Re: on a subcategory of algebras for a monad

Re: on a subcategory of algebras for a monad

[Fred E.J. Linton]

Hi, Emily,

Another example of interesting classes of "objects those X in C such 
that the unit X -> MX is an isomorphism" (C a pleasant category, M a 
(perhaps familiar) monoid on C):

As C, use the category either of vector spaces, or of Banach spaces 
(if the latter, use linear maps of bound not exceeding 1); and as M 
use the corresponding double-dualization monoid. 

Then: among vector spaces, it's the finite-dimensional ones for which 
the unit is an isomorphism; among Banach spaces, it's the reflexive ones.

So again, behavior with respect to limits is, in either case, not 
(TTBOMK) as you[r grad student] might wish.

Cheers, -- Fred

------ Original Message ------
Received: Sun, 12 May 2013 06:48:45 PM EDT
From: "Fred E.J. Linton" <fejlinton@usa.net>
To: Emily Riehl <eriehl@math.harvard.edu>, <categories@mta.ca>
Subject: categories: Re: on a subcategory of algebras for a monad

> Hi, Emily,
> 
> Have you or your grad student noticed the example got by taking as your
> locally presentable category C the category of sets, and as your monad M
the
> ultrafilter or Stone-Cech monad ß (with the Eilenberg-Moore category of
> ß-algebras being the category of compact Hausdorff spaces (and continuous
> maps))?
> 
> Here, your "objects those X in C such that the unit X -> MX is an
isomorphism"
> are just the finite sets, and, unless I misunderstand your limits question,
I
> fear that only finite limits will work as you desire.
> 
> Does that help in any way? Cheers, -- Fred 
> 
> ---
> 
> ------ Original Message ------
> Received: Fri, 10 May 2013 09:23:34 PM EDT
> From: Emily Riehl <eriehl@math.harvard.edu>
> To: categories@mta.ca
> Subject: categories: on a subcategory of algebras for a monad
> 
>> Hi,
>> 
>> I received the following question from a grad student that I was unable 
to
>> answer, but maybe you can (shared with permission). The subcategory
Comp_M
>> he introduces below can equally be defined to be the inverter of the
>> counit of the monadic adjunction. But I don't see how this universal
>> property helps understand limits in the subcategory. We suspect a left
>> adjoint to the inclusion is unlikely.
>> 
>> Can you help? Or have you seen something like this before?
>> 
>> Best,
>> Emily
>> 
>> ***
>> ??
>> Hi folks,
>> ??
>> I'm interested in closure properties of a particular subcategory of the
>> category of algebras of a monad. To be more precise, let C be a locally
>> presentable category and M be a monad on C. The category of algebras
Alg_M
>> has all limits, and they are computed in C. Denote by Comp_M the full
>> subcategory of Alg_M of "M-complete objects" (does anyone have a better
>> name?), with objects those X in C such that the unit X -> MX is an
>> isomorphism, viewed in the natural way as M-algebras (using the inverse 
MX
>> -> X).
>> ??
>> My question: Is Comp_M closed under (actually: sequential) limits,
computed
>> as limits in Alg_M?
>> ??
>> For some examples that come to mind immediately, the answer is clearly
yes,
>> because Comp_M is either trivial (e.g., if M is the free monoid monad  on
>> Sets) or all of Alg_M (i.e., if M is idempotent). A more interesting
> example
>> is Bousfield-Kan R-completion, for which I don't know the answer.
>> ??
>> In fact, I'm interested in left exact monads; in this case, the
idempotent
>> approximation is given by the equalizer of the two natural maps M -> M^2,
>> but I'm not sure if this is relevant. What I'm hoping for is a sufficient
>> criterion or a good counterexample in the abstract situation.
>> ??
>> Many thanks!
>> 




[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: on a subcategory of algebras for a monad

[F. William Lawvere]

