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charset="Windows-1252" Content-Transfer-Encoding: quoted-printable Dear Jonas, Regarding powerset and other unranked monads, I don=92t know if this is qui= te what you=92re seeking, but you might look at our paper on monad tensorab= ility: https://lmcs.episciences.org/740 Section 2 explains the relationship between large theories and monads, and = in particular makes the following point. When considering large theories, = we need to distinguish between the property of having small free algebras a= nd the weaker property of having free small algebras. This leads to two d= istinct notions of presentation of a monad on Set, which we call =93genuin= e=94 and =93spurious=94. Best, Paul From: Jonas Frey Date: Monday, 5 February 2024 at 04:30 To: David Yetter Cc: categories@mq.edu.au Subject: Re: Monadicity questions CAUTION: This email originated from outside the organisation. Do not click = links or open attachments unless you recognise the sender and know the cont= ent is safe. Dear David, Yes, models of single-sorted algebraic theories are always monadic over Set= , and such theories correspond precisely to finitary, ie omega-filtered-col= imit-preserving monads on Set. If we take the correspondence between single= -sorted algebraic theories and Lawvere theories for granted, this is stated= eg in Hyland--Power [1], with further references there. More generally, mo= nads preserving alpha-filtered colimits for a higher regular cardinal alpha= correspond to algebraic theories with alpha-ary operations; unbounded mona= ds (such as the powerset monad) can be viewed as corresponding to "large th= eories" with no bound on their arities. The theory corresponding to the pow= erset monad, eg, is the theory of sup-lattices, ie posets with arbitrary sm= all joins, which is not expressible with operations of bounded arities. Reg= arding references on these generalizations, I would also be curious. Concerning your questions on extensions of theories, and more general kinds= of theories and base categories, there recently was a long thread on the c= ategory theory zulip server [2], which I'll try to summarize: extensions of= single-sorted theories by new operations and/or equations are always monad= ic (regardless of arities); this follows from the fact that monadic functor= s have the left cancellation property (Proposition 3.3 in [3]). Extensions = by new sorts are typically not monadic, eg Set x Set is not monadic over Se= t. The models of many-sorted theories are monadic over powers of Set, and e= xtensions of many-sorted theories by operations and axioms are also monadic= , again by cancellation. Things become more complicated in the generalized/= essential algebraic case, since (in the generalized algebraic, ie dependent= ly typed case), adding new operations can create new sorts by substitution,= which can lead to successive monadic extensions which are not composable, = as Tom Hirschowitz, James Deikun, and possibly others pointed out. In gener= al there's a lot of ongoing work on the dependently typed, ie generalized a= lgebraic case, such as eg Chaitanya Leena Subramaniam's recent PhD thesis r= epresenting dependent algebraic theories by finitary monads on presheaf cat= egories over direct categories [4]. [1] https://www.dpmms.cam.ac.uk/~martin/Research/Publications/2007/hp07.pdf= [2] https://categorytheory.zulipchat.com/#narrow/stream/229199-learning.3A-= questions/topic/distributive.20laws.20and.20monadic.20functors [3] https://ncatlab.org/nlab/show/monadic+functor [4] https://arxiv.org/abs/2110.02804 On Sun, 4 Feb 2024 at 21:00, David Yetter > wrote: We are all, of course, familiar with Beck's Theorem. I'm rather hoping tha= t there are results in the literature that will save me from having to prov= e that the underlying functor U creates coequalizers for U-split pairs in t= he context of two quite different projects I'm working on. Thus, I have tw= o questions for the community: 1. It seems obvious to me that for the same reason categories of models = of finitary algebraic (equational) theories are monadic over Set, if one ha= s (I think I'm using the term correctly here) a conservative extension of a= n equational theory (by which I mean, add operations and equations in such = a way that no new equations are imposed on the operations of the original t= heory), then the category of models of the extension is monadic over the ca= tegory of models of the original theory. Surely this is either explicitly = stated and proved somewhere, or follows easily from some result I simply ha= ve not encountered. Citations? 1. What is the most general sort of theory whose models are monadic ove= r Set? Or if that is not known, what sorts of theories have monadic categor= ies of models over Set? Are there multisorted generalizations of any resu= lts of that sort, ideally not just to Set^\alpha, where \alpha is the cardi= nality of a set of sorts, but to things like Graph? Again some citations w= ould be much appreciated. Best Thoughts, David Yetter University Distinguished Professor Department of Mathematics Kansas State University (That is the first and last time I'll use that signature block in writing t= o the list, but I thought I'd do it once since I thought the community woul= d be gratified that a categorist was so honored. After this it's back to D= avid Y. or D.Y.) You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | Learn = more about Microsoft 365 Groups You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | Learn = more about Microsoft 365 Groups You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | = Learn more about Microsoft 365 Groups --_000_CWXP265MB40585A06F4CEDC1476D64BEFF4442CWXP265MB4058GBRP_ Content-Type: text/html; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable

