Can operads (or category theoretic structures more generally) be compared?

Question

I was reading John Baez’s paper on operads and phylogenetics trees where he formalizes a Jukes–Cantor model of phylogenetics. Because biological questions receive different answers depending on the model used, I was wondering if the operad that represents that model could be compared to an operad representing a different model. i.e., with some distance metric or structural similarity, or perhaps by choosing an appropriate topology on them? So, for example, we might instead adopt a model with more parameters like the Kimura one to better treat the patterns in our data. And it would be interesting to quantify, conceiving of the first model as a predecessor and the second as a recently discovered successor, what was retained in the successor model.

I've been hoping to measure structural continuity across theory change (e.g., the preservation of Newtonian mechanics in a limit of GR, and the same for classical mechanics to quantum which can be measured by deformation theory). But I'm having trouble doing this with biological models, specifically phylogenetic ones formalized in a category theoretic way.

I've talked to Baez about this a little on twitter, so let me know if attaching that would help answer this.

Answer 1

Yes, operads can be compared.

There are lots of kinds of operad (enriched in various categories, symmetric or plain or defined with respect to a monad, one-colored or many-colored, and don't even get me started on models of $\infty$-operads...) and I haven't looked to see which flavor Baez is using.

But one thing they all have in common (and similar to most mathematical structures, for that matter) is that once you've fixed your notion of "operad", there is a natural notion of morphism of operads, so that operads form a category. An operad $O$ can be compared to an operad $P$ via a morphism $O \to P$.

More elaborate sorts of comparison are available in accordance with more elaborate types of operad. For example:

• If your operads are one-colored and enriched in topological spaces, say, then the set of morphisms $Hom(O,P)$ from $O$ to $P$ has the structure of a topological space.

• If your operads are many-colored, then $Hom(O,P)$ will actually itself have the structure of a category (so that many-colored operads form a 2-category).

And so forth.

Similar comments apply to other sorts of categorical structures. This is really one of the nice things about category theory -- categorical structures naturally organize themselves into categories, so you can re-use category-theoretic concepts at higher levels of abstraction. For example, there is a category of categories (with functors as morphisms), which is even a 2-category (with natural transformations as 2-morphisms).

Answer 2

Operads are the one-object version of a multicategory.

This perspective is expanded in the excellent book by Leinster "Higher operads, higher categories"

If you want to be not-so-profound but get the idea, a multicategory is like a category, but instead of having a single domain, morphisms have a tuple of objects for domain; so the generic morphism is a pair $(n,f) : \vec X\to Y$ where $\vec X = (X_1,\dots,X_n)$ is a tuple of objects of $\cal C$. Morphisms of this kind can be composed grafting the "root" of the tree-like shape of this morphism

You have identities, and this botanical composition is associative. Now try to unravel the definition of what such a structure is, when $|\cal C|=*$!

From this, you define a "morphism of operads" as a functor between these gadgets.