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.