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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 21st 2012
   • (edited Jan 21st 2012)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI am wondering whether there is anything out there on a general concept of profinite completion relative to an algebraic (i.e., monadic) functor. The rough idea I have in mind is that if $C$ is locally finitely presentable and $U: A \to C$ is monadic, then in $C$ we might reasonably call a finitely presentable object just "finite", and given an object $a$ of $A$, we might be interested in the profinite completion of $a$ (relative to $U$), the limit over all quotients of $a$ which become finite in $C$ (edit: what should I mean by "limit" here -- do I mean in $pro$-$Fin(C)$ where $Fin(C)$ is the category of finite objects?). The classical case would be $U: Grp \to Set$ of course. But also of interest is the relative monadic functor $U: Alg_k \to Vect_k$, where finite objects in $Vect_k$ are finite-dimensional spaces (so the profinite completion of an algebra would be the inverse limit over all finite-dimensional algebra quotients). This is the type of thing that I was alluding to in [[cofree coalgebra]], where I linked to [[profinite completion]], but so far that page just concentrates on the classical case $U: Grp \to Set$.

   I am wondering whether there is anything out there on a general concept of profinite completion relative to an algebraic (i.e., monadic) functor.

   The rough idea I have in mind is that if $C$ is locally finitely presentable and $U: A \to C$ is monadic, then in $C$ we might reasonably call a finitely presentable object just “finite”, and given an object $a$ of $A$, we might be interested in the profinite completion of $a$ (relative to $U$), the limit over all quotients of $a$ which become finite in $C$ (edit: what should I mean by “limit” here – do I mean in $pro$-$Fin(C)$ where $Fin(C)$ is the category of finite objects?).

   The classical case would be $U: Grp \to Set$ of course. But also of interest is the relative monadic functor $U: Alg_k \to Vect_k$, where finite objects in $Vect_k$ are finite-dimensional spaces (so the profinite completion of an algebra would be the inverse limit over all finite-dimensional algebra quotients). This is the type of thing that I was alluding to in cofree coalgebra, where I linked to profinite completion, but so far that page just concentrates on the classical case $U: Grp \to Set$.

2. • CommentRowNumber2.
   • CommentAuthorTim_Porter
   • CommentTimeJan 22nd 2012
   • (edited Jan 22nd 2012)
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   Author: Tim_Porter
   Format: MarkdownItexPerhaps D. Gildenhuys and J. Kennison, Equational completions, model induced triples and pro-objects , J. Pure Applied Alg., 4, (1971), 317–346 may help. I have added it in as a reference with a stubby subsection on equational completion into [[profinite completion]]. I have also added a bit on pro-C completions.

   Perhaps D. Gildenhuys and J. Kennison, Equational completions, model induced triples and pro-objects , J. Pure Applied Alg., 4, (1971), 317–346 may help. I have added it in as a reference with a stubby subsection on equational completion into profinite completion.

   I have also added a bit on pro-C completions.

3. • CommentRowNumber3.
   • CommentAuthorTim_Porter
   • CommentTimeJan 22nd 2012
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   Author: Tim_Porter
   Format: MarkdownItexOn a slightly different note, have a look at http://www.math.muni.cz/~kunc/conf/Steinberg.pdf for some work of Ben on related constructions and their applications. It would be good to include some of his stuff somewhere relevant, but I do not have time to do it at the moment. (also stuff by Jorge Almeida from Porto.)

   On a slightly different note, have a look at http://www.math.muni.cz/~kunc/conf/Steinberg.pdf for some work of Ben on related constructions and their applications. It would be good to include some of his stuff somewhere relevant, but I do not have time to do it at the moment. (also stuff by Jorge Almeida from Porto.)

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 22nd 2012
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   Author: Todd_Trimble
   Format: MarkdownItexWow, you added a lot, Tim -- thank you! Unfortunately I can't seem to gain access to Gildenhuys-Kennison (Elsevier puts it behind a paywall, and I refuse to pay on principle), so this will have to wait until I can get to the nearest university library. Maybe I could ask here: what's meant by an equational completion? And is there a single core result of that paper which ought to be recorded in the nLab?

   Wow, you added a lot, Tim – thank you! Unfortunately I can’t seem to gain access to Gildenhuys-Kennison (Elsevier puts it behind a paywall, and I refuse to pay on principle), so this will have to wait until I can get to the nearest university library. Maybe I could ask here: what’s meant by an equational completion? And is there a single core result of that paper which ought to be recorded in the nLab?

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 22nd 2012
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   Author: Todd_Trimble
   Format: MarkdownItexAh, well I guess the equational completion is what is also called the monadic completion. I'll see if I can write something up for the nLab.

   Ah, well I guess the equational completion is what is also called the monadic completion. I’ll see if I can write something up for the nLab.

