ScalingStacks

We define a notion of tensor product ${\hskip 1.9919pt\otimes\hskip-10.81218pt\bigcirc}$ of higher representations of $\mathfrak{gl}(1|1)^{+}$, endowing the $2$-category of $2$-representations of ${\mathfrak{gl}}(1|1)^{+}$ on differential categories with a structure of monoidal $2$-category. This does not require working in an $\infty$-categorical setting, as in the Kac–Moody case.

Given $\mathcal{V}_{1}$ and $\mathcal{V}_{2}$ two higher representations, we construct a higher representation $\mathcal{V}_{1}{\hskip 1.9919pt\otimes\hskip-10.81218pt\bigcirc}\mathcal{V}_{2}$. A typical object of this category is a pair $(M_{1}\otimes M_{2},\pi)$ where $M_{1}\otimes M_{2}$ is an object of the ordinary tensor product $\mathcal{V}_{1}\otimes\mathcal{V}_{2}$ and $\pi$ is a closed map $M_{1}\otimes E_{2}(M_{2})\to E_{1}(M_{1})\otimes M_{2}$ compatible with $\tau$. More generally, one considers objects obtained from those by taking cones and direct summands. The image by $E$ of the pair above is $(\mathrm{cone}(\pi),\pi^{\prime})$ for some $\pi^{\prime}$.

This construction generalizes immediately to differential categories endowed with two commuting structures of higher representations, but we need a more general construction dealing with two lax-commuting higher representations to handle general gluings of surfaces. We provide three increasingly subtle versions of such a construction. In general, we obtain a differential category without the (full) structure of higher representation.

Starting with two structures of higher representations given by endofunctors $E_{1}$ and $E_{2}$ on a differential category $\mathcal{W}$ and a map $\sigma:E_{2}E_{1}\to E_{1}E_{2}$ (suitably compatible with $\tau$’s), we define a differential category $\Delta_{\sigma}(\mathcal{W})$ by proceeding as in the tensor product case. It will have a structure of higher representation if $\sigma$ is invertible.

The notion of right higher representation coincides with that of (left) higher representation, but it leads to a different version of the construction above. We start with the same structures as above, but write $F_{1}$ instead of $E_{1}$ and $\lambda$ instead of $\sigma$. We define a differential category $\Delta_{\lambda}(\mathcal{W})$ with typical objects pairs $(M,(\upsilon)_{i\geq 1})$ where $M$ is an object of $\mathcal{W}$ and $\upsilon_{i}:E_{2}^{i}F_{2}^{i}(M)\to M$ is a system of compatible maps with respect to $\lambda$ and $\tau$. In order to define a structure of higher representation, we need $F_{1}$ to have a right adjoint $E_{1}$. Using this adjunction, $\lambda$ gives rise to $\sigma:E_{2}E_{1}\to E_{1}E_{2}$ and, when $\sigma$ is invertible, we obtain a structure of higher representation on $\Delta_{\lambda}(\mathcal{W})$.

Finally, starting with a lax action of $\mathcal{U}\times\mathcal{U}$ on $\mathcal{W}$, we define a differential category $\Delta_{E}(\mathcal{W})$.

Our constructions extend an earlier construction of Douglas-Manolescu [DouMa]. They provided a construction of the category underlying a tensor product.

One of the applications of tensor products in higher representation theory is the construction of complicated categories from simpler ones. This is illustrated below in the reconstruction of partially wrapped Fukaya categories of symmetric powers of surfaces from more basic algebras. We formulate the problem in terms of surfaces with an arc decomposition, and then reformulate it again in terms of certain singular curves. We provide another example, the construction of nil affine Hecke algebras from nil Hecke algebras (in type $A$).