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I’ve added to Eilenberg-Moore category an explicit definition of EM objects in a 2-category and some other universal properties of EM categories, including Linton’s construction of the EM category as a subcategory of the presheaves on the Kleisli category.
Question: can anyone tell me what Street–Walters mean when they say that this construction (and their generalised one, in a 2-category with a Yoneda structure) exhibits the EM category as the ‘category of sheaves for a certain generalised topology on’ the Kleisli category?
I created a page for Paul-André Melliès. I noted that he has a neat paper: that I had not seen.
It’s a nice paper, all right. I’ve put that link to the PDF on the Eilenberg-Moore category page.
can anyone tell me what Street–Walters mean when they say that this construction … exhibits the EM category as the ‘category of sheaves for a certain generalised topology on’ the Kleisli category?
My only guess is that they mean that the EM category is a subcategory of presheaves on the Kleisli category, just as the category of sheaves on a site is a subcategory of the category of presheaves. I don’t see any closer relationship than that, but perhaps they had something else in mind.
Hmm. I’m still trying to understand Melliès’s paper (among many others), but he refers to Berger’s A cellular nerve for higher categories — remark 1.7 there describes a Grothendieck topology on the category $Fin$ of finite sets such that models of a Lawvere theory are the presheaves that restrict to sheaves on $Fin$. I wondered if this (replacing the theory with some $Kl(T)$ and the ‘arities’ $Fin$ with the base category) was what Street & Walters were alluding to. Then again, I’m not even sure if that makes sense…
they mean that the EM category is a subcategory of presheaves on the Kleisli category
Reflective or not ?
I can’t imagine that it would be reflective, since the EM and Kleisli categories are the same “size” whereas the presheaf category is one size bigger. And no, I wouldn’t either be inclined to call the objects of a non-reflective subcategory of a presheaf category any kind of “sheaves;” I was just saying that’s the only relationship I can think of.
@Finn Why not ask Paul-André himself. We were talking on Skype about the area last night and I mentioned that you had some problem understanding the point you mentioned (I gave him a link to the discussion.) Send him an e-mail. Or more generally ask the Cat list for help. They are usually very helpful.
@Tim: Good idea, thanks! I might just do that (in between the other hundred and one things I have to do. Oh, for the life of a Ph.D. student…)
@Finn I can think of worse places to be a PhD student than Baile atha Cliath. I have fond memories of TCD back in the 1970s. I was in UCC in those days and would go up to Dublin for a meeting usually in December (essentially Irish Math Soc., but it did not yet exist.) I stayed several times on a floor in Trinity. Breakfast at Bewley’s … I remember a full Irish breakfast, but that is perhaps a memory only. Then off to the talks.
Yes, I suppose it could be worse. That was just me putting on the ’poor mouth’!
An Béal Bocht by Myles na gCopaleen??
I have never read that. I have the Third Policeman. That would be a good book to mention for its quantum theory!
The ’poor mouth’ as in a particularly Irish kind of maudlin self-pity.
I’ve never read An Béal Bocht either (my Irish was never up to it), but The Third Policeman is one of the funniest books I’ve ever read. Whenever I see a sheep I think
What is a sheep only millions of little bits of sheepness all whirling around and doing intricate convolutions inside the sheep? What is it but that?
But Myles was tongue in cheek. (There is a translation of An Béal Bocht.)
As I understand it, An Béal Bocht is a (pretty vicious) satire on Gaeltacht misery memoirs by people like Peig Sayers and Tomás Ó Criomhthain, and the slum-tourist industry they brought about, hence the title — ’putting on the poor mouth’ means engaging in self-pitying lamentation, usually with an ulterior motive.
I think he could be very vicious at times.
