Categorical Braid Actions and Decategorification

Let $D:=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }\le 1\},\mathrm{\partial }D:=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\},D^{\circ }:=D\setminus \mathrm{\partial }D,$ the closed unit disk with its subspace topology from $\mathbb{C}\cong {\mathbb{R}}^2$ (Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$, Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace), and fix once and for all a set $\mathrm{\Delta }\subset D^{\circ }$ of $m+1\ge 2$ marked points with the induced topology. Fix also an orientation of $D$, namely the standard one. Write $G:={\pi }_0\mathrm{Diff}(D,\mathrm{\partial }D;\mathrm{\Delta })$ for the boundary-fixed mapping class group of the punctured disk of Boundary-fixed mapping class group of a punctured disk, in its smooth version, whose elements are isotopy classes of diffeomorphisms of $D$ fixing $\mathrm{\partial }D$ pointwise and permuting $\mathrm{\Delta }$ setwise. The cited item defines the homeomorphism version; the smooth configuration-space comparison used here is Khovanov–Seidel, Section 3b, equation (3.1), printed p. 19.

Curves. A curve in $(D,\mathrm{\Delta })$ is a subset $c\subseteq D$ of one of the following two kinds.

1. A simple closed curve $c\subset D^{\circ }\setminus \mathrm{\Delta }$ which is essential, that is, not contractible in $D^{\circ }\setminus \mathrm{\Delta }$; it is the image of an embedding $S^1\hookrightarrow D^{\circ }\setminus \mathrm{\Delta }$ (Smooth embeddings).
2. An arc: the image of an embedding $\gamma \text{\hspace{0.17em}⁣}:[0,1]\to D$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length, Smooth embeddings) which is transverse to $\mathrm{\partial }D$, meets the boundary and the marked set exactly in its endpoints, ${\gamma }^{-1}(\mathrm{\partial }D\cup \mathrm{\Delta })=\{0,1\},$ and whose interior lies in $D^{\circ }\setminus \mathrm{\Delta }$. The unoriented image is the curve; a curve may meet $\mathrm{\Delta }$ in one or both of its endpoints, may meet $\mathrm{\partial }D$ in one or both of them, and may have both endpoints in $\mathrm{\Delta }$, both on $\mathrm{\partial }D$, or one of each. Arcs are thus embedded smooth submanifolds with boundary of $D$ (Embedded smooth submanifolds with boundary).

Curves are unoriented: $c$ and its image under any orientation-reversing reparametrization are the same curve. Two curves $c_0,c_1$ are isotopic, written $c_0\simeq c_1$, when one can be deformed into the other by an isotopy in $\mathrm{Diff}(D,\mathrm{\partial }D;\mathrm{\Delta })$; endpoints on $\mathrm{\partial }D$ may not move during an isotopy. Isotopy of curves is the orbit relation for the identity component of $\mathrm{Diff}(D,\mathrm{\partial }D;\mathrm{\Delta })$. The full mapping class group $G$ acts on these isotopy classes and may carry one class to a different class.

Minimal intersection. Let $c_0,c_1$ be curves. They are in minimal intersection when (i) they intersect transversally, (ii) $c_0\cap c_1\cap \mathrm{\partial }D=\mathrm{\varnothing }$, and (iii) the following disk formulation of the bigon condition holds: for any two points $z_{-}\mathrm{\ne }{z}_{+}$ of $c_0\cap c_1$ that do not both lie in $\mathrm{\Delta }$, and arcs ${\alpha }_0\subseteq c_0$, ${\alpha }_1\subseteq c_1$ with endpoints $z_{-},z_{+}$ and ${\alpha }_0\cap {\alpha }_1=\{z_{-},z_{+}\}$, the open Jordan disk $K$ enclosed by ${\alpha }_0\cup {\alpha }_1$ contains a marked point. Other portions of the curves may lie in $K$; it need not be a component of $D\setminus (c_0\cup c_1)$. Thus a transverse interior crossing contributing a removable bigon is forbidden, and a pair of arcs with common endpoints in $\mathrm{\Delta }$ is allowed to bound a marked-free disk only when the two arcs share both endpoints.

