nLab universal algebra in a monoidal category

Initial Purpose of Page

The ideas on this page are things that I, Andrew Stacey, want to use. I suspect that they are well-known and well-developed in the literature but have no idea how to track them down. So I’m expounding on them here in the hope that some kindly soul will fill in the gaps for me.

So should we merge this with PRO now? —Toby

Yes. However, not knowing what is standard and what isn’t, I’m not sure I could do that merge. The stuff on this page is what I want to know about so I don’t want to lose any of it in the merge. —Andrew

Understood. I'm not sure how to do it either, but I'll think about it. —Toby

I should say that if any of this doesn’t fit in the merge then I’m happy for it to find a home elsewhere. My main purpose in scribbling this lot is to find out what is and isn’t known so I don’t have to reprove the known stuff just to make use of it. —Andrew

Now I want to wait until Todd figures out what you really want to do here. (^_^) —Toby

The Basic Idea

The idea is that one tries to “do” universal algebra in a monoidal category by using the monoidal products in place of the standard cartesian ones. The extent to which this is possible is closely related to the extent to which the monoidal product resembles the cartesian product of sets.

In More Detail

Let us consider only finitary theories; that is, with Lawvere theories. In categorical language, recall that a Lawvere theory is a category $T$ with finite cartesian products in which every object is isomorphic to a finite cartesian power of a distinguished object. A model for that theory in some category $C$ is then a product-preserving functor $T \to C$. In a monoidal category, we can ask for this to be a monoidal functor.

The problem with this naive definition is that as the product in $T$ is the cartesian product, it comes along with a whole heap of extra structure that may not be strictly needed. We therefore need to weaken the definition of a Lawvere theory.

Definition

A monoidal Lawvere theory is a monoidal category $(T, \otimes, I)$ in which every object is isomorphic to a finite monoidal product of some distinguished object.

It will be important for later to note what is missing between a monoidal Lawvere theory and an ordinary one. The missing components are:

1. No projection morphisms; e.g. $x^n \to x$
2. No diagonal morphisms; e.g. $x \to x^n$
3. No symmetry morphisms; e.g. $x^2 \to x^2$

With this definition, we can define a model of such a theory in a monoidal category.

Definition

Let $(C, \otimes, I)$ be a monoidal category. Let $(T, \otimes, I)$ be a monoidal Lawvere theory. A monoidal $T$-object in $(C,\otimes, I)$ is a monoidal functor $T \to C$.

The Trade

Any Lawvere theory is automatically a monoidal Lawvere theory and so one can consider monoidal models of an ordinary Lawvere theory in a monoidal category. However, as ordinary Lawvere theories have all the extra structure of projections, diagonals, and symmetry, the underlying object (i.e. the image of the distinguished object in the Lawvere theory) needs to have them as well. Indeed, if one were looking for candidates for such models one would be well-advised to look in the subcategories of objects which have such structure. Let us therefore give these names.

Definition

Let $(C, \otimes, I)$ be a monoidal category. We define the following full subcategories of $C$.

1. $C^{\operatorname{sym}}$ is the largest symmetric monoidal subcategory. That is, it is the full subcategory where the switcheroo functor is naturally isomorphic to the identity.
2. $C^{\operatorname{co}}$ is the subcategory of commutative, associative, unital comonoids. Note that we take morphisms in $C$, not morphisms of comonoids. These are the objects that come equiped with diagonal and projection morphisms.

Todd: Sorry, I don’t know what you mean by the switcheroo functor for a monoidal category.

There is what is called the center of a monoidal category, which is a braided monoidal category. The center of the center should be symmetric monoidal (although I’d have to think this through again carefully). Comonoids in the center or center of center form a monoidal category, and I’m thinking something like this would be a natural habitat to consider.

