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1. • CommentRowNumber1.
   • CommentAuthorTim_Porter
   • CommentTimeNov 16th 2010
   • (edited Nov 16th 2010)
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexBecause of the algebraic Kan complex entry I had a look at the simplicial T-complex page. I am not sure that the current page is quite right in its wording. It is a bit the age old problem of structure or properties. In the algebraic Kan complex, the filler choice function is part of the structure. In a T-complex the thin elements form part of the structure but then properties of the thin elements show that there is a unique choice function taking thin values. They then satisfy some equational conditions. My thought would be that there should be a bit more precision on the differences between them. For instance I think it is true (but I would need to prove it in detail) that any simplicial T-complex gave an algebraic Kan complex, yielding an 'inclusion functor' from SimpT to Alg Kan. That functor should have a left adjoint which kills off the Whitehead products etc, (that need not be trivial for an algebraic Kan complex but are for a simplicial T-complex). I do not see how to construct this explicitly but am sure there must be a simple way of imposing conditions on an alg. Kan complex and looking at 'varieties' in that category. (I have not read Thomas's thesis and he may have done something related to this already.) In other words, can one impose equations on alg. Kan complexes, in this way. The present definition is more or less the free algebras case (?). Before altering the simp. T-complex page, I thought it worth asking this question of 'varieties' as the answer (if it is known) would influence how best to do the edit.

   Because of the algebraic Kan complex entry I had a look at the simplicial T-complex page. I am not sure that the current page is quite right in its wording. It is a bit the age old problem of structure or properties. In the algebraic Kan complex, the filler choice function is part of the structure. In a T-complex the thin elements form part of the structure but then properties of the thin elements show that there is a unique choice function taking thin values. They then satisfy some equational conditions.

   My thought would be that there should be a bit more precision on the differences between them. For instance I think it is true (but I would need to prove it in detail) that any simplicial T-complex gave an algebraic Kan complex, yielding an ’inclusion functor’ from SimpT to Alg Kan. That functor should have a left adjoint which kills off the Whitehead products etc, (that need not be trivial for an algebraic Kan complex but are for a simplicial T-complex). I do not see how to construct this explicitly but am sure there must be a simple way of imposing conditions on an alg. Kan complex and looking at ’varieties’ in that category. (I have not read Thomas’s thesis and he may have done something related to this already.) In other words, can one impose equations on alg. Kan complexes, in this way. The present definition is more or less the free algebras case (?).

   Before altering the simp. T-complex page, I thought it worth asking this question of ’varieties’ as the answer (if it is known) would influence how best to do the edit.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 16th 2010
   • (edited Nov 16th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> I have not read Thomas's thesis... You should! It's well written. > ...and he may have done something related to this already. I does not seem that what he does helps with getting what you are after. In his proof the main step is to establish the left adjoint from ordinary to algebraic Kan complexes, and notably a relative version of that, that freely adds in fillers. You would need something to kill off cells. I think apart from a discussion of the relation to algebraic Kan complexes, the page on [[simplicial T-complex]]es could do with some Examples, some Properties, and some Applications. Currently the page does not make it clear why one should care about simplicial T-complexes.

   I have not read Thomas’s thesis…

   You should! It’s well written.

   …and he may have done something related to this already.

   I does not seem that what he does helps with getting what you are after.

   In his proof the main step is to establish the left adjoint from ordinary to algebraic Kan complexes, and notably a relative version of that, that freely adds in fillers. You would need something to kill off cells.

   I think apart from a discussion of the relation to algebraic Kan complexes, the page on simplicial T-complexes could do with some Examples, some Properties, and some Applications. Currently the page does not make it clear why one should care about simplicial T-complexes.

3. • CommentRowNumber3.
   • CommentAuthorTim_Porter
   • CommentTimeNov 16th 2010
   • (edited Nov 16th 2010)
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexI had skimmed Thomas' thesis and noted its high standard, but had not read it in detail. I know the corresponding structures in simplicial groups. As you say Thomas freely adds in fillers, my thought is that somehow those are the free objects (I am deliberately thinking algebraically rather than homotopically here). The natural algebraic question is to look at the quotients that satisfy interesting families of equations. (In the simplicial group(oid) case, the equations are more or less explicit in order to get the nerves of n-crossed complexes.) When I have time (???) I will overhaul [[simplicial T-complex]] and [[group T-complex]] to fit in with the algebraic Kan complex idea, and to give lots of example.

   I had skimmed Thomas’ thesis and noted its high standard, but had not read it in detail.

   I know the corresponding structures in simplicial groups. As you say Thomas freely adds in fillers, my thought is that somehow those are the free objects (I am deliberately thinking algebraically rather than homotopically here). The natural algebraic question is to look at the quotients that satisfy interesting families of equations. (In the simplicial group(oid) case, the equations are more or less explicit in order to get the nerves of n-crossed complexes.)

   When I have time (???) I will overhaul simplicial T-complex and group T-complex to fit in with the algebraic Kan complex idea, and to give lots of example.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 16th 2010
   • (edited Nov 16th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> As you say Thomas freely adds in fillers, my thought is that somehow those are the free objects (I am deliberately thinking algebraically rather than homotopically here). That's correct, in a precise sense: algebraic Kan complexes are algebras over a free-forgetful monad $U F$ on $sSet$ and so for $X \in sSet$ any simplicial set the algebraic Kan complex $F X$ is indeed the _free $U F$-algebra_ on $X$, yes.

   As you say Thomas freely adds in fillers, my thought is that somehow those are the free objects (I am deliberately thinking algebraically rather than homotopically here).

   That’s correct, in a precise sense: algebraic Kan complexes are algebras over a free-forgetful monad $U F$ on $sSet$ and so for $X \in sSet$ any simplicial set the algebraic Kan complex $F X$ is indeed the free $U F$-algebra on $X$, yes.

5. • CommentRowNumber5.
   • CommentAuthorTim_Porter
   • CommentTimeFeb 17th 2019
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   Author: Tim_Porter
   Format: MarkdownItexI have added a bit more to [[simplicial T-complex]], giving some idea as how the nerve of a crossed complex gives an example of such.

   I have added a bit more to simplicial T-complex, giving some idea as how the nerve of a crossed complex gives an example of such.