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Stream: community: our work

Topic: Claudio Pisani

Claudio Pisani (Sep 19 2024 at 20:02):

I have just put on arXiv a new paper on unbiased multicategories.

Let me highlight the main points.

1. I define "double props" $\mathbb P$ as double categories with (finite) sums such that $\mathbb P_0 = {\bf Set_f}$ (or $\mathbb P_0 = {\bf Set_f}/S$ for "colored double props").

Many classical props arise naturally as the loose part $\mathbb P_l$ of a double prop $\mathbb P$ (the tensor of arrows is the sum in $\mathbb P_1$).

2. A double prop $\mathbb P$ can be seen as a theory in two essentially equivalent ways:

   a) a model for $\mathbb P$ is a product preserving lax functor ${\mathbb P}^{\rm op} \to {\rm \mathbb{S}pan} ({\bf Set})$ (as in this paper).
   b) a model for $\mathbb P$ is a sum preserving discrete fibration ${\mathbb M}\to {\mathbb P}$.

If $\mathbb P = {\mathbb P}{\rm b}$ (the double category of pullback squares in ${\bf Set_f}$), the models are unbiased symmetric multicategories.
Thus in general we call "$\mathbb P$-multicategories" the models of $\mathbb P$.

A $\mathbb P$-multicategory $M:{\mathbb M}\to {\mathbb P}$ is "representable" (respectively, "discrete" or a "$\mathbb P$-monoid") if its loose part $M_l$ is a stable opfibration (respectively, a discrete opfibration).

Furthermore, $\mathbb M$ is itself a double prop and we can consider its own models.
In particular, if $M:\mathbb M \to {\mathbb P}{\rm b}$ is a symmetric multicategory, the $\mathbb M$-monoids correspond to the usual algebras for the multicategory.

3. Important instances of symmetric multicategories are
   a) ${\mathbb B}{\rm ij} \to {\mathbb P}{\rm b}$ (the terminal unary multicategory).
   b) ${\mathbb T}{\rm ot} \to {\mathbb P}{\rm b}$ (the "associative operad").
   ${\mathbb B}{\rm ij}$-multicategories are unary multicategories, while ${\mathbb T}{\rm ot}$-multicategories are plain multicategories.

Other important instances are given in the paper.

4. If we consider sum preserving discrete fibrations of double graphs ${\mathbb G}\to {\mathbb P}{\rm b}$, we get Joyal species, thus enlightening the idea of multicategories as multiplicative species.

5. We define unbiased cartesian multicategories as the algebras for a monad $(-)^{\rm cart}$ on unbiased symmetric multicategories. The loose arrows of $M^{\rm cart}$ are "spans" with a loose leg and a tight leg. One then can prove, in the fibrational framework, the usual theorem on cartesian multicategories: they are representable if and only if they have universal products, if and only if they have algebraic products.

Morgan Rogers (he/him) (Sep 24 2024 at 16:30):

Possibly lazy question: what is a biased (vs. unbiased) multicategory?

Kevin Carlson (Sep 24 2024 at 17:12):

The ordinary kind is biased in this context, I take it because a multi-morphism $(a_1,\ldots,a_n)\to b$ in the usual formalism comes equipped with the bias of a total ordering on its domains. Thus a symmetric multi-category becomes a planar multicategory equipped with the extra structure of group actions imposing irrelevance of that choice of ordering. I've always thought that's rather ugly, so I was really pleased to see this presentation of how to define a symmetric multicategory to have morphisms $(a_s)_{s\in S}$ with domains indexed by a set, thus not equipped with a distinguished order.

Jean-Baptiste Vienney (Sep 24 2024 at 17:15):

I think it should be indexed by a multiset rather than a set, no?

Mike Shulman (Sep 24 2024 at 17:20):

The domain itself could be said to be a "multiset" (although that word has so many different possible meanings that I usually avoid it), but it's indexed by a set.

Jean-Baptiste Vienney (Sep 24 2024 at 17:27):

Oh ok, thanks.

Morgan Rogers (he/him) (Sep 25 2024 at 05:21):

That use of "unbiased" aligns very poorly with its use for monoids...

Mike Shulman (Sep 25 2024 at 16:58):

Yes, I would say that both kinds of multicategory are "unbiased", since they have multimorphisms of all arity. I think Leinster called the arbitrary-finite-sets version a "fat symmetric multicategory".

Morgan Rogers (he/him) (Sep 25 2024 at 17:01):

The representable ones would be the commutative monoidal categories (ones where different permutations of censored objects are equal on the nose) so I would have expected these to be called commutative rather than unbiased

Kevin Carlson (Sep 25 2024 at 17:04):

Why do you say that? That's not the impression I have; I think they're supposed to be equivalent to symmetric multicategories which ought to include what the representables are.

Kevin Carlson (Sep 25 2024 at 17:06):

(Yes, the paper does confirm that representable "unbiased" symmetric multicategories are symmetric monoidal categories, not commutative ones.)

Kevin Carlson (Sep 25 2024 at 17:20):

Mike Shulman said:

Yes, I would say that both kinds of multicategory are "unbiased", since they have multimorphisms of all arity. I think Leinster called the arbitrary-finite-sets version a "fat symmetric multicategory".

Both kinds of multicategory are unbiased in the sense of arity, i.e. not biased toward binary multimorphisms. This is good since the conceptual content of associativity is that you can multiply a sequence of $n$ things together indifferently. It seems like the conceptual content of symmetry is exactly that you can multiply a set of $n$ things together indifferently, so the generalized use of "unbiased" feels pretty sensible to me. Certainly I don't much like "fat" for at least two independent reasons.

Morgan Rogers (he/him) (Sep 25 2024 at 19:49):

Ah right you did mention that the group action was still there without the ordering, hence symmetric rather than commutative.
But the notion of "unbiased" for monoids is not indifferent to the ordering of the arguments. Associativity doesn't imply commutativity :face_with_raised_eyebrow: