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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeSep 15th 2010
   • (edited Sep 15th 2010)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI am still not happy with my rudimentary understanding of the [[characteristic class]]es of homotopy algebras, e.g. [[A-infinity algebra]]s as presented by Hamilton and Lazarev. Kontsevich had shown how to introduce [[graph complex]]es in that setup, almost 20 years ago, but in his application to Rozansky-Witten theory he has shown the relationship to the usual Gel'fand-Fuks cohomology and usual characteristic classes of foliations. On the other hand all the similar applications are now systematized in the kind of theory Lazarev-Hamilton present. Their construction however does not seem to directly overalp but is only analogous to the usual charactersitic classes. These two points of view I can not reconcile. So I started a stub for the new entry [[Feynman transform]]. The Feynman trasnform is an operation on twisted modular operads which is Feynman graph expansion-motivated construction at the level of operads and unifies variants of graph complexes which are natural recipients of various characteristic classes of homotopy algebras.

   I am still not happy with my rudimentary understanding of the characteristic classes of homotopy algebras, e.g. A-infinity algebras as presented by Hamilton and Lazarev. Kontsevich had shown how to introduce graph complexes in that setup, almost 20 years ago, but in his application to Rozansky-Witten theory he has shown the relationship to the usual Gel’fand-Fuks cohomology and usual characteristic classes of foliations. On the other hand all the similar applications are now systematized in the kind of theory Lazarev-Hamilton present. Their construction however does not seem to directly overalp but is only analogous to the usual charactersitic classes. These two points of view I can not reconcile. So I started a stub for the new entry Feynman transform. The Feynman trasnform is an operation on twisted modular operads which is Feynman graph expansion-motivated construction at the level of operads and unifies variants of graph complexes which are natural recipients of various characteristic classes of homotopy algebras.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeSep 15th 2010
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   Author: zskoda
   Format: MarkdownItexAdded [[cyclic operad]].

   Added cyclic operad.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 15th 2010
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   Author: Todd_Trimble
   Format: MarkdownItexGood to see you added [[cyclic operad]]. However, my impression is that the foundational theory for cyclic operads needs to be seriously examined by category theorists. The article by Getzler and Kapranov is far from optimal in this regard. Particularly, it should be examined from the POV of differential calculus of species. I hope to have more to say about this in the not-too-distant future.

   Good to see you added cyclic operad. However, my impression is that the foundational theory for cyclic operads needs to be seriously examined by category theorists. The article by Getzler and Kapranov is far from optimal in this regard.

   Particularly, it should be examined from the POV of differential calculus of species. I hope to have more to say about this in the not-too-distant future.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeSep 15th 2010
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   Author: zskoda
   Format: MarkdownItexGo on :)

   Go on :)

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 15th 2010
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   Author: Todd_Trimble
   Format: MarkdownItexI'll only say a little now, merely to whet the appetite. :-) There are various ways of thinking about operads; one is by defining a suitable monoidal product on the category of species (modules of the permutation groupoid $\mathbf{P}$), called substitution product, so that operads are identified with monoids in this monoidal category. However, for purposes of discussing fundamental constructions like the bar construction for operads, this is IMO not the most suitable formulation. For the bar construction, it is convenient to discuss a different lax monoidal product which I call "graft product". In the differential calculus of species, where $F'$ denotes the species derivative, the graft product of species $F, G$ may be defined by the formula $$F # G \coloneqq F' \otimes G$$ and it has a non-invertible associator $F # (G # H) \to (F # G) # H$. The structure of a (non-unital) operad $T$ may then be specified by a multiplication $$m: T # T \to T$$ or $m: T' \otimes T \to T$. The operad axioms imply that the derivative $T'$ is an algebra (with respect to the monoidal product $\otimes$), and that the structure $m$ makes $T$ a module over the algebra $T'$. There is a final operad axiom, making operads the same thing as special sorts of modules over their derivatives. A cyclic module $X$ is a $\mathbf{P}$-module equipped with a cyclic structure, which can be understood as a second $\mathbf{P}$-module $Y$ together with a module isomorphism $\phi: X \to Y'$. (Of course, as in calculus, such an antiderivative $Y$ is determined up to a constant. We can choose $Y$ to that $Y[0] = 0$, in other words so that the object of 0-ary operations = constants of $Y$ is trivial.) A structure of cyclic operad can then be understood as being given by a map $Y' \otimes Y' \to Y$, a kind of bilinear pairing on the derivative. By taking the derivative of this map, we get $$Y'' \otimes Y' + Y' \otimes Y'' \to Y'$$ This breaks into two pieces, one for each of the two summands. The first is a map $$Y'' \otimes Y' \to Y'$$ or in other words $m: X' \otimes X \to X$ (recalling that $X$ is the derivative of $Y$). This map is the structure of the ordinary operad that underlies the cyclic operad, according to the alternative formulation of operads given above. The development in Getzler and Kapranov is founded on tree combinatorics for free operads and free cyclic operads, but I think there is some foundational work to be done in understanding these structures in terms of species calculus, a categorification of ordinary calculus. So far I am only scratching the surface.

