The n-Category Café

Products don’t distribute over reflexive coequalizers either, in $Top$. There’s a cute easy result that you can construct general coequalizers from binary coproducts and reflexive coequalizers (see here, just before the corollary), and if products distribute over coproducts (as they do in $Top$), then they distribute over reflexive coequalizers iff they distribute over general coequlizers.

But this reminds me that one way of dealing with John’s question is to invoke the crude monadicity theorem – which crucially requires in this case that finite products distribute over reflexive coequlizers. I think the main hypothesis to check for invoking crude monadicity in this case is the existence of a left adjoint to the forgetful functor, and I believe this is rather easy, but maybe I should write this down when I get a chance. Better yet, someone will supply a good reference.

I think multi-sorted operads (aka “colored operads” or “multicategories”) actually have been worked out in more detail than multi-sorted algebraic theories (aka “cartesian operads”).

Mike wrote:

I think multi-sorted operads (aka “colored operads” or “multicategories”) actually have been worked out in more detail than multi-sorted algebraic theories (aka “cartesian operads”).

Okay, I’m not fussy—I just want to get the job done.

Can you point me to a general theorem implying that given a set $\Lambda$ of ‘colors’ or ‘sorts’ and an $\Lambda$-colored topological operad $O$,

1) the forgetful functor from the category of $O$-algebras to $Top^\Lambda$ has a left adjoint, and

2) this forgetful functor is monadic?

(Here as mentioned before I’m perfectly happy to replace the old-fashioned category $Top$ with one of the standard ‘convenient categories of topological spaces’. Also, I need my operads to be symmetric.)

Since all I really care about is the special case where $\Lambda = \mathbb{N}$ and $O$ is the colored operad whose algebras are operads, I’d be equally happy to get straight to the point and find a reference proving this:

1) the forgetful functor from the category of topological operads to $Top^{\mathbb{N}}$ has a left adjoint, and

2) this forgetful functor is monadic.

And if necessary, instead of using collections (objects $O \in Top^{\mathbb{N}}$) I could probably get away with using symmetric collections (objects $O \in Top^{\mathbb{N}}$ where $O(n)$ has an action of $S_n$). So I’d also be content, in a grudging sort of way, to learn that someone has shown:

1) the forgetful functor from topological operads to symmetric collections has a left adjoint, and

2) this forgetful functor is monadic.

My collaborator just pointed out that this result is almost proved:

If $C$ is any symmetric monoidal category in which the tensor product preserves colimits on both sides, then the free-forgetful adjunction between the category of symmetric operads in $C$ and the category of symmetric collections in $C$ is monadic. Benoit Fresse proved this for what he calls connected operads (which are operads with no term of arity zero and such that the term of arity one is the unit of the monoidal category) in Theorem B.4.3 of Appendix B of his book Homotopy of Operads and Grothendieck-Teichm¼ller Groups. He also mentions at the beginning of Appendix B that the same is true for not necessarily connected operads.

This may be good enough. However, it would be a bit odd if nobody had ever really written up a proof of any of my desired results.

Okay, thanks for the additional information, John. There seem to be a lot of accounts of trees and free operads and so forth, but I’ve often been unhappy with the accounts I’ve seen, for one reason or another. A lot of the time it has to do with a somewhat cavalier attitude towards nullary operations, or with general hand-waviness, or something. I don’t wish to name names here; suffice it to say that I expect a good account from you!

Of course, you and Jim Dolan famously worked on typed operads in your development of opetopes, and so all the stuff we’re discussing (multisorted cartesian operads) should be old familiar ground, in some sense, even though you guys were dealing with the doctrine of symmetric monoidal categories rather than cartesian monoidal categories.

In case it’s useful to you, I’ve written down one possible account which proves the theorem I asserted earlier in this thread:

If $C$ is cocomplete, and has finite products that distribute over colimits, then for any $\Lambda$-sorted Lawvere theory $\Theta$, the forgetful functor $U: Mod_C(\Theta) \to C^\Lambda$ is monadic.

I chose to prove this by going through and checking the hypotheses of the crude monadicity theorem. This is not actually the route that I favor (you don’t have to deal with technical hypotheses of crude monadicity at all – one can check monadicity quite directly!), but I thought at least the buzzwords would be more familiar in such an approach, more so than in the other approach that I’d prefer.

Anyway, it’s all laid out here.

Todd wrote:

Anyway, it’s all laid out here.

Wow, great!

Would you prefer that I cite this, or include it as an appendix to our paper, authored by you?