Hi, Emily and Fred
This definition of finiteness can be extended to a definition of smallness.
Double dualisation and coadequacy monads have a big overlap,related to
the Isbell conjugacy between Csheaves and Calgebras where  C is a given
category. Instead of taking C to consist only of 2 and 3 element sets,as in the 
Stone case, let C be a full category of countable sets and ask if C is coadequate
among all sets. Then the fixed algebras for the resulting monad are small in the 
sense of being smaller than the UlamTarski number. That they are sufficiently big 
for all reasonable constructions of geometry and analysis concentrates in the fact
that they are closed under the formation of Hurewicz exponentials, ie that their
  category is Cartesian closed, indeed a topos (the subobject classifier is tautological
since it is in C). It would be desirable to have a general proof of this fact.
         The intuitions connected with historical cases of Mcompleteness require that. 
M is already idempotent, ie does not cry out for transfinite iteration. Thus indeed
a suggestive and appropriate name for these fixed algebras is needed.

> Date: Sat, 11 May 2013 01:09:32 -0400
> From: fejlinton@usa.net
> To: eriehl@math.harvard.edu; categories@mta.ca
> Subject: categories: Re: on a subcategory of algebras for a monad
> 
> Hi, Emily,
> 
> Have you or your grad student noticed the example got by taking as your
> locally presentable category C the category of sets, and as your monad M the
> ultrafilter or Stone-Cech monad ß (with the Eilenberg-Moore category of
> ß-algebras being the category of compact Hausdorff spaces (and continuous
> maps))?
> 
> Here, your "objects those X in C such that the unit X -> MX is an isomorphism"
> are just the finite sets, and, unless I misunderstand your limits question, I
> fear that only finite limits will work as you desire.
> 
> Does that help in any way? Cheers, -- Fred 
> 
> ---
> 
> ------ Original Message ------
> Received: Fri, 10 May 2013 09:23:34 PM EDT
> From: Emily Riehl <eriehl@math.harvard.edu>
> To: categories@mta.ca
> Subject: categories: on a subcategory of algebras for a monad
> 
>> Hi,
>> 
>> I received the following question from a grad student that I was unable   to
>> answer, but maybe you can (shared with permission). The subcategory Comp_M
>> he introduces below can equally be defined to be the inverter of the
>> counit of the monadic adjunction. But I don't see how this universal
>> property helps understand limits in the subcategory. We suspect a left
>> adjoint to the inclusion is unlikely.
>> 
>> Can you help? Or have you seen something like this before?
>> 
>> Best,
>> Emily
>> 
>> ***
>> ??
>> Hi folks,
>> ??
>> I'm interested in closure properties of a particular subcategory of the
>> category of algebras of a monad. To be more precise, let C be a locally
>> presentable category and M be a monad on C. The category of algebras Alg_M
>> has all limits, and they are computed in C. Denote by Comp_M the full
>> subcategory of Alg_M of "M-complete objects" (does anyone have a better
>> name?), with objects those X in C such that the unit X -> MX is an
>> isomorphism, viewed in the natural way as M-algebras (using the inverse  MX
>> -> X).
>> ??
>> My question: Is Comp_M closed under (actually: sequential) limits, computed
>> as limits in Alg_M?
>> ??
>> For some examples that come to mind immediately, the answer is clearly yes,
>> because Comp_M is either trivial (e.g., if M is the free monoid monad  on
>> Sets) or all of Alg_M (i.e., if M is idempotent). A more interesting
> example
>> is Bousfield-Kan R-completion, for which I don't know the answer.
>> ??
>> In fact, I'm interested in left exact monads; in this case, the idempotent
>> approximation is given by the equalizer of the two natural maps M -> M^2,
>> but I'm not sure if this is relevant. What I'm hoping for is a sufficient
>> criterion or a good counterexample in the abstract situation.
>> ??
>> Many thanks!
>> 

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: on a subcategory of algebras for a monad

[Fred E.J. Linton]

Hi, Emily,

Have you or your grad student noticed the example got by taking as your
locally presentable category C the category of sets, and as your monad M the
ultrafilter or Stone-Cech monad ß (with the Eilenberg-Moore category of
ß-algebras being the category of compact Hausdorff spaces (and continuous
maps))?

