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charset="utf-8" Content-Transfer-Encoding: quoted-printable 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 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 equa= tional 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 theory)= , then the category of models of the extension is monadic over the category= of models of the original theory. Surely this is either explicitly stated= and proved somewhere, or follows easily from some result I simply have not= encountered. Citations? 2. 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 --000000000000a17d4806109acb5d Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable
Dear David,

Yes, models of singl= e-sorted algebraic theories are always monadic over Set, and such theories = correspond precisely to finitary, ie omega-filtered-colimit-preserving mona= ds 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. Mor= e generally, monads preserving alpha-filtered colimits for a higher regular= cardinal alpha correspond to algebraic theories with alpha-ary operations; unbounded monads (such as the powerset= monad) can be viewed as corresponding to "large theories" with n= o bound on their arities. The theory corresponding to the powerset monad, e= g, is the theory of sup-lattices, ie posets with arbitrary small joins, which is not expressible with operations of bo= unded arities. Regarding references on these generalizations, I would also = be curious.

Concerning your ques= tions on extensions of theories, and more general kinds of theories and bas= e categories, there recently was a long thread on the category theory zulip= server [2], which I'll try to summarize: extensions of single-sorted theories by new operations and/or equations ar= e always monadic (regardless of arities); this follows from the fact that m= onadic functors 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 Set. The models of many-sort= ed theories are monadic over powers of Set, and extensions of many-sorted t= heories by operations and axioms are also monadic, again by cancellation. T= hings become more complicated in the generalized/essential algebraic case, since (in the generalized algebr= aic, ie dependently 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 general there's a lot of= ongoing work on the dependently typed, ie generalized algebraic case, such= as eg Chaitanya Leena Subramaniam's recent PhD thesis representing depende= nt algebraic theories by finitary monads on presheaf categories 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.20an= d.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 Ye= tter <dyetter@ksu.e= du> wrote:
We are all, of course, familiar with Beck's Theorem.  I'm rather hopin= g that there are results in the literature that will save me from having to= prove that the underlying 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 communit= y:

  1. It seems ob= vious to me that for the same reason categories of models of finitary algeb= raic (equational) theories are monadic over Set, if one has (I think I'm using the term correctly here) a co= nservative extension of an equational theory (by which I mean, add operatio= ns and equations in such a way that no new equations are imposed on the ope= rations of the original theory), then the category of models of the extension is monadic over the category of mo= dels of the original theory.  Surely this is either explicitly stated = and proved somewhere, or follows easily from some result I simply have not = encountered.   Citations?

  2.  What is th= e most general sort of theory whose models are monadic over Set? Or if that= is not known, what sorts of theories have monadic categories of models over Set?   Are there multisorted g= eneralizations of any results of that sort, ideally not just to Set^\alpha,= where \alpha is the cardinality of a set of sorts, but to things like Grap= h?  Again some citations would be much appreciated.

Best Though= ts,
David Yette= r
University = Distinguished Professor
Department = of Mathematics
Kansas Stat= e University

(That is th= e first and last time I'll use that signature block in writing to the list,= but I thought I'd do it once since I thought the community would be gratified that a categorist was so honore= d.  After this it's back to David 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.
 
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