ScalingStacks

We gather in §2 a number of basic definitions and facts involving differential categories and bimodules. Most differential vector spaces we encounter come with bases, and we formalize this aspect in the notion of “differential pointed sets” and corresponding differential pointed categories.

We consider Hecke algebras in §3. We study in §3.1 the differential algebra structure on nil Hecke algebras of Coxeter groups over a field of characteristic $2$ and we describe adjunctions for induction and restriction functors, in the case of finite Coxeter groups. An important fact is that those Hecke algebras are the graded algebras associated with the filtration of the group algebra with respect to the length function. The remainder of §3 is devoted to the case of symmetric groups and their affine versions. We introduce in §3.2.6 positive submonoids of the affine symmetric groups and we provide a description by generators and relations of their nil Hecke algebras.

Section §4 is devoted to the development of the $2$-representation theory of ${\mathfrak{gl}}(1|1)^{+}$. We introduce the monoidal category ${\mathcal{U}}$. Our main construction is that of a tensor product operation on $2$-representations, and more generally, of a diagonal action given two (lax) commuting $2$-representation structures. We also consider a more complicated “dual” construction in §4.4. In §5, we recast our functorial constructions into bimodule constructions. We formulate our constructions in the differential ungraded setting.

In §6, we construct bimodules and $2$-representations associated with nil Hecke algebras. In §6.1, we describe explicitly the structures of $2$-representation coming from the left and the right action of the monoidal category ${\mathcal{U}}$ on itself and we show that the diagonal category arising from these commuting left and right actions corresponds to Hecke algebras of positive affine symmetric groups. It is a remarkable fact that those can be recovered from the Hecke algebras of the ordinary symmetric groups. We introduce in §6.2 a categorical version of affine symmetric groups and their Hecke algebras.

We develop in §7 an extension of Lipshitz-Ozsváth-Thurston [LiOzTh1] and Zarev’s [Za] theory of strand algebras associated with matched circles and intervals. Instead of considering curves with matchings, we consider the corresponding quotient spaces, where the matched points are identified. We start in §7.1 with $1$-dimensional spaces, which we define as complements of a finite set of points in a $1$-dimensional finite CW-complex. In §7.2, we define our objects of interest, the singular curves. They are $1$-dimensional spaces together with an additional structure at singular points, and a partially defined orientation. They arise as quotients of smooth curves, or, equivalently, as curves in ${\mathbf{R}}^{n}$ with transverse intersections of branches. This leads to a notion of admissible paths, those paths that lift to a smooth model for the curve (§7.3). We introduce in §7.4 the differential categories of strands associated to a curve. They are defined as graded categories associated with a filtered category, in a way similar to the constructions of §3.1. We show in §7.4.3 that strand categories on unoriented $S^{1}$ correspond to the categories built from nil Hecke algebras of affine symmetric groups.

The final section §8 shows that the strand category of a glued curve is obtained as a tensor (or more general diagonal) construction from the strand category of the original curve. This provides some sort of $1$-dimensional field theory, which is really part of a $2$-dimensional field theory for surfaces with extra structure. This gives a categorical mechanism by which strand categories can be computed by cutting the curve into basic building blocks. We start in §8.1 by constructing a structure of $2$-representation associated with an unoriented “end” of a curve. We describe in §8.2 how the strand categories behave under the gluing of two ends of a curve. This requires to solve a combinatorial generators and relation problem generalizing Proposition 3.2.9. When the gluing operation does not create an $S^{1}$, we show in §8.3 that the resulting $2$-representation is the one obtained from the diagonal action.