Jason Parker, Brandon University
Isotropy of algebraic theories
In , my PhD supervisors (Pieter Hofstra and Philip Scott) and I studied the new topos-theoretic phenomenon of isotropy (as introduced in ) in the context of single-sorted algebraic theories, and we gave a logical/syntactic characterization of the isotropy group of any such theory, thereby showing that it encodes a notion of inner automorphism or conjugation for the theory. In the present talk, I will summarize the results of my recent PhD thesis, in which I build on this earlier work by studying the isotropy groups of (multi-sorted) quasi-equational theories (also known as essentially algebraic, cartesian, or finite limit theories). In particular, I will show how to give a logical/syntactic characterization of the isotropy group of any such theory, and that it encodes a notion of inner automorphism or conjugation for the theory. I will also describe how I have used this characterization to exactly characterize the ‘inner automorphisms’ for several different examples of quasi-equational theories, most notably the theory of strict monoidal categories and the theory of presheaves valued in a category of models. In particular, the latter example provides a characterization of the (covariant) isotropy group of a category of set-valued presheaves, which had been an open question in the theory of categorical isotropy.
 J. Funk, P. Hofstra, B. Steinberg. Isotropy and crossed toposes. Theory and Applications of Categories 26, 660-709, 2012.
 P. Hofstra, J. Parker, P.J. Scott. Isotropy of algebraic theories. Electronic Notes in Theoretical Computer Science 341, 201-217, 2018.