Semantics-structure adjunction

Question

In the discussion on the nLab article for monadic adjunctions, John Baez suggests and Mike Shulman confirms that the relationship between adjunctions and monads itself constitutes an adjunction called the semantics-structure adjunction.

It is later asked what exactly "semantics" and "structure" mean in this context and Mike suggests that someone named Daniel has written an unpublished paper giving the correct answer -- does anyone here know what that answer is or have access to the paper in question?

There is an arxiv paper from 2017 titled 'Structure and Semantics' which appears to be a thesis however it is by Tom Avery, ~230 pages long and the section on this adjunction is 60 pages deep and saturated in the language of proto-theories which makes it hard to read for me. (also the history on the nlab page suggests that Mike left his last comment in 2018 which would rule this paper out)

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Ivan Di Liberti seems to give a correct answer below to what exactly semantics and structure mean in the above context, however I was imprecise phrasing the question (apologies Ivan for creating a moving target).

I would like to understand intuitively why we should think of codensity monads parametrized by functors as 'structure' and forgetful functors from the Eilenberg-Moore category of a monad to its underlying category as semantics. I can understand viewing monads as structure since they offer syntax independent presentations of equational theories (why codensity monads in particular though?), but why should forgetful functors from categories of algebras be thought of as semantics?

Here is Mike Shulman's comment from the nlab where he poses the question I would like addressed:

If anyone can give a nice conceptual explanation of the terms “semantics” and “structure” in this context, Daniel Schaeppi is currently writing a paper which could benefit from such an explanation. I’ve never found anyone who really explains the words (nor is it entirely clear who pioneered their use in this context—Dubuc perhaps?) It makes sense to me that the E-M category can be called the “semantics” of a monad, but my intuition for “structure” is fuzzier, except that of course any monad can be regarded as a notion of structure with which one can equip objects. But why is that “the” structure associated to an adjunction?

Answer 1

I am not sure about what prof. Shulman refers to, but I am pretty confident that the answer to your question is on page 74 of Kan Extensions in Enriched Category Theory, Lecture Notes in Mathematics 145, by Eduardo Dubuc:

There is an adjunction $\text{Str} \dashv \text{Sem}$,$$\text{Str}:\text{Cat}^*_{/A} \leftrightarrows \text{Monad}(A)^{\circ} : \text{Sem}$$ moreover $\text{Monad}(A)^{\circ}$ is reflective in $\text{Cat}^*_{/A}$ via this adjunction.

Sem maps a monad $T$ to the forgetful functor $U: \text{Alg}(T) \to A$, while Str maps a functor to its codensity monad. Observe that by $\text{Cat}^*_{/A}$ I mean the full subcategory of $\text{Cat}_{/A}$ of those functors that admit a codensity monad.

Of course, the same result can be obtained with density comonads and comonads.

Answer 2

My comment "Daniel has come up with what seems to be the right answer to this question" was written on revision 8 in 2009. I think the answer I had in mind appears in Tannaka duality for comonoids in cosmoi (arXiv:0911.0977, hence Nov. 2009), specifically this passage from section 5:

These names go back to Lawvere (see [Law04, p. 77]). A monad can be viewed as a sort of logical theory, and from this viewpoint the semantics functor sends it to its category of models; the study of the models of a logical theory is generally called its semantics. On the other hand, a monad can also be seen as a ‘type of structure’ with which objects of $C$ can be equipped. From this point of view, the structure functor sends a category $A$ over $C$ to the universal structure with which the objects of $A$ can be equipped.