Mentioned the pullback theorem for relative monads.

]]>Translated diagram to tikzcd

Bartosz Milewski

]]>Added a diagram

Bartosz Milewski

]]>Cite voutas.

]]>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)

Added reference to *Structures pseudo-algébriques (1ère partie)*.

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.

I have completed the publication data for

- {#Street72} Ross Street,
*The formal theory of monads*, Journal of Pure and Applied Algebra**2**2 (1972) 149-168 [doi:10.1016/0022-4049(72)90019-9]

and added pointer to:

- Stephen Lack, Ross Street,
*The formal theory of monads II*, Journal of Pure and Applied Algebra**175**1–3 (2002) 243-265 [doi:10.1016/S0022-4049(02)00137-8]

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?

]]>Added redirects for name variants with en-dashes.

]]>@varkor Thanks, this reference has been helpful for me; in particular the discussion on page 169.

]]>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.

]]>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?

]]>Added $Rel$ as an example where not all monads have Eilenberg-Moore objects.

]]>Fixed two faulty links

]]>True, I have fixed the wording.

]]>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.

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.

That is also fair enough :=)

]]>@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.”

]]>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.

]]>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.

]]>I think he could be very vicious at times.

]]>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.

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