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Suppose that $X$ and $Y$ are two categories. Let $\operatorname{Funny}(X,Y)$ denote the category whose objects are functors $X\to Y$ and whose morphisms are unnatural transformations $F\to G$, where an unnatural transformation $F\to G$ is given by an $\operatorname{Ob}(X)$-indexed family of arrows $\gamma_x:F(x)\to G(x)$, and that's it.

It is easy to see that its left adjoint gives a symmetric monoidal closed structure with unit object the terminal category. We call this monoidal product the funny tensor product and denote it by $\Box$.

What I'd like to know is if the funny tensor product satisfies any algebraic "laws" with the ordinary cartesian product, by which I mean, if we have some kind of sequence of tensor products and cartesian products together with parentheses, for example, something like the sequence $$(((A\Box B)\times C)\Box D)\times E,$$ there is a way to either rearrange the parentheses or rewrite the expression with fewer or no parentheses.

For instance, we can think of distributivity as an algebraic "law" between two operations. We can see associativity as an algebraic "law" between an operation and itself. However, I have never seen a nontrivial "law" between two associative operations having the same unit. Is there an extended form of the Eckmann-Hilton argument which shows that no nontrivial relationship of this type can ever be satisfied unless the two operations are equal (given that they are both associative and share a unit)?

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Can you describe $\square$ explicitly? – Martin Brandenburg Jul 23 '12 at 9:19
Not completely explicitly. Given two finite directed graphs (by which I mean actual directed graphs, not all quivers) $X$ and $Y$, we define the tensor product of digraphs to be the digraph with $(X \otimes Y)_0 = X_0 × Y_0$ and $(X\otimes Y)_1 = X_0 \times Y_1\coprod X_1 \times Y_0$ where the sources are given by $s(x,g)=(x,s(g))$ and $s(f,y)=(s(f),y)$. Then in the category of free categories on finite directed graphs, we define the tensor product of $F(X)\Box F(Y)$ to be $F(X\otimes Y)$. Since free categories on finite directed graphs are dense in $\mathbf{Cat}$, we can extend it everywhere. – Harry Gindi Jul 23 '12 at 10:16
(In this setup, you have to define $\Box$ on arrows between on free categories on finite directed graphs, which amounts defining, for any maps $f:[1]\to F(X)$ and $g:[1] \to F(Y)$, a map $f\Box g$, but such maps classify arrows, which are specified uniquely by finite-length sequences of "composable" arrows $f_0,\dots,f_n$ and $g_0, \dots, g_m$ in the generating digraph. Then we send $[1]\Box x$ for $y\in \{0,1\}$ to the composite of $(f_0,y),\dots,(f_n,y)$ and $x \Box [1]$ for $x\in \{0,1\}$ to the composite of the $(x,g_0)\dots (x, g_m)$. However, these generate $[1]\Box[1]$, so we win.) – Harry Gindi Jul 23 '12 at 10:44
You also need that the digraphs have "no direct loops", which is a condition I forgot to impose. – Harry Gindi Jul 23 '12 at 11:31

In a recent article (arxiv:0909.4722v1) Mark Weber gives a nice description of □ in terms of a pushout of the span $A_0\times B\leftarrow A_0\times B_0\rightarrow A\times B_0$. This works as well for the presheaf category of graphs (directed multigraphs, if you like). This pushout description yields canonical functors $A□(B\times C)\rightarrow(A□B)\times C$ that are easily seen not to be isomorphisms (both categories or graphs have the same objects, but different morphisms). In fact, □ as tensor and $\times$ as "par" provide a "linearly distributive structure" on both, the category of small categories and the category of small graphs. The latter notion is due to Robin Cockett and Robert Seely and was initially called "weak distributivity" (JPAA 114, (1997) 133--173, plus corrected version on Robert Seely's web page).

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