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created – for our derived seminar – the entry homotopy T-algebra with the main result by Badzioch and interlinked it with model structure on simplicial T-algebras and (infinity,1)-algebraic theory
How do the definitions under homotopy T-algebra fit with the definitions from Leinster http://arxiv.org/abs/math/0002180 ?
Does Tom check how his notion compares with the ordinary notion (by cofibrant resolution of operads)?
In my memory (I travel to Wien the day after tomorrow so I can not now reread details), Tom was not doing comparison with any model theoretic nonsense. He is considering something like colax monoidal functors from free monoidal category on the original operad to what he calls "cartesian monoidal category with equivalences". So the colaxness does all the homotopy thing if I understood the idea right. In any case, Leinster's approach covers the standard examples like A infinity, L infinity and Gerstenhaber infinity algebras; the domain of his functors (representing infinity algebras) is in rather algebraic setup, and all the information about weak equivalences is in the domain monoidal category.
Added: The question if the homotopy algebras over original operad are the usual algebras about some kind of some resolution is a question of representability which is not needed for his general concept, I think.
Zoran,
I think the relation of homotopy T-algebras to algebras over $(\infty,1)$-operads is clear:
by Bandzioch, Bergner and Lurie we have homotopy T-algebras model the algebras over $T$ regarded as an $\infty$-algebraic theory. These $\infty$-algebraic theories must have the same kind of relation to $\infty$-operads as ordinary algebraic theories have to ordinary operads. So in as far as an $\infty$-operad corresponds to an $\infty$-algebraic theory, its $\infty$-category of algebras should be equivalent to that of the $\infty$-algebraic theory.
So Tom’s notion should be equivalent to this to the extent that it is equivalent to algebras over $\infty$-operads. Which we know are modeled by ordinary algbras over cofibrant topological/simplicial operads.
Urs, you assume that every case which Tom covers is representable in a sense. I am not sure, because the Lurie and others consider operads in more specific context. Leinster takes any monoidal category with a choice of class of morphisms called equivalences satifsying some axioms. he does not take operads in dg-category or in stable infinity category or in similar more specific context. So it may be that it is not quite the same level of generality, but one should check.
Leinster takes any monoidal category with a choice of class of morphisms called equivalences satifsying some axioms.
Similarly, in
Berger, Moerdijk,
The Boardman-Vogt resolution of operads in monoidal model categories (arXiv)
Axiomatic homotopy theory for operads (pdf)
operads over any symmetric monoidal model category with suitable interval object are considered.
We happen to know for the case that the enriching category is $sSet_{Quillen}$ how this relates to $\infty$-operads, hence to $\infty$-algebraic theories, hence to homotopy $T$-algebras.
For other enriching categories less is known about the general abstract context. But since you asked about the relation of Tom Leinster’s construction to homotopy $T$-algebras: if one knew how his construction relates to operads enriched in $sSet_{Quillen}$, we knew how it relates to homotopy $T$-algebas.
So, the question on enrichment in sSet is more or less if there is nice "simplicial nerve" of his "monoidal categories with equivalences", something like simplicial localization. This is an interesting question.
Yes, for instance consider the example of enrichment in chain complexes: if we take non-negatively graded chain complexes, then we know that their simplicial nerve, yes, injects them into $sSet_{Quillen}$ and we may think this way of a $Ch_\bullet^+$-enriched operad as a certain $(\infty,1)$-operad.
Another interesting choice to consider is enriching in $sSet_{Joyal}$. If that goes through, it would be a natural candidate for a model for $(\infty,2)$-operads.
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