Alex Simpson: Equality, Equivalence and Atomic Sheaves (Part 2)
Abstract: Equality and equivalence can be subtle notions in mathematics. Sometimes there is more than one interesting notion of equivalence between values, and whether or not equivalent entities can be treated as identical may depend on the context. For example, in probability theory, one has (at least) three fundamental notions of equivalence between random variables: actual equality (equality as functions), almost-sure equality, and equality in distribution.
In these two seminars, I shall show how sheaves for atomic Grothendieck topologies provide a systematic framework for explaining the interrelationships between a 'local’ equality and a 'global’ equivalence; notions which, in the case of “probability sheaves”, specialise to almost-sure equality and equality in distribution respectively.
In the first seminar, I shall introduce atomic sheaves, using two main examples, the well-known “Schanuel topos” and the category of “probability sheaves”, both of which I shall define from first principles. In the latter, almost-sure equality is enshrined as the fundamental equality between random variables, and it arises inevitably from actual equality via an application of Grothendieck’s associated-sheaf functor.
In the second seminar, I shall consider an intrinsic notion of equivalence on atomic sheaves, which, in the case of probability sheaves, coincides with equality in distribution. The general ambience of atomic sheaves allows one to identify logical principles regarding the interaction between equality and equivalence. In particular, an “invariance axiom” expresses that equivalent values cannot be distinguished by properties corresponding to subsheaves. By specialising this axiom in the case of probability sheaves, we derive core results from the infrastructure of probability theory.