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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 22nd 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## Idea Analytic monads are [[monads]] on [[Set]] that correspond to [[operads]] in [[Set]]. ## Definition More precisely, an [[operad]] $O$ in [[Set]] induced a [[monad]] $T$ on [[Set]]: $$T(S)=\coprod_{n\ge0} O_n \times_{\Sigma_n} S^n.$$ Such a monad $T$ is equipped with a canonical weakly [[cartesian natural transformation]] to the [[moand]] $Sym$ arising from the [[commutative operad]]. ## Properties A theorem of Joyal \cite{Joyal} states that there is a [[monoidal equivalence]] between the [[monoidal category]] of [[endofunctors]] $Set\to Set$ that admits a weakly [[cartesian natural transformation]] to $Sym$ and the [[monoidal category]] of [[species]], i.e., symmetric sequences in [[Set]] with the [[substitution product]]. In particular, the category of analytic [[monads]] on [[Set]] is equivalent to the category of [[operads]] in [[Set]]. ## The colored case The correspondence carries over to [[colored operads]] (with a [[set]] of colors $C$) if we use the [[slice category]] $Set/C$ instead of [[Set]]. ## The nonsymmetric case A similar correspondence can be established for [[nonsymmetric case]], except that we must include the data of a transformation to $Sym$, which is no longer unique. ## The homotopical case The correspondence generalizes to [[(∞,1)-categories]], with some statements becoming more elegant. See Gepner–Haugseng–Kock \cite{GHK}. ## Related concepts * [[species]] * [[operad]] * [[monad]] ## References * [[André Joyal]], _Foncteurs analytiques et espèces de structures_, Combinatoire énumérative (Montréal/Québec, 1985), Lecture Notes in Mathematics 1234 (1986), 126-159. [doi](http://dx.doi.org/10.1007/bfb0072514). * [[Mark Weber]], _Generic morphisms, parametric representations and weakly Cartesian monads_, Theory Appl. Categ. 13 (2004), 191–234. * [[David Gepner]], [[Rune Haugseng]], [[Joachim Kock]], _∞-Operads as Analytic Monads_, [arXiv:1712.06469](https://arxiv.org/abs/1712.06469). <a href="https://ncatlab.org/nlab/revision/analytic+monad/1">v1</a>, <a href="https://ncatlab.org/nlab/show/analytic+monad">current</a>

   Created:

   Idea

   Analytic monads are monads on Set that correspond to operads in Set.

   Definition

   More precisely, an operad $O$ in Set induced a monad $T$ on Set:

   $$!A = \sum_{E: V} (E \to A}T(S)=\coprod_{n\ge0} O_n \times_{\Sigma_n} S^n.$$

   Such a monad $T$ is equipped with a canonical weakly cartesian natural transformation to the moand $Sym$ arising from the commutative operad.

   Properties

   A theorem of Joyal \cite{Joyal} states that there is a monoidal equivalence between the monoidal category of endofunctors $Set\to Set$ that admits a weakly cartesian natural transformation to $Sym$ and the monoidal category of species, i.e., symmetric sequences in Set with the substitution product.

   In particular, the category of analytic monads on Set is equivalent to the category of operads in Set.

   The colored case

   The correspondence carries over to colored operads (with a set of colors $C$) if we use the slice category $Set/C$ instead of Set.

   The nonsymmetric case

   A similar correspondence can be established for nonsymmetric case, except that we must include the data of a transformation to $Sym$, which is no longer unique.

   The homotopical case

   The correspondence generalizes to (∞,1)-categories, with some statements becoming more elegant. See Gepner–Haugseng–Kock \cite{GHK}.

   Related concepts

   • species

   • operad

   • monad

   References

   • André Joyal, Foncteurs analytiques et espèces de structures, Combinatoire énumérative (Montréal/Québec, 1985), Lecture Notes in Mathematics 1234 (1986), 126-159. doi.

   • Mark Weber, Generic morphisms, parametric representations and weakly Cartesian monads, Theory Appl. Categ. 13 (2004), 191–234.

   • David Gepner, Rune Haugseng, Joachim Kock, ∞-Operads as Analytic Monads, arXiv:1712.06469.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorvarkor
   • CommentTimeJun 22nd 2021
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   Author: varkor
   Format: MarkdownItex> A similar correspondence can be established for nonsymmetric case, except that we must include the data of a transformation to Sym, which is no longer unique. What is meant by this sentence, considering that there may be nonisomorphic nonsymmetric operads that induce isomorphic monads (as proved in Leinster's *Are Operads Algebraic Theories?*).

