Date: Thu, 2 Feb 1995 04:01:28 -0400 (AST) Subject: Algebraic structures for Eilenberg-Moore algebras Date: Wed, 01 Feb 1995 11:51:57 +0000 From: Andrea Schalk I'm looking for some pointer to literature where I can find the following (or a similar) Theorem: Let $(T,\eta,\mu)$ be a monad. Assume we have an algebraic structure such that every free algebra for that monad carries one of those algebraic structures and such that all morphisms of the form $Tf$ and all $\mu_C$ preserve it. Then all Eilenberg-Moore algebras carry such a structure and all morphisms between them preserve it. Thanks Andrea Schalk University of Cambridge Computer Lab Andrea.Schalk@cl.cam.ac.uk Date: Thu, 2 Feb 1995 23:02:17 -0400 (AST) Subject: Re: Algebraic structures for Eilenberg-Moore algebras Date: Thu, 2 Feb 95 08:39:39 EST From: Michael Barr - - Date: Wed, 01 Feb 1995 11:51:57 +0000 - From: Andrea Schalk - - - I'm looking for some pointer to literature where I can find the - following (or a similar) Theorem: - - Let $(T,\eta,\mu)$ be a monad. Assume we have an algebraic structure - such that every free algebra for that monad carries one of those - algebraic structures and such that all morphisms of the form $Tf$ - and all $\mu_C$ preserve it. Then all Eilenberg-Moore algebras carry - such a structure and all morphisms between them preserve it. - - Thanks - - Andrea Schalk - University of Cambridge - Computer Lab - - Andrea.Schalk@cl.cam.ac.uk - - I don't know of an explicit reference, but things of this sort are certainly familiar. Look, for example, at the proof in TTT that toposes are cartesian closed. For that matter, the proof that a slice of a topos is a topos uses the same idea. Michael Barr Date: Thu, 2 Feb 1995 23:04:41 -0400 (AST) Subject: re Algebraic structures for Eilenberg-Moore algebras Date: Thu, 02 Feb 1995 11:27:38 -0500 (EST) From: MTHISBEL@ubvms.cc.buffalo.edu ''From: Andrea Schalk I'm looking for some pointer to literature where I can find the following (or a similar) Theorem: Let $(T,\eta,\mu)$ be a monad. Assume we have an algebraic structure such that every free algebra for that monad carries one of those algebraic structures and such that all morphisms of the form $Tf$ and all $\mu_C$ preserve it. Then all Eilenberg-Moore algebras carry such a structure and all morphisms between them preserve it. Thanks'' I don't know where to look for such a theorem but would be more optimistic if I knew what 'carries' means. Harvey Friedman published something of this tendency about 1977, I think with semantic hypothesis and syntactic conclusion. John Isbell Date: Sun, 5 Feb 1995 20:39:50 -0400 (AST) Subject: Re: Algebraic structures for Eilenberg-Moore algebras Date: Fri, 3 Feb 1995 15:52:31 +0000 (GMT) From: Dusko Pavlovic According to categories: > > Date: Wed, 01 Feb 1995 11:51:57 +0000 > From: Andrea Schalk > > > I'm looking for some pointer to literature where I can find the > following (or a similar) Theorem: > > Let $(T,\eta,\mu)$ be a monad. Assume we have an algebraic structure > such that every free algebra for that monad carries one of those > algebraic structures and such that all morphisms of the form $Tf$ > and all $\mu_C$ preserve it. Then all Eilenberg-Moore algebras carry > such a structure and all morphisms between them preserve it. > Manes' book "Algebraic Theories" contains many propositions and exercises of this kind, relating monadic and equational presentations of algebraic theories. I think something of this kind should be there. Regards, -- Dusko Pavlovic Date: Sun, 5 Feb 1995 20:41:36 -0400 (AST) Subject: Re: Algebraic structures for Eilenberg-Moore algebras Date: Fri, 3 Feb 1995 16:44:33 +0000 (GMT) From: Dusko Pavlovic > > Let $(T,\eta,\mu)$ be a monad. Assume we have an algebraic structure > such that every free algebra for that monad carries one of those > algebraic structures and such that all morphisms of the form $Tf$ > and all $\mu_C$ preserve it. Then all Eilenberg-Moore algebras carry > such a structure and all morphisms between them preserve it. > PS On a second thought, the matter of monadic vs. equational presentation just complicates things here. If we begin by presenting this algebraic structure, carried by all free T-algebras, as another monad, say (S,\eta,\mu), the proof boils down to two diagrams. The assumption that each free T-algebra is an S-algebra means that there is a natural transformation s:ST->T: its components are the given S-algebras on TX; the assumption that each Tf preserves the structure is just the naturality of s. On the other hand, the assumption that \mu of T preserves it means that \mu s = s S\mu. Using this and the naturalities, one directly checks that if a:TX->X is a T-algebra, then SX --S\eta--> STX --s--> TX --a--> X must be an S-algebra --- clearly preserved by T-morphisms. All the best, -- Dusko Date: Sun, 5 Feb 1995 20:37:12 -0400 (AST) Subject: query of A. Schalk Date: Sat, 4 Feb 95 09:26:05 EST From: Ernie Manes February 3, 1995 To: Andrea Schalk, Andrea.Schalk@cl.cam.ac.uk From: Ernie Manes, manes@math.umass.edu Re: Your query yesterday on cat-dist The question you asked was solved in my 1967 thesis and appears as Exercise 9 on page 217 of my book on Algebraic Theories, Springer-Verlag GTM 28, 1976 where the reference to the thesis is also given. In effect, if T is your monad and C is the category of algebraic structures, you give yourself a functor J : F ---> C where F is the full subcategory of free T-algebras. This is because a typical homomorphism between free T-algebras has the form of a composition Tf followed by a mu. Your question then amounts to asking for a canonical extension of J to the category of all T-algebras. This construction, based on contractible coequalizers, is given in the references cited in the first paragraph. All the best, egm Date: Fri, 10 Feb 1995 22:57:36 -0400 (AST) From: categories To: categories Subject: The recent question of Andrea Schalk, and replies to it. Date: Fri, 10 Feb 95 18:21:27 +1100 From: Max Kelly The best answer was that of Ernie Manes. What is at stake is one very simple observation: if A is the category of algebras for a monad (perhaps in the context of enriched category theory) and F is the full subcategory given by the free algebras, then F is dense in A, and moreover this density may be "presented" by coequalizers. For the meaning of "density presentation", a notion due to Brian Day, see p.172 of my book "Basic Concepts of Enriched Category Theory", CUP 1982. Here it means that every algebra is a coequalizer of a pair of maps between free algebras, IN SUCH A WAY THAT this colimit is preserved by the representables A(f,-) where f is a free algebra; and this is very easy to see. Now Thm 5.30 of my book gives a very simple proof that the left Kan extension along J: F --> A of any functor T: F --> B exists, provided only that B has coequalizers. The observation of Dusko Pavlovic, that an elegant argument is to hand when B too is monadic, gives too SPECIAL a result. The others that have answered don't seem to have pointed out that the "canonical" extension of T they refer to IS indeed the left Kan extension. Max Kelly.