What non-monoidal functors on monoidal categories are used "in nature"?

Question

Background

For my PhD dissertation, I've developed a categorical generalization of many different systems of denotational semantics for light linear logic (LLL). I'd like to see if I can use this generalization to find a more "natural" (in the colloquial sense of the word) denotational semantics for LLL. At its core is a symmetric monoidal closed category with two functors on it. One of the functors is monoidal, and the other is not (well, it could be monoidal, but then it's a trivial example). There are some other requirements, but for the moment, I'm mostly curious about how common it is to have non-monoidal functors in the first place.

Question

If you know of an example where someone uses a non-monoidal functor $T$ on a symmetric monoidal closed category $\mathbb{C}$, I'd like to hear about it. If you know of such a $T$ with natural transformations $d_A:TA \to TA\otimes TA$ and $e_A:TA \to 1$ forming comonoids for every object $A$, even better. If the category $\mathbb{C}$ also comes with a monoidal functor $S$, that would be even more fantastic. And if there's a natural transformation $T\Rightarrow S$, then I'll buy you dinner.

I've got examples (fibered phase spaces, stratified coherent spaces and locally bounded stratified cliques, games and discreet strategies, light length spaces), but they're all specifically created for this purpose, and I'm curious to see just how natural this kind of construction is.

Answer 1

As in my comment above, I'm not sure precisely what's being asked, so the following might or might not be a useful answer. In any case, it doesn't get me dinner.

Let $\mathbf{D}$ be the category of finite totally ordered sets, which is monoidal under disjoint union. It doesn't matter whether you take "all" finite totally ordered sets or just a skeleton, but the important thing is that the empty set is included — so $\mathbf{D}$ is not $\Delta$.

Small theorem: colax monoidal functors $\mathbf{D} \to \mathbf{Set}$ (yes, covariant) are the same thing as simplicial sets.

Generally, for any category $\mathcal{E}$ with finite products, colax monoidal functors $\mathbf{D} \to \mathcal{E}$ amount to simplicial objects in $\mathcal{E}$. (Proof: Proposition 3.1.7 of this, where I'm afraid $\mathbf{D}$ is called $\Delta$ and $\Delta$ is called $\Delta^+$.)

For a functor $T$ to be colax monoidal means that it comes equipped with maps $$ T(A \otimes B) \to TA \otimes TB, \qquad T(I) \to I $$ satisfying coherence axioms. In this case, they're not invertible unless the corresponding simplicial set is the nerve of a monoid. So, whether by "monoidal" you meant "lax monoidal" or "strong monoidal", the functors $\mathbf{D} \to \mathbf{Set}$ corresponding to simplicial sets are not usually monoidal.

Edit I see that Erik wanted examples of non-monoidal functors on symmetric monoidal categories. The monoidal category $\mathbf{D}$ isn't symmetric, so my example won't do. But there's something analogous in the symmetric world, concerning not simplicial sets but the $\Gamma$-sets of Segal.

Let $\mathbf{F}$ be the category of finite sets (including $\emptyset$), which is symmetric monoidal under disjoint union. Then a symmetric colax monoidal functor $\mathbf{F} \to \mathbf{Set}$ turns out to be the same thing as a $\Gamma$-set. Again, you can replace $\mathbf{Set}$ by any other category with finite products, and again these functors are not in general monoidal.

Answer 2

It seems like many of the standard examples of monads in functional programming can be transported to linear logic to produce examples of non-monoidal functors.

E.g., the linear state monad $T_S(A) = S \multimap S \otimes A$ has two evident natural transformations $T_S(A) \otimes T_S(B) \to T_S(A \otimes B)$ (corresponding to evaluating the left or the right argument first), but neither one will satisfy the coherence properties needed to be a monoidal functor. Likewise, the linear exception monad $E(A) = 1 \oplus A$ doesn't even have a natural transformation of the right type, and is not even strong.

Is there some extra condition you want? Perhaps if you could say something about the operational intuition I could be more helpful (your $T$ looks like the restricted exponential of light logic, and I guess $S$ is the "paragraph" modality?)

Answer 3

Consider the category of algebras over a fixed field k. Taking the dual as a k-vectorspace gives a (contravariant) monoidal functor that can be your S. On the other hand, you can also consider the ''finite dual'', usually denoted as ^\circ. As the finite dual turns every algebra into a coalgebra, this can act as your functor T. Moreover, the finite dual is a subspace of the dual vectorspace, this gives you your natural transformation T\to S. As you want to obtain cocommutative coalgebras, you can restrict to commutative algebras in the first place. The problem in this example is to get the functors as endofunctors on a suitable (sub)category. I don't know how essential this point is for you. If you start with (all) algebras, or (all) commutative algebras as above, then taking the linear dual or finite dual gets you to vectorspaces or coalgebras. The finite dual is however an endofunctor on the category of Hopfalgebras, so here you are out of problems. For the linear dual, I would suggest to replace it by the ''restricted dual'' on the category of multiplier Hopf algebras, and I think on this category you will have your example.