Andrew: Knowing that none of my terminology would be right, I let myself get carried away there. Apologies. I mean the functor that swaps the order of the product, namely that takes $A \otimes B$ to $B \otimes A$. (Actually, now I come to think about it I’m not convinced that such must exist in an arbitrary monoidal category.) What I mean is that if I have a monoidal category then even if it isn’t symmetric, there may still be an object that is itself symmetric. I guess I don’t need the category of such to be symmetric monoidal and I don’t need it to commute with products of other objects, just with itself. So I could still have a commutative monoid in an arbitrary monoidal category.

Or maybe I can’t. Now I write it, it seems a lot more tenuous than when I thought it. It’d be useful to get that clear.

Todd: Not sure I can add anything else useful right now. I don’t think there’s a general switcheroo functor. In some sense a monoidal functor $Fin^{op} \to M$ to a monoidal category $M$ ought to simulate a cocommutative comonoid in $M$, but I’m not sure how useful an observation this is for you.

Is there a specific setting you have in mind where what you want to do is unproblematic?

Andrew: Yes. I want to do it in the category of models in set of a commutative algebraic theory. Over there, it doesn’t say anything about the monoidal structure being symmetric but I have an inkling that it is. However, not every object is a coalgebra.

(As a little background: I’m returning to some stuff that I thought about a while ago and am trying to finish off, but I don’t remember all of what I thought when I originally thought about it which is why I’m not as clear as I would like to be)

Toby: Even in a symmetric monoidal category, I don't know how to define a functor that takes $A \otimes B$ to $B \otimes A$. Of course, there is a functor $C \times C \to C$ that takes $(A,B)$ to $B \otimes A$, but that doesn't help as far as I can see.

There's an inherent problem with considering the largest (or any) symmetric subcategory of a given monoidal category, which is that being symmetric is not a property but a structure on a monoidal category. So it's more like looking for the largest commutative submonoid of a given set than looking for the largest commutative submonoid of a given monoid.

Even if that worked, I wouldn't understand the full subcategory of CAU comonoids, since again being a CAU comonoid is not a property but a structure on an object in a given symmetric monoidal category. This is like asking for the full subcategory of $Set$ consisting of groups; it doesn't parse. At best, I would understand this as the full subcategory of sets that have at least one group structure, but you're no longer keeping track of that structure (and in fact, at least assuming the axiom of choice, this is just $Set \setminus \{\empty\}$, which has nothing much to do with groups at all).

Andrew: Okay, let me be absolutely specific here. I was trying to get at the general story first, but that’s the wrong way round for here (it’s the right way round for the paper, but that’s once I’ve understood what the general story is!).

This is tied up with Tall-Wraith monoids. If you start with a set and take the free ring on that set, then that’s again a biring (co-ring object in rings). Now start with abelian groups instead. If you take the free ring on an abelian group, then that’s not a biring, but still has a certain amount of co-structure because a ring is a monoid in $(Ab, \otimes, \mathbb{Z})$. To make it a biring, I need to start with a coalgebra - that’s enough to supply the missing pieces of the co-structure.

• Toby: OK, this makes sense. In general, you start with a comonoid in your monoidal category; a comonoid in $(Ab,\otimes)$ is a coalgebra, while a comonoid in $(Set,\times)$ is simply a set (by an argument that applies whenever the monoidal category is cartesian).

Furthermore, if I wish to interpret the resulting biring as representing a functor from abelian groups to rings, then I can only do so on the subcategory of coalgebras - but I need to take abelian group homs not coalgebra homs.

• Toby: H'm, this is odd. I still don't know exactly what the ‘subcategory’ of coalgebras is, since being a coalgebra is not a property of an abelian group. Since the forgetful functor from coalgebras to abelian groups is faithful, you can make it a literal subcategory up to equivalence, then consider the corresponding full subcategory; or (equivalently) just take the category whose objects are coalgebras and whose morphisms are homomorphisms of the underlying abelian groups. (In general, this concept is the full image of a functor; here we are taking the full image of the forgetful functor from coalgebras to abelian groups.)

