# A family of groups as a monoidal category

1.Context
My lecture notes present the following example of a monoidal category:

Let $$G:=(G_n)_{n\in \mathbb {N_0}}$$ be a family of groups with $$G_0$$ the trivial group with one element. We define a category $$C_G$$ with:

• $$Obj(C_G)= \mathbb {N_0}$$
• $$Hom(m,n)= \begin{cases} G_m & m=n \\ \emptyset & m \neq n \\ \end{cases}$$
• Composition of two morphisms $$f,h: m \rightarrow m$$ is given by the (associative) multiplication in the respective group $$G_m$$. The identity morphism for each $$m\in \mathbb {N_0}$$ is given by the neutral element of $$G_m$$.

Now suppose that for any $$m,n,k \in \mathbb {N_0}$$ we have a group homomorphism $$\rho_{m,n}: G_m \times G_n \rightarrow G_{m+n}$$ such that $$\rho_{m+n,k} \circ (\rho_{m,n} \times id_{G_k})= \rho_{m,n+k} \circ (id_{G_m} \times \rho_{n,k}).$$ Then we can equip the category $$C_G$$ with a strict monoidal structure in the following (natural) way:
Define a functor $$\otimes: C_G \times C_G \rightarrow C_G$$ with $$m \otimes n := m + n$$ and $$f \otimes h := \rho_{m,n}(f,h)$$ for $$f \in G_m, h \in G_n$$. The monoidal unit is given by $$0 \in \mathbb N_0$$.

The essential property of this category seems to be, that it allows two types of operations (on certain families of groups): The standard multiplication within a group ("vertical composition") and a operation between elements of different groups ("horizontal composition").

By the way, unless I am overlooking something, a similar construction to the one above should work for certain families of associative monoids as well, i.e. we didn't use the inverse of a group element in the construction.

2. Questions

• Does the above category (its construction) have a name?
• My lecture notes mention the category in passing. It seems like a cute example of a monoidal category, that illustrates the versatility of the definition. Is there more to it? Is the category, in particular considered as a monoidal category, used anywhere? That is, are there any results where it is of interest?
• What (interesting or enlightening) examples of the above category exist?
An example presented in my lecture notes is $$G:=(B_n)_{n\in \mathbb {N_0}}$$ where $$B_n$$ denotes the braid group with $$n$$ strands. The group homomorphism $$\rho_{m,n}$$ is given by $$\rho_{m,n} (\sigma_i, \sigma_j)= \sigma_i \sigma_{m+j}$$ for each $$m,n \in \mathbb {N_0}$$, i.e. by the juxtaposition of braids. Here, $$\sigma_i$$ denotes one of the respective canonical generators of the braid group. Obviously, a similiar construction works for the family of symmetric groups $$G:=(S_n)_{n\in \mathbb {N_0}}$$.
The family $$G:=(C_n)_{n\in \mathbb {N_0}}$$ of cyclic groups (or dihedral groups, for that matter) does not seem to work. I wouldn't know what the group homomorphisms $$\rho_{m,n}$$ should be. (One might modify the above construction so that $$\rho_{m,n}: G_m \times G_n \rightarrow G_{lcm(m,n)}$$. However, I would like to stick to above construction).
• Although it is not the same thing as what you define, your understanding of this question might be improved by reading what an "operad" is. – Fosco Aug 11 at 15:59

This is not a full answer, but too long for a comment.

I don't know if that category has a name.

A place where it's used is the following : take $$G_n = GL_n(F)$$, for a field $$F$$. The morphisms $$GL_n(F)\times GL_m(F)\to GL_{n+m}(F)$$ are block sums; that is, if you have an automorphism $$f:F^n\to F^n$$, and $$g:F^m\to F^m$$, then you get $$f\oplus g : F^{n+m}\to F^{n+m}$$.

The category you get is often denoted $$\coprod_n BGL_n(F)$$ (because the one-object category associated to a group $$G$$ can be denote $$BG$$), and if you take its geometric realization, you get a nice space (the space associated to $$BGL_n(F)$$ has $$\pi_1 = GL_n(F)$$ and not other homotopy group).