6. • CommentRowNumber6.
   • CommentAuthorTim_Porter
   • CommentTimeJan 22nd 2012
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexEquational category is meant in the sense of Linton: Some aspects of equational categories. La Jolla. Also in the triples seminar collection. The idea in Gildenhuys and Kennison is to have a functor (e.g. the forgetful functor $I: FinGrps\to Sets$) and to find an equational category (approximately a category of algebras over a possibly infinitary theory or monad) so that $I$ factors through th forgetful functor from that category to sets and is universal with that property. Linton shows that the class of $n$-ary operations for the completion of $I$ can be regarded as $Nat(I^n,I)$ for $n$ a set. (This needs to have some 'tractability' conditions on it to ensure smallness etc.) The problem is thus to find enough information on those natural transformations so that you get a reasonable description of the 'algebras'. For $I: FInSets\to Sets$ the inclusion, a natural transformation from $I^n$ to $I$ corresponds to an ultrafilter on $n$ (and the equational completion is that category of compact Hausdorff spaces). many of the linear compactness conditions on algebras are related to similar algebraic situations. (I am copying out or adapting from the intro to the paper so hope this makes sense!!!!) Much of the terminology may have changed since 1971 so you may know of these results in a different situation.

   Equational category is meant in the sense of Linton: Some aspects of equational categories. La Jolla. Also in the triples seminar collection.

   The idea in Gildenhuys and Kennison is to have a functor (e.g. the forgetful functor $I: FinGrps\to Sets$) and to find an equational category (approximately a category of algebras over a possibly infinitary theory or monad) so that $I$ factors through th forgetful functor from that category to sets and is universal with that property. Linton shows that the class of $n$-ary operations for the completion of $I$ can be regarded as $Nat(I^n,I)$ for $n$ a set. (This needs to have some ’tractability’ conditions on it to ensure smallness etc.) The problem is thus to find enough information on those natural transformations so that you get a reasonable description of the ’algebras’.

   For $I: FInSets\to Sets$ the inclusion, a natural transformation from $I^n$ to $I$ corresponds to an ultrafilter on $n$ (and the equational completion is that category of compact Hausdorff spaces). many of the linear compactness conditions on algebras are related to similar algebraic situations. (I am copying out or adapting from the intro to the paper so hope this makes sense!!!!)

   Much of the terminology may have changed since 1971 so you may know of these results in a different situation.

7. • CommentRowNumber7.
   • CommentAuthorTim_Porter
   • CommentTimeJan 22nd 2012
   • (edited Jan 22nd 2012)
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexas to 6., I was double checking and that looks right. Equational = monadic in this case. There is also a paper by Diers: in the [Cahiers](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1976__17_4/CTGDC_1976__17_4_363_0/CTGDC_1976__17_4_363_0.pdf). As to adding a lot, I have a draft monograph of 500 + pages on profinite algebraic homotopy so a bit of copy paste edit works wonders!!!

   as to 6., I was double checking and that looks right. Equational = monadic in this case. There is also a paper by Diers: in the Cahiers.

   As to adding a lot, I have a draft monograph of 500 + pages on profinite algebraic homotopy so a bit of copy paste edit works wonders!!!

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeJan 23rd 2012
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   Author: Mike Shulman
   Format: MarkdownItexLooks like the free monadic functor on a tractable functor in Gildenhuys-Kennison is what you'd expect: its [[codensity monad]]. Their goal is then: if $M\to Set$ is a tractable functor (meaning, I guess, that it has a codensity monad — it suffices for M to be small), with codensity monad $T$, then is there a small understandable (perhaps finitary) submonad $T_0 \subseteq T$ such that $T$-algebras can be identified with certain topological $T_0$-algebras, with topology induced by all maps into the objects of $M$ (considered as discrete)?

   Looks like the free monadic functor on a tractable functor in Gildenhuys-Kennison is what you’d expect: its codensity monad. Their goal is then: if $M\to Set$ is a tractable functor (meaning, I guess, that it has a codensity monad — it suffices for M to be small), with codensity monad $T$, then is there a small understandable (perhaps finitary) submonad $T_0 \subseteq T$ such that $T$-algebras can be identified with certain topological $T_0$-algebras, with topology induced by all maps into the objects of $M$ (considered as discrete)?

9. • CommentRowNumber9.
   • CommentAuthorTim_Porter
   • CommentTimeJan 23rd 2012
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   Author: Tim_Porter
   Format: MarkdownItexThere is a link between the codensity monad and categorical shape theory. If $K:A\to B$ is a functor, then the shape category for $K$ is the Kleisli category for the codensity monad (in profunctors) provided the counit transformation is invertible. (I am abbreviating things here and it is a long time since I worked on this!)

   There is a link between the codensity monad and categorical shape theory. If $K:A\to B$ is a functor, then the shape category for $K$ is the Kleisli category for the codensity monad (in profunctors) provided the counit transformation is invertible. (I am abbreviating things here and it is a long time since I worked on this!)

10. • CommentRowNumber10.
    • CommentAuthorTim_Porter
    • CommentTimeJan 24th 2012
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    Author: Tim_Porter
    Format: MarkdownItexAnother interesting result is that they establish a direct link with pro-objects in their section 2.

    Another interesting result is that they establish a direct link with pro-objects in their section 2.

11. • CommentRowNumber11.
    • CommentAuthorTim_Porter
    • CommentTimeSep 3rd 2012
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    Author: Tim_Porter
    Format: MarkdownItexI have created a stub on [[profinite completion of a space]], which is not about the profinite completion of the homotopy type, but is more purey topological. It is needed if one is to discuss Fabien Morel's work or Geirion Quick's on simplicial profinite spaces and their relationship with étale homotopy theory.

    I have created a stub on profinite completion of a space, which is not about the profinite completion of the homotopy type, but is more purey topological. It is needed if one is to discuss Fabien Morel’s work or Geirion Quick’s on simplicial profinite spaces and their relationship with étale homotopy theory.