  But I don't understand why you need to use this instead of simply the category of coalgebras. After all, if you have a functor from the full image, then you certainly have a functor from the category of coalgebras by composing with the inclusion functor of coalgebras into the full image. On the other hand, it seems very strange that you can define this functor on arbitrary abelian group homomorphisms. Applying this to abelian group isomorphisms, you are basically saying that, if you start with two different coalgebra structures on a given abelian group, the resulting birings will be isomorphic. Is that true?

  Andrew — Okay, I still wasn’t being completely clear. I do mean the category of coalgebras. But the functor is constructed by taking the composition of the functor from coalgebras to abelian groups and then taking the hom-functor in abelian groups with a biring. So if $C$ is a coalgebra and $B$ is a biring then I want to consider $Hom_{Ab}(C,B)$. That’s again a ring and it’s functorial for coalgebra morphisms.

  Phew! “Oh what a tangeled web we weave, when first we practise to understand category theory.”

  Toby: OK, so when you wrote ‘I need to take abelian group homs not coalgebra homs’, you weren't still specifying the source of the functor but were instead specifying how that biring represents the functor. Now it makes sense!

This all generalises quite nicely: replace abelian groups by a commutative algebraic theory and replace rings (rather, replace monoids) by a PRO. Then the PRO-objects in the commutative algebraic theory will again be an algebraic theory and the free functor from the commutative theory to the other one has lots of nice properties.

But this felt like it was a specialisation of an even more general story. My mistake was to try to guess the general story first rather than tell you the special case.

(Not that I regard this particularly as a mistake! It was enough to learn about PROs and it’s all part of learning how to ask good questions.)

Todd: Okay, so to pick up the thread again, here is a preliminary observation. Suppose we have an operad $P$ (giving rise to a prop $Pr(P)$) and a symmetric monoidal closed category $K$ (such as the category of abelian groups, or the category of algebras of a commutative monad on $Set$), then a $P$-algebra is essentially the same as a symmetric monoidal functor $A: Pr(P) \to K$, and we can define a $P$-coalgebra to be a symmetric monoidal functor $C: Pr(P)^{op} \to K$. Let $A_1$ and $C_1$ denote the underlying $K$-objects (= values of $A$ and $C$ at tensor-generating object 1 of $Pr(P)$), and let $K(-, -)$ denote internal hom in $K$. Then we can define a $P$-algebra structure with underlying object $K(C_1, A_1)$ whose operations

$$\widehat{\theta}: K(C_1, A_1)^{\otimes n} \to K(C_1, A_1)$$

are obtained as composites

$$K(C_1, A_1)^{\otimes n} \to K(C_{1}^{\otimes n}, A_{1}^{\otimes n}) \overset{K(C(\theta), A(\theta))}{\to} K(C_1, A_1)$$

where the first map is a canonical map available in any symmetric monoidal closed category. If we denote this $P$-algebra by $K(C, A)$, then this gives us an actegory action

$$K(-, -): Coalg_P^{op} \times Alg_P \to Alg_P$$

and in this framework one could go on to consider $P$-bialgebras, etc. I think this is what you were getting at or had observed yourself; Barr touches upon this sort of thing in the paper I mentioned over at CommCoalg or whatever we’re calling it now. I don’t know how well it generalizes to general props or pros.

Thus to be able to consider arbitrary Lawvere theories we need to be in the intersection of $C^{\operatorname{sym}}$ and $C^{\operatorname{co}}$. However, there are many Lawvere theories that do not need such structure. Accordingly we can interpose a finer classification on Lawvere theories according to whether or not they need the symmetric or comonoid structure.

If one is given a presentation of a Lawvere theory in terms of operations and identities, it is quite simple to identify what level of structure it requires. The key is to look at the identities and express them in terms of elements (this, of course, violates the principle of equivalence). An identity consists of two ways of applying operations starting with an ordered list of distinct elements.

