Added remark about essential injectivity

]]>This functor $Z$ goes $\mathcal{C}' \to \underline{\mathcal{C}}$. So I guess to get the desired functor on EM-categories $\underline{(-)}$ we need to assume that $U$ is monadic.(?)

Yes, that’s right. The presentation of the result in Maranda is certainly more awkward than one would hope, though I think the essence of the idea is contained there.

I think I am out of energy with editing on this point now, but this would be good to clarify in the entry.

I will try to find some time soon to clarify these aspects.

]]>added pointer to today’s

- Nicholas Agia,
*Massive Type IIB Superstrings Part I: 3- and 4-Point Amplitudes*[arXiv:2309.11538]

Thanks for the heads-up. This formula is from revision 9.

I am not sure what the intention of this category of “pre-gvs” really is.

]]>Is the formula $(f\otimes g) (v\otimes w) = (-1)^{|g||f|}(f(v) \otimes g(w))$ the intended one (last section on the tensor product)? For me the formula $(f\otimes g) (v\otimes w) = (-1)^{|g||v|}(f(v) \otimes g(w))$ is the standard one, and the associated category is not monoidal, but rather supermonoidal.

]]>The part I was referring to was this line in Theorem 2:

Thanks, I see now.

This functor $Z$ goes $\mathcal{C}' \to \underline{\mathcal{C}}$. So I guess to get the desired functor on EM-categories $\underline{(-)}$ we need to assume that $U$ is monadic.(?)

There are two relevant functors here:

Right, thanks.

I think I am out of energy with editing on this point now, but this would be good to clarify in the entry.

]]>brief `category:people`

-entry for hyperlinking references

brief `category:people`

-entry for hyperlinking references

Re. 85: There are two relevant functors here: one $EM \colon Mnd(C)^\circ \to CAT/C$ which is the one currently described in the article as Example 3.3 (though I notice this functor is not explicitly mentioned); and one $Kl \colon Mnd(C) \to C/CAT$ sending each monad to its Kleisli category (which is currently referenced in Example 3.2, but again not explicitly written down). It is the full faithfulness of this latter functor that I meant when I wrote “the corresponding statement for Kleisli” (rather than a restriction of the former to free algebras).

]]>brief `category:people`

-entry for hyperlinking references

added pointer to:

- Vivien M. Kendon, Kae Nemoto, William J. Munro,
*Quantum Analogue Computing*, Phil. Trans. R. Soc. A**368**1924 (2010) 3621-3632 [arXiv:1001.2215, doi:10.1098/rsta.2010.0017]

Re. 83:

However, while this article defines monad morphisms (in the “right” direction! :-) and discusses them in the context of initiality of the Kleisli category, I haven’t spotted a declaration of the induced functor between categories of modales.

The part I was referring to was this line in Theorem 2:

such that if is an adjoint morphism defining the fundamental construction $(T, U, k', u) \colon \mathscr C(S', p', k')$ and if $m \colon (S, p, k) \to (S', p', k')$, then there exists a unique functor […]

While Maranda doesn’t actually show this assignment of a functor from a monad morphism is functorial, he gives its action on morphisms.

]]>The sentence starting with

More slickly…

(here, originating from revision 2)

is wrong with the evident understanding of “groupal category” and the relevant clarification is missing by the underlying link remaining broken.

That’s probably the reason why revision 6 added a remark right below that line, clarifying the situation.

But with that remark in place, the sentence starting with “More slickly…” could/should just be deleted.

]]>Actually, could you maybe say what you mean by:

The corresponding statement for Kleisli

?

The functor induced from a monad morphism $\phi$ preserves free modales (up to iso) iff $\phi$ is an isomorphism, no?

]]>have now also added pointer to:

- Jean-Marie Maranda,
*Sur les Proprietes Universelles des Foncteurs Adjoints*, In:*Études sur les Groupes abéliens*/*Studies on Abelian Groups*Springer (1968) [doi:10.1007/978-3-642-46146-0_16]

As before, I haven’t yet spotted a definition of the induced functor on EM-categories in there.

(But it’s strainful reading, and not due to the French. If I missed it and you show me it’s somewhere in there, I won’t be surprised.)

]]>Added a couple of references.

]]>Thanks for this. I have added pointer to

- Jean-Marie Maranda,
*On Fundamental Constructions and Adjoint Functors*, Canadian Mathematical Bulletin**9**5 (1966) 581-591 [doi:10.4153/CMB-1966-072-9]

(also at *monad morphism* and at *Kleisli category*).

However, while this article defines monad morphisms (in the “right” direction! :-) and discusses them in the context of initiality of the Kleisli category, I haven’t spotted a declaration of the induced functor between categories of modales.

(I admit that I haven’t read line-by line, but such a functor would need to involve the symbol “$\overline{\mathcal{C}'}$” in Maranda’s notation, and that seems to never appear.)

]]>added pointer to:

- J.-M. Maranda, Thm. 1 in:
*Some remarks on limits in categories*, Canadian Mathematical Bulletin**5**2 (1962) 133-146 [doi:10.4153/CMB-1962-015-0]

Added the result that the Banach Lie group $U(\mathcal{H})_{norm}$ is metrizable hence paracompact, citing Nikolaus–Sachse–Wockel.

]]>Examples

]]>I’m travelling at the moment, so it’s not convenient for me to double check right now, but I believe functoriality of the EM construction (and Kleisli construction) is due to Maranda’s 1966 “On fundamental constructions and adjoint functors”. This paper is also the one where the universal property of the Kleisli category as the initial resolution of the monad is first proven. In the 1968 “Sur les propriétés universelles des foncteurs adjoints”, Maranda then proves a 2-categorical universal property of the Kleisli and EM constructions. However, I don’t believe it follows immediately from the results in these papers that monad morphisms are (dually) in bijection with functors between the EM categories: as far as I’m aware, the Frei paper is the earliest reference for this fact. The corresponding statement for Kleisli doesn’t explicitly appear in the literature for some time, but it is implicit in Linton’s early papers (e.g. the 1969 “An outline of functorial semantics”).

]]>made explicit (here) the equivalence between dual objects and their adjunctable tensor product functors *for specific hom-isomorphism* ( Dold & Puppe 1984, Thm. 1.3 (b) and (c))

and mentioned that the resulting adjunctions are amazing ambidextrous:

$(\text{-}) \otimes A \;\; \dashv \;\; (\text{-}) \otimes A^\ast \;\; \dashv \;\; (\text{-}) \otimes A \,.$ ]]>and to this thesis

Thomas S. Ligon,

*Galois-Theorie in monoidalen Kategorien*, Munich (1978) [pdf, doi:10.5282/edoc.14958]transl:

*Galois theory in monoidal categories*(2019) [pdf, doi:10.5282/edoc.24952]

following Pareigis

]]>added pointer to

- Bodo Pareigis,
*Non-additive ring and module theory IV: The Brauer group of a symmetric monoidal category*, Lecture Notes in Mathematics**549**(1976) [doi:10.1007/BFb0077339]

which has early discussion of dualizable objects (calling them “finite objects”)

]]>