Dear Jonas,

Regarding powerset and other unranked monads, I don=92t know if thi= s is quite what you=92re seeking, but you might look at our paper on monad = tensorability:

https://lmcs.episciences.org/740

Section 2 explains the relationship between large theories and mona= ds, and in particular makes the following point.  When considering lar= ge theories, we need to distinguish between the property of having small free algebras and the weaker property of havi= ng free small algebras.   This leads to two distinct notions of p= resentation of a monad on Set, which we call  =93genuine=94 and =93spu= rious=94.

Best,

Paul

 

 

From: Jonas Frey <jonas743@gmail.com>= ;
Date: Monday, 5 February 2024 at 04:30
To: David Yetter <dyetter@ksu.edu>
Cc: categories@mq.edu.au <categories@mq.edu.au>
Subject: Re: Monadicity questions

CAUTION: This email originated from outside the organisation. Do not click links= or open attachments unless you recognise the sender and know the content i= s safe.

 

Dear David,

 

Yes, models of single-sorted algebraic theories are = always monadic over Set, and such theories correspond precisely to finitary= , ie omega-filtered-colimit-preserving monads on Set. If we take the corres= pondence between single-sorted algebraic theories and Lawvere theories for granted, this is stated eg in Hyland--Po= wer [1], with further references there. More generally, monads preserving a= lpha-filtered colimits for a higher regular cardinal alpha correspond to al= gebraic theories with alpha-ary operations; unbounded monads (such as the powerset monad) can be viewed as= corresponding to "large theories" with no bound on their arities= . The theory corresponding to the powerset monad, eg, is the theory of sup-= lattices, ie posets with arbitrary small joins, which is not expressible with operations of bounded arities. Regarding ref= erences on these generalizations, I would also be curious.

 

Concerning your questions on extensions of theories,= and more general kinds of theories and base categories, there recently was= a long thread on the category theory zulip server [2], which I'll try to s= ummarize: extensions of single-sorted theories by new operations and/or equations are always monadic (regardless= of arities); this follows from the fact that monadic functors have the lef= t cancellation property (Proposition 3.3 in [3]). Extensions by new sorts a= re typically not monadic, eg Set x Set is not monadic over Set. The models of many-sorted theories are mona= dic over powers of Set, and extensions of many-sorted theories by operation= s and axioms are also monadic, again by cancellation. Things become more co= mplicated in the generalized/essential algebraic case, since (in the generalized algebraic, ie dependently typed = case), adding new operations can create new sorts by substitution, which ca= n lead to successive monadic extensions which are not composable, as Tom Hi= rschowitz, James Deikun, and possibly others pointed out. In general there's a lot of ongoing work on the depend= ently typed, ie generalized algebraic case, such as eg Chaitanya Leena Subr= amaniam's recent PhD thesis representing dependent algebraic theories by fi= nitary monads on presheaf categories over direct categories [4].

 

 

On Sun, 4 Feb 2024 at 21:00, David Yetter <dyetter@ksu.edu> wrot= e:

We are all, of course, f= amiliar with Beck's Theorem.  I'm rather hoping that there are results= in the literature that will save me from having to prove that the underlyi= ng functor U creates coequalizers for U-split pairs in the context of two quite different projects I'm working on. = Thus, I have two questions for the community:

 =

  1. It seems obvious to me that for the same reason categories of models of fin= itary algebraic (equational) theories are monadic over Set, if one has (I t= hink I'm using the term correctly here) a conservative extension of an= equational theory (by which I mean, add operations and equations in such a way that no new equations are impos= ed on the operations of the original theory), then the category of models o= f the extension is monadic over the category of models of the original theo= ry.  Surely this is either explicitly stated and proved somewhere, or follows easily from some result I simply h= ave not encountered.   Citations?

 

  1.  What is the most general sort of theory whose models are monadic over= Set? Or if that is not known, what sorts of theories have monadic categori= es of models over Set?   Are there multisorted generalizations of= any results of that sort, ideally not just to Set^\alpha, where \alpha is the cardinality of a set of sorts, but to thin= gs like Graph?  Again some citations would be much appreciated.

 

Best Thoughts,

David Yetter=

University Distinguished= Professor

Department of Mathematic= s

Kansas State University<= /span>

 

(That is the first and l= ast time I'll use that signature block in writing to the list, but I though= t I'd do it once since I thought the community would be gratified that a ca= tegorist was so honored.  After this it's back to David Y. or D.Y.)

 =

 

 

 

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