# Is there a monoidal analogue of equalizers?

There are three different kinds of finite limits in categories: terminal objects, binary products, and equalizers. In a category $$C$$, these define functors $$1_{C},\times,\mathrm{Eq}\colon\mathrm{Fun}(I,C)\to C$$ where $$I=\emptyset,\{\bullet\ \ \bullet\}$$, and $$\{\bullet\rightrightarrows\bullet\}$$ respectively.

Monoidal categories generalise the first two in that we now have a functor $$1_{C}$$ from $$\mathcal{C}^{\emptyset}=*$$ to $$\mathcal{C}$$ and a functor $$\otimes_C$$ from $$C^{\{\bullet\ \ \bullet\}}=C$$ to $$C$$, i.e.\ functors \begin{align*} 1_C &\colon * \to \mathcal{C}\\ \otimes_C &\colon \mathcal{C}\times\mathcal{C} \to \mathcal{C} \end{align*} together with associativity and unitality natural isomorphisms satisfying compatibility conditions.

What about equalizers? Has the notion of a category $$C$$ equipped with a functor $$\rm{Eq}\colon\rm{Fun}(\{\bullet\rightrightarrows\bullet\},C)\to C$$, a unit functor, and unitality/associativity natural isomorphisms satisfying coherence conditions been studied before? Moreover, are there any examples of such structures "found in nature"?

• How do you formulate associativity for this kind of structure? I think the concept of a 2-monad is relevant here - basically you see monoidal categories as algebras for the 2-monad you have sketched, and want to change that 2-monad so that the "cartesian algebras" are not categories with finite products, but rather categories with binary equalizers (maybe more?). But right now your description has no evident 2-monad structure, at least I don't see it. May 20, 2021 at 10:36
• In my brain, the vague idea is that product maps have something to do with the fact the three-element set is a pushout of a span of two-element sets. One can imagine analogues of this for the walking parallel pair of arrows, for instance to get a parallel triple, or to get a three-object version, with two sets of parallel pairs that are composable. Just something to consider! May 20, 2021 at 10:48

Here is a way to write down what a symmetric monoidal category is starting just from the idea of finite (co)products. We have $$SymMonCat = Fun^\times_{(2,1)}(Span^{fin}_{(2,1)}, Cat_{(2,1)})$$ That is, a symmetric monoidal category is equivalent to the data of a finite-product-preserving $$(2,1)$$-functor from the $$(2,1)$$-category $$Span^{fin}_{(2,1)}$$ of finite sets and spans between them, to the $$(2,1)$$-category $$Cat_{(2,1)}$$ of categories.

I can think of at least two ways to understand the significance of $$Span^{fin}_{(2,1)}$$ from the point of view of finite (co)products.

1. There is a functor $$Span^{fin}_{(2,1)} \to Cat^{\amalg}_{(2,1)}$$ (where $$Cat^{\amalg}_{(2,1)}$$ is the $$(2,1)$$-category of categories with finite coproducts and finite-coproduct preserving functors). The functor sends $$n \mapsto Set^{fin}/n$$. This functor is fully faithful. Moreover, $$Set^{fin}/n$$ is itself the free coproduct completion of the discrete category $$n$$.

So from this perspective, we have boiled it all down to finite (co)products, but the picture is a bit odd and I'm not sure how to complete the thought from this perspective. Alternatively,

1. Note that the $$(2,2)$$-category $$Span^{fin}_{(2,2)}$$ of finite sets, spans between them, and maps of spans has the following universal property. For any 2-category $$\mathcal K$$ with finite products, we have that the 2-category $$Fun_{(2,2)}^\times(Span^{fin}_{(2,2)}, \mathcal K)$$ of all finite-product preserving 2-functors, is equivalent to $$FinProd(\mathcal K)$$, the 2-category of objects of $$\mathcal K$$ which internally have finite products.

(2) suggests the following generalization. Let $$J$$ be a set of small categories, and let $$2Cat(J)$$ be the category of 2-categories with $$J$$-limits. I believe that the functor $$Comp_J: 2Cat(J) \to 2Cat$$ carrying $$\mathcal K$$ to the 2-category of objects in $$\mathcal K$$ which internally have $$J$$-limits, is corepresentable by some $$\mathcal J \in 2Cat(J)$$. Define a $$J$$-symmetric monoidal category to be an object of $$Fun^J_{(2,1)}(\mathcal J_{(2,1)}, Cat)$$, where $$\mathcal J_{(2,1)}$$ is the maximal sub $$(2,1)$$-category of the $$(2,2)$$-category $$\mathcal J$$, and $$Fun^J_{(2,1)}$$ means we take $$(2,1)$$-functors which preserve $$J$$-limits in the $$(2,1)$$-categorical sense.

Then in the case where $$J$$ consists of the finite discrete categories, a $$J$$-symmetric monoidal category is a usual symmetric monoidal category. When $$J$$ consists of just the empty category, a $$J$$-symmetric monoidal category is an $$E_0$$-category. You could try out other classes of $$J$$, such as equalizers, but I'm not sure what you get!

• There's an issue with this answer: in order to say that $K \in \mathcal K$ internally has, say, equalizers, you need $\mathcal K$ to have certain $Cat$-cotensors -- a 2-categorical limit rather than a 1-categorical one. After modifying the above story to account for this, the analogy with symmetric monoidal categories is no longer perfect. Dec 27, 2021 at 19:07