   I’ll only say a little now, merely to whet the appetite. :-) There are various ways of thinking about operads; one is by defining a suitable monoidal product on the category of species (modules of the permutation groupoid $\mathbf{P}$), called substitution product, so that operads are identified with monoids in this monoidal category.

   However, for purposes of discussing fundamental constructions like the bar construction for operads, this is IMO not the most suitable formulation. For the bar construction, it is convenient to discuss a different lax monoidal product which I call “graft product”. In the differential calculus of species, where $F'$ denotes the species derivative, the graft product of species $F, G$ may be defined by the formula

   $$F # G \coloneqq F' \otimes G$$

   and it has a non-invertible associator $F # (G # H) \to (F # G) # H$. The structure of a (non-unital) operad $T$ may then be specified by a multiplication

   $$m: T # T \to T$$

   or $m: T' \otimes T \to T$. The operad axioms imply that the derivative $T'$ is an algebra (with respect to the monoidal product $\otimes$), and that the structure $m$ makes $T$ a module over the algebra $T'$. There is a final operad axiom, making operads the same thing as special sorts of modules over their derivatives.

   A cyclic module $X$ is a $\mathbf{P}$-module equipped with a cyclic structure, which can be understood as a second $\mathbf{P}$-module $Y$ together with a module isomorphism $\phi: X \to Y'$. (Of course, as in calculus, such an antiderivative $Y$ is determined up to a constant. We can choose $Y$ to that $Y[0] = 0$, in other words so that the object of 0-ary operations = constants of $Y$ is trivial.)

   A structure of cyclic operad can then be understood as being given by a map $Y' \otimes Y' \to Y$, a kind of bilinear pairing on the derivative. By taking the derivative of this map, we get

   $$Y'' \otimes Y' + Y' \otimes Y'' \to Y'$$

   This breaks into two pieces, one for each of the two summands. The first is a map

   $$Y'' \otimes Y' \to Y'$$

   or in other words $m: X' \otimes X \to X$ (recalling that $X$ is the derivative of $Y$). This map is the structure of the ordinary operad that underlies the cyclic operad, according to the alternative formulation of operads given above.

   The development in Getzler and Kapranov is founded on tree combinatorics for free operads and free cyclic operads, but I think there is some foundational work to be done in understanding these structures in terms of species calculus, a categorification of ordinary calculus. So far I am only scratching the surface.

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeSep 16th 2010
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   Author: zskoda
   Format: MarkdownItexIndeed very intringuing :)

   Indeed very intringuing :)

7. • CommentRowNumber7.
   • CommentAuthorGuest
   • CommentTimeSep 29th 2010
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   Author: Guest
   Format: Textwould this paper http://arXiv.org/abs/0912.1243 help for the present discussion?

   would this paper http://arXiv.org/abs/0912.1243 help for the present discussion?
8. • CommentRowNumber8.
   • CommentAuthorzskoda
   • CommentTimeSep 29th 2010
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   Author: zskoda
   Format: MarkdownItexThank you for the reference which is indeed relevant and very interesting, though not exactly in the language closest to most of the people in this forum. I will list in the nlab.

   Thank you for the reference which is indeed relevant and very interesting, though not exactly in the language closest to most of the people in this forum. I will list in the nlab.

9. • CommentRowNumber9.
   • CommentAuthorTobyBartels
   • CommentTimeSep 30th 2010
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   Author: TobyBartels
   Format: MarkdownItexHandy link: <http://arXiv.org/abs/0912.1243> (Tip for our welcome Guest: Pick ‘Markdown+Itex’ to format comments, then write `<http://arXiv.org/abs/0912.1243>` to make the handy link.)

   Handy link: http://arXiv.org/abs/0912.1243

   (Tip for our welcome Guest: Pick ‘Markdown+Itex’ to format comments, then write <http://arXiv.org/abs/0912.1243> to make the handy link.)