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Stream: theory: type theory & logic

Topic: atomization / Morleyzation

Matteo Capucci (he/him) (Mar 07 2026 at 11:37):

Atomization (see §3 of Hodge's big red book) is the process whereby we replace a theory $T$ with $T^{atm}$ which is built from $T$ by adding, for all sentences $\varphi$, a relation symbol $R_\varphi$ and an axiom $R_\varphi \leftrightarrow \varphi$.
I have a feeling this has a neat categorical description: if we take $T$ to be an algebra $DA \to A$ for some doctrine (= 2-monad) $D$, then its atomization is the equivalent algebra $DA / \cong$ obtained as the strict [[coisoinserter]] (aka pseudocoequalizer)
image.png
In fact I'm not sure precomposing by $i$ is even necessary. Without that, that is a categorified version of the usual way of presenting algebras as quotients of free ones.

Anyway, does anyone know if this has been noticed before?

John Baez (Mar 07 2026 at 16:10):

I don't understand this stuff at all, but I notice your title is "atomization / Morleyzation", and I remember that Olivia Caramello has written about Morleyization from a categorical point of view. If your question is related to Morleyization (you didn't say anything about that, but it seems a lot like atomization), maybe her work is relevant? Just a wild thought.

She talks about Morleyization in her Cambridge topos theory course— specifically Lectures 21 and 22, available as a PDF here. Her book Theories, Sites, Toposes covers the same material in a more polished form. And in her paper The unification of mathematics via topos theory) she notes that any first-order theory has a Morleyization.

I guess your viewpoint is different, so probably these remarks of mine don't help at all. And you seem to be working more generally, where she's probably focued on toposes. (But is the 2-category of toposes with geometric morphisms the 2-category of algebras of some doctrine? I don't know.)

Matteo Capucci (he/him) (Mar 09 2026 at 16:10):

I forgot to address that, but yeah, atomization is just another name for Morleyzation

Matteo Capucci (he/him) (Mar 09 2026 at 16:10):

Thanks for the pointer, I didn't remember this reference at all

Matteo Capucci (he/him) (Mar 09 2026 at 16:13):

She deals with the consequences and usefulness of atomization, namely the fact that the resulting theory has the same models as the one we started with. If I can entertain a wild thought, I bet one can show $Mod : Th \to \bf Cat$ (where here $Th=Alg(D)$ from above) is a stack of sorts, for a topology on $Th$ that makes the map $DA \to DA/\cong\$ a [[local isomorphism]].