The latter might be good, because there seems to be a kind of hole in the literature around this point, and not everyone is comfortable citing websites yet, so putting this result in a journal might be helpful. Of course you might be embarrassed to have a smallish result with a deliberately ‘crude’ proof appear under your name… you could obviously write something much grander… but what you did is really just the right level for the context of our paper, and we can add suitable disclaimers if it helps. (Also, I’d make some of the language a bit more formal and stiff, as befits a journal.)

There seem to be a lot of accounts of trees and free operads and so forth, but I’ve often been unhappy with the accounts I’ve seen, for one reason or another. A lot of the time it has to do with a somewhat cavalier attitude towards nullary operations, or with general hand-waviness, or something.

Yes, I’ve been feeling the same way as Nina and I try to put this paper together. I’d hoped that all the infrastructure had been adequately developed and we could just use it. But the foundations feel a bit shaky. Some results have unncessary restrictive assumptions on nullary and unary operations, and many authors use the concept of ‘tree’ without defining it, which seems fine until you notice that different authors are using different notions.

We want to describe free operads and coproducts of operads using trees. It’s possible to do this ‘directly’, by hand, but this becomes quite fussy. So it seems better to start by constructing these structures ‘abstractly’, applying general results on multi-sorted Lawvere theories to the theory of operads, and then prove that the structures thus constructed can be describe in terms of trees. And indeed, the really good concepts of ‘tree’ fall out as consequences of operad theory, as sketched here.

I don’t wish to name names here; suffice it to say that I expect a good account from you!

I doubt we’ll do things in an optimal way, since I don’t have the patience and it’s all a big digression from the original goal of the paper. In particular, our definition of ‘planar tree’ is a longish nuts-and-bolts affair, which we retroactively make elegant by showing how planar trees are precisely operations in the free operad on the terminal collection. I bet someone here could polish the initial definition to a fine sheen of perfection. But I don’t want to… I want the proofs to be actual proofs, but I don’t want this foundational material to utterly take over what started as a paper on phylogenetic trees!

Although I had slightly “dissed” the approach I wrote up (towards answering John’s query) as maybe needlessly complicated, since the monadicity can be established directly without verifying the hypotheses of the crude monadicity theorem, one bright side of this approach did occur to me a little while ago.

The point is that while it’s nice to know that $Op \to C^\mathbb{N}$ (for a cartesian closed cocomplete $C$) is monadic, this by itself doesn’t give you everything you’d like to know. It’s a notorious fact about monads $T$ and their algebras that even if your base category $B$ is cocomplete, the category of algebras $B^T$ might not be. Barr and Wells have a lot of material to address this very point.

For example, maybe you want to convince yourself that $Op$ has colimits. Even just coproducts: I’ve seen papers that go through some machinations just to prove that the category of $C$-valued operads has coproducts.

But happily, all this comes out of the wash when you take the “crude monadicity theorem” approach. Let me recall that theorem:

Crude monadicity theorem. A functor $U: A \to B$ is monadic if

• $U$ has a left adjoint $F$,
• $A$ has reflexive coequalizers,
• $U$ preserves reflexive coequalizers, and reflects isomorphisms.

Here’s the extra bonus:

Under the hypotheses of the crude monadicity theorem, if $B$ is cocomplete, then so is $A$.

Proof: Let $T$ be the monad $U F$; let $\eta: 1_B \to U F$ be the unit and $\varepsilon: F U \to 1_A$ be the counit of the adjunction. First, a family of free $T$-algebras $\{F(c_i)\}_{i \in I}$ has a coproduct: it’s just $F(\sum_i c_i)$ since the left adjoint $F$ preserves coproducts.

Next, under our hypotheses, the category of $T$-algebras $A$ has arbitrary coproducts, because the coproduct of a family $\{a_i\}_{i \in I}$ of algebras can be exhibited as a coequalizer of a reflexive fork diagram consisting of coproducts of free algebras:

$$\sum_i F U a_i \stackrel{\sum_i F \eta U a_i}{\to} \sum_i F U F U a_i \stackrel{\overset{\sum_i \varepsilon F U a_i}{\to}}{\underset{\sum_i F U \varepsilon a_i}{\to}} \sum_i F U a_i$$

Finally, coequalizers (not just reflexive coequalizers) exist in $A$, because $A$ has binary coproducts, and the coequalizer of a general pair of arrows $a' \stackrel{\overset{f}{\to}}{\underset{g}{\to}} a$ can be computed as the coequalizer of the reflexive fork

$$a \stackrel{incl}{\to} a + a' \stackrel{\overset{(1_a, f)}{\to}}{\underset{(1_a, g)}{\to}} a.$$