1. If an identity involves changing the order of the elements, it needs the symmetric structure.
2. If an identity involves duplicating an element, it needs the diagonal structure.
3. If an identity involves ignoring an element, it needs the projection structure.

A Worked Example

The passage from the theory of unital, associative monoids to that of abelian groups is a good illustration of all of this. The Lawvere theory of unital, associative monoids has a presentation with 2 operations: one zeroary, $1$, and one binary, $\mu$, together with the identities:

1. Associativity
   $$\mu(a,\mu(b,c)) = \mu(\mu(a,b),c)$$
2. Unit
   $$\mu(a,1) = a, \qquad \mu(1,a) = a$$

None of these identities involve swaps, repetitions, or omissions and so this presentation is also of a monoidal Lawvere theory (note that $1$ is an operation not an element).

Now let us add in an inverse. That is, a unary operation $\iota$ with the identities:

1. Inverses
   $$\mu(\iota(a),a) = 1, \qquad = \mu(a, \iota(a)) = 1$$

These involve both a duplication and an omission. The duplication is on the left-hand side in each case, the omission on the right. Note that the starting point for each is the ordered list $(a)$. Therefore a group object in a monoidal category must lie in the subcategory of comonoids.

Let us forget the inverse for a moment and add in the commutativity identity instead. This is the identity:

1. Commutativity
   $$\mu(a,b) = \mu(b,a)$$

This obviously involves a swap, and thus requires a symmetric object as its model.

Finally, for an abelian group object we need both: symmetry and comonoid.

Representing Functors

One of the applications of Lawvere theories is to the theory of representing objects. Under mild conditions, a model of a Lawvere theory is equivalent to a lift of a representable contravariant functor $C \to \operatorname{Set}$ to a functor $C \to \operatorname{Set}T$, where $\operatorname{Set}T$ is the category of models of $T$ in $\operatorname{Set}$. However, except for very simple theories (i.e. those with no binary or higher operations), this does not carry over to the monoidal case because the construction of the operations on the Hom sets involves extensive use of diagonal morphisms.

However, where it does work is when one restricts to the subcategory of comonoid objects with comonoid-structure preserving morphisms. That is, if $C^{\operatorname{co}}_{c}$ denotes the subcategory of $C$ of comonoids with such morphisms then a model $G$ for $T$ in $(C,\otimes,I)$ represents a functor $C^{\operatorname{co}}_{c} \to \operatorname{Set}T$ via

$$C^{\operatorname{co}}_c \to C \overset{Hom(-,G)}{\to} \operatorname{Set}T$$

Questions

Apart from the obvious question of finding the literature behind all of this (and thus getting the terminology correct), certain other questions spring to mind. Well, okay, one question springs to mind.

1. Suppose that I have two monoidal Lawvere theories, $T_1$ and $T_2$, that are equivalent as ordinary Lawvere theories. Are they equivalent as monoidal Lawvere theories? More concretely, suppose that I have two presentations for the same Lawvere theory in terms of operations and identities and both define monoidal Lawvere theories (so neither has identities involving repetition, omission, or exchanging¹), are the monoidal Lawvere theories equivalent?

Todd: My previous answer (which has been deleted) was based on a too-hasty reading of the question. I’ll try to set aside some time to find a better answer.

Todd: The time set aside turned out to be time last night when I had trouble falling asleep. I believe the answer is no. If for example $T_1$ is the pro for comonoids and $T_2$ is the initial pro (so equivalent to the natural numbers as discrete monoidal category), then the “Lawverifications” of those pros ought to be equivalent, both being equivalent to the initial Lawvere theory $FinSet^{op}$. At the semantic level, this boils down to the easy observation that in a cartesian monoidal category, there is a unique comonoid structure on every object.

References

S. Lack, Composing PROPS, Theory Appl. Categ. 13 (2004), No. 9, 147–163.

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1. I keep wanting to write “repetition, hesitation, or deviation”. ↩