question about monoidal structure of a 2-category

Question

Consider an extension of the 1-category of vector spaces and linear maps down to a 2-category $\mathcal{C}$ whose objects are $k$-linear categories. What is the symmetric monoidal structure on the objects, and how does one interpret a morphism of unit $k$-linear categories as a vector space?

In other language, consider a d-dimensional once-extended TFT $Z:\text{Bord}_d^2\rightarrow\mathcal{C}$ that assigns $k$-linear categories to $(d-2)$-manifolds and etcetera. A bordism of of closed $(d-1)$-manifolds maps under $Z$ to a linear map between vector spaces. In particular, a bordism of empty sets is a closed $d$-manifold and maps to an endomorphism of the unit vector space $k$, i.e. an element of $k^\times$, as expected. I wish to understand the lower dimensional analog: a bordism $B$ of closed $(d-2)$-manifolds $M$ and $N$ maps to a functor $Z(B):Z(M)\rightarrow Z(N)$ of $k$-linear categories. If $M$ and $N$ are empty, $B$ is a closed $(d-1)$-manifold. How can I see that $Z(B)$ is a vector space?

Answer 1

Najib Idrissi's answer has a subtle error which I'd like to address. I'd also like to give some context.

It's easiest to restrict our attention to the nicest $k$-linear categories, namely those which arise as the categories $\text{Mod}(A)$ of (right) $A$-modules for $A$ a $k$-algebra. In this context one attractive candidate for the tensor product of $\text{Mod}(A)$ and $\text{Mod}(B)$ is $\text{Mod}(A \otimes_k B)$. In fact there is a lovely symmetric monoidal $2$-category, the Morita $2$-category over $k$, which can be described in the following equivalent ways: it has

• objects the $k$-linear categories $\text{Mod}(A)$, equipped with the above tensor product,
• morphisms the cocontinuous $k$-linear functors $\text{Mod}(A) \to \text{Mod}(B)$,
• $2$-morphisms the ($k$-linear? This might be automatic) natural transformations of such functors.

Equivalently, it has

• objects the $k$-algebras $A$, equipped with the usual tensor product,
• morphisms the $k$-bimodules, or equivalently the right modules over $A^{op} \otimes_k B$ (which compose under the tensor product of bimodules),
• $2$-morphisms the morphisms of $k$-bimodules.

These $2$-categories are equivalent by the Eilenberg-Watts theorem, which identifies the category of cocontinuous $k$-linear functors $\text{Mod}(A) \to \text{Mod}(B)$ with the category of $k$-bimodules over $A$ and $B$. Given a bimodule $M$ the corresponding cocontinuous functor is $(-) \otimes_A M$, and conversely given a cocontinuous $k$-linear functor $F$ the corresponding bimodule is $F(A)$. Now:

The unit object is $\text{Mod}(k)$ in the first description of this $2$-category and $k$ in the second description, and by Eilenberg-Watts we have an identification between the category of cocontinuous $k$-linear functors $\text{Mod}(k) \to \text{Mod}(k)$ and $\text{Mod}(k)$ itself.

You should think of this result as a categorification of the Yoneda lemma result that endomorphisms of a ring $R$ as a right $R$-module is the same as $R$ itself, thinking of $\text{Mod}(k)$ as a categorified ring with multiplication the tensor product and thinking of the Morita $2$-category as a $2$-category of (nice) module categories over $\text{Mod}(k)$. (The module structure is given not by the enrichment but by the tensoring of $\text{Mod}(A)$ over $\text{Mod}(k)$.)

That $k$-linearity condition is essential (and this is the error in Najib Idrissi's answer): without it, the category of cocontinuous (this implies $\mathbb{Z}$-linear but not necessarily $k$-linear) functors $\text{Mod}(k) \to \text{Mod}(k)$ is more complicated, and is instead identified with the category of $(k, k)$-bimodules, or equivalently modules over $k \otimes_{\mathbb{Z}} k$. In general this category contains many interesting bimodules which are not equivalent to direct sums of copies of $k$ as a bimodule. For example, any automorphism of $k$ lets you twist the usual bimodule structure on $k$, and leads to an interesting automorphism of $\text{Mod}(k)$ that you can think of as acting by that automorphism on matrices.

In any case, one of the many attractive things about this symmetric monoidal $2$-category is that every object $A$ is automatically dualizable, with dual $A^{op}$, hence automatically defines a $1$-dimensional field theory. The value of this field theory on a circle turns out to be a version of Hochschild homology, although to get Hochschild homology in the ordinary sense we need to work in a derived version of the Morita $2$-category. It's a nice exercise to work out which $A$ are $2$-dualizable, so can define a $2$-dimensional field theory.

Answer 2

When $M=N=\emptyset$, then $Z(\emptyset) = \mathsf{Vec}_f$ is the category of finite dimensional vector spaces, aka the unit for the tensor product. In the end the question becomes: how does one identify functors $\mathsf{Vec}_f \to \mathsf{Vec}_f$ with finite dimensional vector spaces? (We need finite dimensionality here I think, and vector spaces are finite dimensional in this context anyway.)

Define a functor $\operatorname{ev} : \mathsf{Fun}(\mathsf{Vec}_f, \mathsf{Vec}_f) \to \mathsf{Vec}_f$ by $\operatorname{ev}(F) = F(k)$, where $k$ is the ground field. Then $\operatorname{ev}$ is an equivalence of categories. Indeed, by this MO answer of Todd Trimble, additive covariant functors preserve direct sums (this is the kind of functors we're studying here). This means that every $F \in \mathsf{Fun}(\mathsf{Vec}_f, \mathsf{Vec}_f)$ is uniquely determined by its value on $k$, because every $V \in \mathsf{Vec}_f$ is a (finite) direct sum of copies of $k$.

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As for the actual symmetric monoidal structure, I think the best you can do is read Remark 1.2.7 in Lurie's article On the classification of topological field theories. Roughly speaking, if $\mathsf{C,D}$ are cocomplete $k$-linear categories, then $\mathsf{C} \otimes \mathsf{D}$ is the category representing "bilinear" functors $\mathsf{C} \times \mathsf{D} \to \mathsf{E}$.