Timeline for Maps between operads of different arities
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 20, 2012 at 20:30 | comment | added | David White | @Peter May: Thanks for your comment. I agree that I was too hasty in my choice of language. Also, I very much like your second comment as a way of understanding why the definition "morphism of operads" is what it is | |
| Dec 20, 2012 at 17:35 | comment | added | Poisson | @Peter May: Thanks, that's a good comment. My construction is fundamentally nonlinear so perhaps I should not expect it to behave well. | |
| Dec 20, 2012 at 17:00 | comment | added | Tom Leinster | David, the category of operads isn't a subcategory of the category of collections (at least, not in the obvious way), since one collection can carry many different operad structures. | |
| Dec 20, 2012 at 16:56 | answer | added | Tom Leinster | timeline score: 5 | |
| Dec 20, 2012 at 15:59 | comment | added | Peter May | Mathematically, the question implicitly highlights one designed in limitation to the generality of the kinds of algebraic structure that operads define: they are not designed (or rather designed not) to deal with identities with repeated variables, such as x^2 = 1 or even x x^{-1} = 1. There are several reasons for that choice. | |
| Dec 20, 2012 at 15:56 | comment | added | Peter May |
David, while your comments are otherwise reasonable, the idea that there is an approach that deserves to be called the right one'' to operads is anathema to me. It is like saying, for example, that groupoids give the right approach'' to groups. For some purposes yes, in general, ex-cathedra, obviously no. Operads, like groups, give a fundamental algebraic structure that appears in a variety of contexts and admits a variety of generalizations and elaborations.
|
|
| Dec 20, 2012 at 15:46 | comment | added | David White | Here's another reason why we shouldn't allow such maps. An operad is a special case of a colored operad, which is the same thing as a multicategory. Maps of colored operads are functors of multicategories. They are defined in Section 1 of "Localization of Algebras over Coloured Operads" by Casacuberta et al. In that definition, the maps are levelwise, and the authors remark that this matches with the notion of a functor of multicategories. Many who study operads think this approach via enriched categories and multicategories is the right one, so "morphism of operads" should match it | |
| Dec 20, 2012 at 15:35 | comment | added | David White | Well, if you think of the category of operads as a subcategory of the category of Collections then the answer is no. If $C$ is your base category then the category of Collections is $\prod_n{C^{\Sigma_n}}$ and the morphisms in this category are levelwise. There are good reasons to think of Operads inside $Coll(C)$. In particular, it gives you the notion of $\Sigma$-cofibrant operad (i.e. cofibrant in the projective model structure on $Coll(C)$) which is very useful for doing homotopy theory | |
| Dec 20, 2012 at 15:18 | history | asked | Poisson | CC BY-SA 3.0 |