. To a first approximation, it has exactly those properties which are shared by all groups. The generic group and other generic models such as the generic monoid, the generic ring and the generic poset were discovered by Monique Hakim in her 1972 algebraic geometry thesis, have subsequently been studied in topos theory and type theory and have recently found their way back into algebraic geometry and commutative algebra.
Apart from their interest to philosophy of mathematics, these generic models are useful in mathematics, because they — in refinement to the first approximation stated above — tend to enjoy better properties than most ordinary objects. For instance, the generic ring turns out to be a field.
The talk will be light on technical details, which can be found in a forthcoming Agda formalization. Instead the talk will focus on examples and case studies illustrating the use of generic models. The shapeshifting nature of the generic group and other generic models is facilitated by “arithmetic universes”, a certain flavor of category which has recently gained increased attention, and the talk will give a gentle introduction to these as well.