And no, I wouldn’t either be inclined to call the objects of a non-reflective subcategory of a presheaf category any kind of “sheaves;”
Well, there are cases where this is justified, namely the sheaves on noncommutative spaces and, similarly, sheaves on a Q-category. The sheaf condition is over there not with respect to covers which are cones over discrete set of objects but rather over cones over more general diagrams. This is alike the situation in enriched category theory where conical diagrams are replaced by weighted limits, and they are still called limits. In noncommutative geometry, the sheaf condition is always more general than one coming from Grothendieck topology. There are lots of examples of noncommutative sheaves and bundles which are rich enough and behave too well not to justify the name of sheaf.
Sheaves are about 1-categorical local gluing conditions. The fact that for the sheaves of sets on commutative spaces, and on sites in particular one has a general nonsense characterization of sheafification is not in my opinion more important than the original motivation of gluing from local patches. Now every sensible notion of local has appropriate version of gluing, hence appropriate kind of sheaf theory.
@Zoran: fair enough. I should have said “I wouldn’t be inclined to call the objects of a category ’sheaves’ just because they are a full subcategory of some presheaf category, without some additional reason to believe that they behave in a sheaf-like manner.”
That is also fair enough :=)
At Eilenberg-Moore category I have tried to make the paragraph on the relation of $T$-algebras to free $T$-algebras more explicit and more comprehensive, now a small new subsection titled As a colimit completion of the Kleisli category.
First of all I added the statement of the universal Beck coequalizer, for completeness, and then I edited the formatting and the citations for the characterization via presheaves on the Kleisli category a bit.
Similarly I have touched the Definition section, trying to edit a bit for readability.
Proposition 1 is a bit ambiguous. It is an absolute coequaliser in the base but not in the category of algebras. Also, if you choose the definitions of $C_T$ and $F_T$ right, the pullback diagram in Proposition 2 is also a bicategorical pullback diagram.
True, I have fixed the wording.
I’m a little confused by the discussion of ’universal $T$-module’ — I assume it’s just the metaphor that I’m having trouble grasping. A (left) $T$-module in the category $C$ is defined toward the beginning; then later we are suggested to view the forgetful functor $C^T\to C$ as the “universal (left) $T$-module”, but we have not yet been instructed on how to think of this as a module of any kind, much less a universal one.
Can someone help me understand what is meant here?
Any adjunction $L \dashv R$, which induces a monad $T$, induces a left $T$-module whose underlying 1-cell is $R$, and whose action is induced by the counit of the adjunction. The universal left $T$-module is the left $T$-module induced by $F_T \dashv U_T$. It is universal in the sense that it is induced by the representing object for the 2-presheaf sending each object $T\text{-Alg} : \mathcal KX$ to the category of $X$-indexed left $T$-modules. One place to read about this is §3 of Kelly–Street’s Review of the elements of 2-categories.
@varkor Thanks, this reference has been helpful for me; in particular the discussion on page 169.
It is claimed that the forgetful functor $U : C^T \to C$ is a terminal object in the full subcategory of $\mathrm{Cat} / C$ of right-adjoint functors whose induced monad is $T$. I am having trouble seeing why this is the case. I do know that, given some right adjoint $R : D \to C$ whose corresponding monad is $T$, we can construct a functor $F : D \to C^T$ such that $UF = R$, where the algebras are induced by the counit of the adjunction. But I don’t see why this functor is unique. For instance, take $C^T$ to be the category of bi-pointed sets (i.e., $T X = X + 2$). There is a functor $G : C^T \to C^T$ that swaps the two points of a bi-pointed set. This functor satisfies $U G = U$, but it is not the same as the canonical functor $F$ constructed above, which in this case would be the identity. Am I missing something?
I have completed the publication data for
and added pointer to:
I am pretty sure there was an erroneous dualization here in the treatment of comonads. The construction of the co-EM category is still a “limit-type” construction, so I am pretty sure that its universal property involves mapping in from other categories, rather than mapping out as was indicated here. I have taken the liberty of changing what the page says to reflect this, but it is possible I am mistaken.
Added the original reference for the Eilenberg-Moore category: Samuel Eilenberg, John Moore, Adjoint functors and triples, Illinois J. Math. 9 (3), pp. 381 - 398, September 1965. (doi:10.1215/ijm/1256068141)
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