Title: The etale category associated to an algebraic theory
Abstract: Equational algebraic theories are typically used in
mathematics to specify notions such as group, ring, lattice, and so on.
In computer science, by contrast, they are often used to specify
programming language features such as input/output, state, stack, and so
on.
For the mathematical applications, one typically cares about set-based
models. For the computer science applications, greater weight is placed
on the set-based comodels (i.e., models in Set^op), which can be seen as
state machines providing an environment for interpreting the given
language features. This is an idea due to Plotkin, Power and Shkaravska
(in various combinations).
The objective of this talk is to explain that:
1) the category of set-based comodels of an algebraic theory is always a
presheaf category;
2) the category of topological comodels of an algebraic theory is always
a category of continuous presheaves on an etale topological category.
By way of illustration, we explain how certain etale topological
categories, well-known from the study of C*-algebras, arise in this
manner.