This space is very closely related to the algebraic $$K$$-theory of $$F$$, and actually the monoidal structure on your category is a way to define an additive structure on the $$K$$-theory at the level of spaces.

(there are variants when you replace $$F$$ with an arbitrary ring)

The case of the family of symmetric groups $$(S_n)$$ is related to stable homotopy theory (specifically to the sphere spectrum and therefore the stable homotopy groups of spheres), and allegedly to the $$K$$-theory of "the field with one element"

• I don't know anything about algebraic $K$-theory, the geometric realization of a category, nor about homotopy groups. Thanks anyway, guess I will try to learn at least the basics. – M.C. Aug 12 at 13:05

Not a full answer either but here is what I can make out of it:

I don't know if this construction has a name, neither if it is used in any example, but I can provide you with some intuition about that:

First we need to start with the "delooping" of a monoid: this is kind of standard in category theory. The general takeaway is that monoids really are the same as categories with a single object. I won't go too much in details about this as it is really the same as what you have presented just the notations: given a monoid $$M$$, you can define the category $$\mathbf{B}M$$, which has a single object $$\ast$$ and the morphisms are given by $$\mathbf{B}M(\ast,\ast) = M$$. The composition of morphisms is given by the composition in $$M$$. In case you have not encountered this already, you can prove that $$\mathbf{B}$$ is an equivalence of categories between the category of monoids and the category of categories with a single object, and this has super slick generalizations to higher categories, but that's kind of a (very interesting) rabbit hole (see https://ncatlab.org/nlab/show/k-tuply+monoidal+n-category if you want more!).

Now what you define (without considering the monoidal at first) is like the delooping of monoid, except that you do it for a family of monoids. Luckily, these deloopings don't talk to each other (you have $$\operatorname{Hom}(m,n)=\emptyset$$ is $$m\neq n$$). So we can express it using the sum of the delooping. The sum of two categories $$C$$ and $$D$$, that I denote $$C+D$$, is the category whose objects is the disjoint union of the objects of $$C$$ and the object of $$D$$, and the morphisms are given by the moprhisms in $$C$$ and the morphisms in $$D$$, and there is no extra morphisms in between pairs of objects that come from a mix of $$C$$ and $$D$$. It is the categorical sum inside the category of categories, and you can picture it as haveing $$C$$ and $$D$$ side by side, but not interacting. Now, given a family of monoids $$M_0,M_1,M_2,\ldots$$, you can define the category you consider as the infinite sum $$C_M = {\large+}_{i=0}^\infty \mathbf{B}M_i$$. (Actually you would need to define this $$\large{+}$$ operation, but that's similar to the case where there are only 2 monoids).

Another way of picturing this is that $$C_M$$ is that it is a category that has an countable number of connected components, such that the connected components are exactly given by the list $$M_0,M_1,M_2,\ldots$$ An extra image you can adopt is that a category is kind of a big collection of monoids that interact with each other is a nice way, but here you just have monoids that forgot to interact with each other. I am giving as much images as I can, as maybe some of them will be more intuitive, but up until now I have only said one thing.

Now we can take a look at the additional condition, that there is a family of homomorphisms satisfying the identities you have mentioned. Then you can construct a monoidal structure on $$C_M$$, as you mentioned. But you can alo go the other way and check that a monoidal structure on $$C_M$$ is exactly the same as a family of homomorphisms satisfying your condition gives back a family of homomorphisms. Simply define $$\rho_{m,n}(f,g) = f\otimes g$$, and this family satisfies the equations you asked for. So in the end, a family of morphisms such as the one you asked for really is a synonymous to a monoidal structure on $$C_M$$. Now my opinion about this is that being a monoidal structure on $$C_M$$ is a much simpler and natural condition, that tells you everything about a family monoids equipped with homomorphisms that interact in a nice way in one go. So I would argue that this is the more primitive notion and that, if you ever encounter such families of morphisms, you should really understand them as a monoidal structure on a category $$C_M$$.

That being said, I don't know of any example of such structure, so I trust the example to be relevant!