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    • CommentRowNumber1.
    • CommentAuthorFosco
    • CommentTimeJun 11th 2014

    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!

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 11th 2014
    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.
    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJun 12th 2014

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

    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 12th 2014
    Of course. But only the proof of Theorem 2 is currently expanded, so it was the only possible reference.
    • CommentRowNumber5.
    • CommentAuthorFosco
    • CommentTimeJun 12th 2014
    • (edited Jun 13th 2014)

    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:𝒞𝒞F\colon \mathcal{C}\to \mathcal{C} and consider for any object the ω\omega-chain

    XηFXFηFFXFFη 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) XT F(X)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 FF; but a rapid computation gives a transformation in the wrong direction, T FT FT FT_F\to T_F T_F.

    Maybe one is able to obtain T T FT FT_{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! :)

    • CommentRowNumber6.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 22nd 2014
    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.