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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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
created vDbl
added the raw definition and the reference and virtual equipment.
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)?
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?
I added a reference to a recent paper of Hyland to generalized multicategory.
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