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1. • CommentRowNumber1.
   • CommentAuthorFosco
   • CommentTimeJun 11th 2014
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   Author: Fosco
   Format: MarkdownItexI began adding proofs of Lemma 1-4 to the page [[transfinite construction of free algebras]]. The layout of the two array environment has to be fixed; proof of 3-4 to be added. Any help/suggestion is extremely appreciated!

   I began adding proofs of Lemma 1-4 to the page transfinite construction of free algebras. The layout of the two array environment has to be fixed; proof of 3-4 to be added.

   Any help/suggestion is extremely appreciated!

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 11th 2014
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   Author: Dmitri Pavlov
   Format: TextNice work! Do you know if similar constructions also hold for _pushouts_ of free algebras? Specifically, what I would like to know is whether one can use Kelly's results to obtain the following statement about pushouts of algebras over operads: If C is a locally presentable closed symmetric monoidal category and O is an operad in C, then for any morphism f: X→Y in C and any cocartesian square of the form O∘X → A   ↓      ↓ O∘Y → B where O∘− is the free O-algebra functor, the map A→B is the transfinite composition of maps A=A_0→A_1→⋯→A_∞=B, where the map A_{t−1}→A_t fits into a cocartesian square O_A[t] ⊗_{Σ_t} X_t → A_{t−1}             ↓                       ↓ O_A[t] ⊗_{Σ_t} Y_t → A_t, where X_t→Y_t is the t-fold pushout product of f with itself, which can be computed by considering the cubical diagram {0,1}^t→C induced by f and taking the canonical morphism from the colimit of the punctured cube to the terminal vertex 1^t (whose value is simply Y^{⊗t}) and the symmetric sequence O_A is defined by the universal property O_A∘Z=A⊔(O∘Z). At least formally this seems to be similar to the explicit description following the proof of Theorem 2, so I wonder if Kelly's arguments could be made directly applicable in this case.

   Nice work! Do you know if similar constructions also hold for _pushouts_ of free algebras?
   Specifically, what I would like to know is whether one can use Kelly's results to obtain the following statement about pushouts of algebras over operads:
   If C is a locally presentable closed symmetric monoidal category and O is an operad in C,
   then for any morphism f: X→Y in C and any cocartesian square of the form
   
   O∘X → A
     ↓      ↓
   O∘Y → B
   
   where O∘− is the free O-algebra functor,
   the map A→B is the transfinite composition of maps A=A_0→A_1→⋯→A_∞=B,
   where the map A_{t−1}→A_t fits into a cocartesian square
   
   O_A[t] ⊗_{Σ_t} X_t → A_{t−1}
               ↓                       ↓
   O_A[t] ⊗_{Σ_t} Y_t → A_t,
   
   where X_t→Y_t is the t-fold pushout product of f with itself,
   which can be computed by considering the cubical diagram {0,1}^t→C induced by f
   and taking the canonical morphism from the colimit of the punctured cube to the terminal vertex 1^t (whose value is simply Y^{⊗t})
   and the symmetric sequence O_A is defined by the universal property O_A∘Z=A⊔(O∘Z).
   
   At least formally this seems to be similar to the explicit description following the proof of Theorem 2,
   so I wonder if Kelly's arguments could be made directly applicable in this case.
3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJun 12th 2014
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   Author: Mike Shulman
   Format: MarkdownItexWell, Theorem 2 is about algebras for a pointed endofunctor rather than for a monad. In the monad case, you'd need to $\beta$-reduce the proof of Theorem 3.

   Well, Theorem 2 is about algebras for a pointed endofunctor rather than for a monad. In the monad case, you’d need to $\beta$-reduce the proof of Theorem 3.

4. • CommentRowNumber4.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 12th 2014
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   Author: Dmitri Pavlov
   Format: TextOf course. But only the proof of Theorem 2 is currently expanded, so it was the only possible reference.

   Of course. But only the proof of Theorem 2 is currently expanded, so it was the only possible reference.
5. • CommentRowNumber5.
   • CommentAuthorFosco
   • CommentTimeJun 12th 2014
   • (edited Jun 13th 2014)
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   Author: Fosco
   Format: MarkdownItexDmitri: I don't know the answer but I'm interested. I began interesting myself in the topic because of this question: Take a (well-)pointed endofunctor $F\colon \mathcal{C}\to \mathcal{C}$ and consider for any object the $\omega$-chain $$ X\overset{\eta}\to F X\overset{F\eta}\to F F X\overset{F F\eta}\to\dots $$ its colimit should define a new pointed (because of the UMP of colimit) $X\mapsto T_F(X)$, which I expect to have some "nice" property. Initially I thought it was sort of the free monad attached to $F$; but a rapid [computation](http://math.stackexchange.com/questions/815462/colimit-of-the-sequence-x-to-fx-to-ffx-to-dots) gives a transformation in the wrong direction, $T_F\to T_F T_F$. Maybe one is able to obtain $T_{T_F}\to T_F$? If yes, in which way? Before adding this new question I decided to go deeper in the study of Kelly's "And so on" paper, and here am I! :)

   Dmitri: I don’t know the answer but I’m interested.

   I began interesting myself in the topic because of this question:

   Take a (well-)pointed endofunctor $F\colon \mathcal{C}\to \mathcal{C}$ and consider for any object the $\omega$-chain

   $$X\overset{\eta}\to F X\overset{F\eta}\to F F X\overset{F F\eta}\to\dots$$

   its colimit should define a new pointed (because of the UMP of colimit) $X\mapsto T_F(X)$, which I expect to have some “nice” property. Initially I thought it was sort of the free monad attached to $F$; but a rapid computation gives a transformation in the wrong direction, $T_F\to T_F T_F$.

   Maybe one is able to obtain $T_{T_F}\to T_F$? If yes, in which way? Before adding this new question I decided to go deeper in the study of Kelly’s “And so on” paper, and here am I! :)

6. • CommentRowNumber6.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 22nd 2014
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   Author: Dmitri Pavlov
   Format: TextLemma 4 seems to have several problems in the formulation. For example, G goes from B to A, yet the statement says that GA is an S'-algebra, where S' is an endofunctor on B.

   Lemma 4 seems to have several problems in the formulation.
   For example, G goes from B to A, yet the statement says that GA is an S'-algebra, where S' is an endofunctor on B.