   A similar correspondence can be established for nonsymmetric case, except that we must include the data of a transformation to Sym, which is no longer unique.

   What is meant by this sentence, considering that there may be nonisomorphic nonsymmetric operads that induce isomorphic monads (as proved in Leinster’s Are Operads Algebraic Theories?).

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 23rd 2021
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   Author: Dmitri Pavlov
   Format: MarkdownItex> What is meant by this sentence, considering that there may be nonisomorphic nonsymmetric operads that induce isomorphic monads (as proved in Leinster’s Are Operads Algebraic Theories?). Exactly what is written: the date of a transformation to Sym must be included together with the monad, to get an equivalence of categories.

   What is meant by this sentence, considering that there may be nonisomorphic nonsymmetric operads that induce isomorphic monads (as proved in Leinster’s Are Operads Algebraic Theories?).

   Exactly what is written: the date of a transformation to Sym must be included together with the monad, to get an equivalence of categories.

4. • CommentRowNumber4.
   • CommentAuthorRuneHaugseng
   • CommentTimeJun 23rd 2021
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   Author: RuneHaugseng
   Format: TextIt's not the transformation to Sym that must be included (that is still unique), but the transformation to the monad for associative algebras - nonsymmetric operads are equivalent to symmetric operads over the associative operad, but a given symmetric operad can have distinct maps to it (which must be the case in Leinster's counterexample).

   It's not the transformation to Sym that must be included (that is still unique), but the transformation to the monad for associative algebras - nonsymmetric operads are equivalent to symmetric operads over the associative operad, but a given symmetric operad can have distinct maps to it (which must be the case in Leinster's counterexample).
5. • CommentRowNumber5.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 23rd 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexReplaced Sym by the monad of the associative operad in the nonsymmetric case. <a href="https://ncatlab.org/nlab/revision/diff/analytic+monad/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/analytic+monad/3">v3</a>, <a href="https://ncatlab.org/nlab/show/analytic+monad">current</a>

   Replaced Sym by the monad of the associative operad in the nonsymmetric case.

   diff, v3, current

6. • CommentRowNumber6.
   • CommentAuthorvarkor
   • CommentTimeJun 23rd 2021
   • (edited Jun 23rd 2021)
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   Author: varkor
   Format: MarkdownItexThanks for the clarification. Does "transformation" here refer to a monad morphism, so that the precise statement is $\mathrm{Operad} \simeq \mathrm{Mnd}_{\mathrm{analytic}}(\mathrm{Set})/\mathrm{Assoc}$?

   Thanks for the clarification. Does “transformation” here refer to a monad morphism, so that the precise statement is $\mathrm{Operad} \simeq \mathrm{Mnd}_{\mathrm{analytic}}(\mathrm{Set})/\mathrm{Assoc}$?

7. • CommentRowNumber7.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 23rd 2021
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   Author: Dmitri Pavlov
   Format: MarkdownItexRe #6: Operads are by definition symmetric (as defined originally by May), so the left side should say NonsymmetricOperad, not Operad. For the right side, the transformations must be cartesian, unlike the symmetric case, in which they are only weakly cartesian.

   Re #6: Operads are by definition symmetric (as defined originally by May), so the left side should say NonsymmetricOperad, not Operad.

   For the right side, the transformations must be cartesian, unlike the symmetric case, in which they are only weakly cartesian.

8. • CommentRowNumber8.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 23rd 2021
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   Author: Dmitri Pavlov
   Format: MarkdownItexAdded a reference to Leinster's book. <a href="https://ncatlab.org/nlab/revision/diff/analytic+monad/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/analytic+monad/4">v4</a>, <a href="https://ncatlab.org/nlab/show/analytic+monad">current</a>

   Added a reference to Leinster’s book.

   diff, v4, current

9. • CommentRowNumber9.
   • CommentAuthorSam Staton
   • CommentTimeJun 24th 2021
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   Author: Sam Staton
   Format: MarkdownItexStart to mention the kinds of theory that correspond to analytic monads. I hope I am not mixing up or confusing terminology here. <a href="https://ncatlab.org/nlab/revision/diff/analytic+monad/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/analytic+monad/5">v5</a>, <a href="https://ncatlab.org/nlab/show/analytic+monad">current</a>

   Start to mention the kinds of theory that correspond to analytic monads. I hope I am not mixing up or confusing terminology here.

   diff, v5, current