Here, your "objects those X in C such that the unit X -> MX is an isomorphism"
are just the finite sets, and, unless I misunderstand your limits question, I
fear that only finite limits will work as you desire.

Does that help in any way? Cheers, -- Fred 

---

------ Original Message ------
Received: Fri, 10 May 2013 09:23:34 PM EDT
From: Emily Riehl <eriehl@math.harvard.edu>
To: categories@mta.ca
Subject: categories: on a subcategory of algebras for a monad

> Hi,
> 
> I received the following question from a grad student that I was unable  to
> answer, but maybe you can (shared with permission). The subcategory Comp_M
> he introduces below can equally be defined to be the inverter of the
> counit of the monadic adjunction. But I don't see how this universal
> property helps understand limits in the subcategory. We suspect a left
> adjoint to the inclusion is unlikely.
> 
> Can you help? Or have you seen something like this before?
> 
> Best,
> Emily
> 
> ***
> ??
> Hi folks,
> ??
> I'm interested in closure properties of a particular subcategory of the
> category of algebras of a monad. To be more precise, let C be a locally
> presentable category and M be a monad on C. The category of algebras Alg_M
> has all limits, and they are computed in C. Denote by Comp_M the full
> subcategory of Alg_M of "M-complete objects" (does anyone have a better
> name?), with objects those X in C such that the unit X -> MX is an
> isomorphism, viewed in the natural way as M-algebras (using the inverse  MX
> -> X).
> ??
> My question: Is Comp_M closed under (actually: sequential) limits, computed
> as limits in Alg_M?
> ??
> For some examples that come to mind immediately, the answer is clearly yes,
> because Comp_M is either trivial (e.g., if M is the free monoid monad on
> Sets) or all of Alg_M (i.e., if M is idempotent). A more interesting
example
> is Bousfield-Kan R-completion, for which I don't know the answer.
> ??
> In fact, I'm interested in left exact monads; in this case, the idempotent
> approximation is given by the equalizer of the two natural maps M -> M^2,
> but I'm not sure if this is relevant. What I'm hoping for is a sufficient
> criterion or a good counterexample in the abstract situation.
> ??
> Many thanks!
> 



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

on a subcategory of algebras for a monad

[Emily Riehl]

Hi,

I received the following question from a grad student that I was unable to
answer, but maybe you can (shared with permission). The subcategory Comp_M
he introduces below can equally be defined to be the inverter of the
counit of the monadic adjunction. But I don't see how this universal
property helps understand limits in the subcategory. We suspect a left
adjoint to the inclusion is unlikely.

Can you help? Or have you seen something like this before?

Best,
Emily

***
??
Hi folks,
??
I'm interested in closure properties of a particular subcategory of the
category of algebras of a monad. To be more precise, let C be a locally
presentable category and M be a monad on C. The category of algebras Alg_M
has all limits, and they are computed in C. Denote by Comp_M the full
subcategory of Alg_M of "M-complete objects" (does anyone have a better
name?), with objects those X in C such that the unit X -> MX is an
isomorphism, viewed in the natural way as M-algebras (using the inverse MX
-> X).
??
My question: Is Comp_M closed under (actually: sequential) limits, computed
as limits in Alg_M?
??
For some examples that come to mind immediately, the answer is clearly yes,
because Comp_M is either trivial (e.g., if M is the free monoid monad on
Sets) or all of Alg_M (i.e., if M is idempotent). A more interesting example
is Bousfield-Kan R-completion, for which I don't know the answer.
??
In fact, I'm interested in left exact monads; in this case, the idempotent
approximation is given by the equalizer of the two natural maps M -> M^2,
but I'm not sure if this is relevant. What I'm hoping for is a sufficient
criterion or a good counterexample in the abstract situation.
??
Many thanks!



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]