Example of non-symmetric biclosed monoidal category

Question

Does there exists any example of non-symmetric biclosed monoidal category ? By biclosed, I mean right-closed and left-closed.

Answer 1

The categories of $(R,R)$-bimodules over a ring $R$ give many examples. There is in general no bimodule isomorphism whatsoever between the bimodule tensor products $A\otimes_R B$ and $B\otimes_R A$. However, this tensor product is biclosed, with the right exponential $C^B$ given by the $(R,R)$-bimodule of right $R$-module homomorphisms $B\to C$, while the left exponential ${}^AC$ has for elements the left $R$-module homomorphisms.

This example is most naturally seen as the endomorphism category of $R$ in the bicategory $\mathbf{Bimod}$ of rings, bimodules, and bimodule homomorphisms, which is locally biclosed in that all composition functors are two-variable left adjoints. Of course this can all be generalized to the bicategory of $A$-algebras and their bimodules where $A$ is any commutative ring.

More significantly, we can view $\mathbf{Bimod}$ as an example of a bicategory $\mathcal{V}-\mathbf{Prof}$ of categories enriched over a suitable closed monoidal category $\mathcal V$ and profunctors between them. The case above is for $\mathcal V$ being abelian groups; we could also consider ordinary categories, getting for instance a biclosed product on functors $C\times C^{\mathrm{op}}\to \mathbf{Set}$ by looking again at endomorphism categories.

There are more examples of this kind of "closed bicategory", in which the endomorphism categories will usually give asymmetric biclosed monoidal categories; Wood in his abstract proarrows series suggests toposes with left exact functors, although I don't know exactly how one constructs the exponentials.

Answer 2

Such examples are aplenty.

(1) Take any braided closed monoidal category, which is not symmetric. For instance, you can take modules over a generic quasitriangular Hopf algebra.

(2) Take any coboundary closed monoidal category, which is not symmetric. For instance, the category of finite crystals over a simple Lie algebra will do.

(3) There are a plenty of random examples as well. The easiest one is $G$-graded vector spaces over a non-abelian group $G$.

Answer 3

An important example is the lax Gray tensor product $\otimes^{\mathsf{lax}}_{\mathsf{Gray}}$. It equips $\mathsf{2Cats}$ with the structure of a non-symmetric monoidal biclosed category having

• Left internal homs $[\mathcal{C},\mathcal{D}]^{\mathrm{L}}_{\mathsf{2Cats}}$ given by the $2$-category $\mathsf{2Fun}^{\mathsf{lax}}(\mathcal{C},\mathcal{D})$ of $2$-functors from $\mathcal{C}$ to $\mathcal{D}$, lax transformations between them, and modifications between these;
• Right internal homs $[\mathcal{C},\mathcal{D}]^{\mathrm{R}}_{\mathsf{2Cats}}$ given by the $2$-category $\mathsf{2Fun}^{\mathsf{oplax}}(\mathcal{C},\mathcal{D})$ of $2$-functors from $\mathcal{C}$ to $\mathcal{D}$, oplax transformations between them, and modifications between these.

In other words, the lax Gray tensor product gives us isomorphisms of $2$-categories $$ \begin{align*} \mathsf{2Fun}(\mathcal{C}\otimes^{\mathsf{lax}}_{\mathsf{Gray}}\mathcal{D},\mathcal{E}) &\cong \mathsf{2Fun}(\mathcal{D},\mathsf{2Fun}^{\mathsf{lax}}(\mathcal{C},\mathcal{E})),\\ \mathsf{2Fun}(\mathcal{C}\otimes^{\mathsf{lax}}_{\mathsf{Gray}}\mathcal{D},\mathcal{E}) &\cong \mathsf{2Fun}(\mathcal{C},\mathsf{2Fun}^{\mathsf{oplax}}(\mathcal{D},\mathcal{E})). \end{align*} $$

This was discussed on the $n$-Category Café a while ago.