You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
Fawzi Hreiki (Jan 10 2021 at 17:26):In categorical model theory, for any given notion of logic, there is (or should be) a dual adjunction between structure and semantics.
For example, Stone duality says that the space of models of a classical propositional theory (a Boolean algebra) is a Stone space and that every Stone space arises in this way. Another example is Gabriel-Ulmer duality which sets up a dual equivalence between essentially algebraic theories (categories with finite limits) and locally finitely presentable categories.
What other examples are there of such dualities? I know of Lawvere duality for finite product categories, and Makkai duality for Boolean categories. Is there a more general framework encapsulating all the known examples?
Fawzi Hreiki (Jan 10 2021 at 17:36):Also, what needs to be assumed about a subcategory so that the presheaf functor on small categories has an adjoint?
Amar Hadzihasanovic (Jan 10 2021 at 19:36):Do you know Yoshihiro Maruyama's phd thesis? That should be a pretty good survey of everything around these dualities.
Fawzi Hreiki (Jan 10 2021 at 20:07):Oh nice. No I haven't seen this before.