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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 12th 2010
• (edited Nov 12th 2010)

There is a student here who is thinking about how to relate $\infty$-algebraic theories with $\infty$-operads and in the course of that also dendroidal sets with Lurie-type “$\infty$-categories of operations” of $\infty$-operads.

I am trying to help a bit where I can. First of all I thought I’d need to get a better idea of how the triangle

$\array{ && monads \\ \\operads &&&& algebraic theories }$

works in 1-category theory. So I am naturally looking at Mike’s Generalized Virtuology to get some hints.

I had planned to typeup the little that I understand about the relation between Lawvere theories and generalized multicategories, but now I ended up spending some time just on the entry on virtual double categories. Here is what I did

• created a subsection “Monads on virtual double categories” with the basic definitions

• and further subsections on “Monoids and modules” (this existed as an empty stub before)

• and “Generalized multicategories” (with the basic definition, then pointing over to generalized multicategory of course).

I also

I took the liberty of mentioning the term “fc-multicategory” at the beginning of virtual double category (because that happens to remind me easier of what the term refers to) and at virtual equipment I said that this term is short for “proarrow equipment of a virtual double category”.

(Hm, that summary of what i did is almost longer than the little bit of text that I acutally added! :-)

Mike, here is a question:

when I read your article with Cruttwell, I have slight trouble when it comes to definition 8.2 of normalized monoids. It has a horizontal morphism labeled $U_A$ where definition 4.2 has an equality. It seems. Is there a typo in one of these or did I miss some further definition in the intermediate four sections (not unlikely)?

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeNov 12th 2010

Thank you! Geoff and I have been meaning to improve those pages for a while, but haven’t had time yet.

Definition 8.2 only makes sense in a virtual equipment, or at least a virtual double category with units (defined in section 5). When you have units, there is a natural bijection (essentially by definition of “units”) between cells of nullary source starting at an object A, and cells of unary source whose source is a unit horizontal arrow $U_A$. Definition 4.2 uses the former, since at that point we are only in a virtual double category, while 8.2 refers (implicitly) to the equivalent cell of the latter form, since only a cell of unary source can be said to be cartesian. Does that clarify?

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeFeb 6th 2014

I added a reference to a recent paper of Hyland to generalized multicategory.