Existence of minimal representatives. Given curves $c_0,c_1$ with $c_0\cap c_1\cap \mathrm{\partial }D=\mathrm{\varnothing }$ there is a curve $c_1^{\mathrm{\prime }}\simeq c_1$ in minimal intersection with $c_0$: first perturb $c_1$ into transverse position with distinct endpoint germs, giving finitely many intersections. If the disk condition fails, an innermost marked-free bigon can be removed by pushing $c_1$ across it, with support near the bigon and fixing the marked points and the boundary (the bigon removal of Khovanov–Seidel, Section 3a); as each such move decreases the finite number $\mathrm{\mid }{c}_0\cap c_1\mathrm{\mid }$ of intersection points, the process terminates. In the case $c_0\cap c_1\cap \mathrm{\partial }D\mathrm{\ne }\mathrm{\varnothing }$ one first applies the flow extension below.

Choice scope for geometric intersection numbers. Assume AC (The Axiom of Choice) for the representative-independence, isotopy invariance and auxiliary-choice independence of $I$ supplied by Geometric intersection numbers are isotopy invariants ↗. That supplier retains AC for its topological relative-arc inputs. The definitions of the marked disk, curves, isotopy and minimal intersection, and the finite perturbation and bigon-removal construction above do not invoke that input.

The geometric intersection number. Under this assumption, let $c_0,c_1$ be curves with $c_0\cap c_1\cap \mathrm{\partial }D=\mathrm{\varnothing }$, and choose a minimal-intersection representative $c_1^{\mathrm{\prime }}\simeq c_1$ of $c_1$. The geometric intersection number is the half-integer $I(c_0,c_1):=\mathrm{\mid }(c_0\cap c_1^{\mathrm{\prime }})\setminus \mathrm{\Delta }\mathrm{\mid }+\frac{1}{2}\mathrm{\mid }{c}_0\cap c_1^{\mathrm{\prime }}\cap \mathrm{\Delta }\mathrm{\mid },I(c_0,c_1)\in \frac{1}{2}\mathbb{Z},$ except in the exceptional case that $c_0,c_1$ are simple closed curves with $c_0\simeq c_1$, where one sets $I(c_0,c_1):=2$. Interior intersection points count once and common marked endpoints count one half, so that in particular $I(b_i,b_i)=1$ for an arc $b_i$ joining two marked points. Isotopic arcs have the same endpoint set, since an isotopy fixes every marked point and every boundary point. The number is independent of the chosen representative $c_1^{\mathrm{\prime }}$: this is proved in Geometric intersection numbers are isotopy invariants ↗, together with the invariance of $I$ under isotopies of both arguments. The value is finite because two curves in minimal intersection meet in finitely many points after a small perturbation, the arcs involved being compact.

The flow extension. The definition above requires $c_0\cap c_1\cap \mathrm{\partial }D=\mathrm{\varnothing }$. For the source's $I$ on arbitrary pairs one fixes a nonvanishing smooth vector field on $\mathrm{\partial }D$ which is positively oriented with respect to the fixed orientation, extends it to a smooth vector field $Z$ on $D$ which vanishes on $\mathrm{\Delta }$, and lets $(f_t)$ be the flow of $Z$. For $t>0$ small enough that the endpoints of $c_0$ on $\mathrm{\partial }D$ are moved along $\mathrm{\partial }D$ past no endpoint of $c_1$, set $I(c_0,c_1):=I(c_0^{+},c_1),c_0^{+}:=f_t(c_0)\mathrm{.}$ This extension is independent of the auxiliary choices by the isotopy invariance proved in Geometric intersection numbers are isotopy invariants ↗; it depends on the orientation of $D$ and is not symmetric, since it removes the common boundary endpoints of $c_0$ with $c_1$ by pushing $c_0$ off them.

Standing conventions. Under the stated AC hypothesis, throughout this page $I$ always denotes this half-integer valued function of isotopy classes of curves, with the exceptional value $2$ for isotopic simple closed curves, and with the flow extension whenever a pair meets on $\mathrm{\partial }D$. The basic arcs $b_0,\dots ,b_m$ fixed in Basic arcs, admissible curves and the standard normal form form a chain: $b_0$ joins a fixed boundary point to the first marked point, and $b_i$ for $1\le i\le m$ joins the $(i-1)$st marked point to the $i$th marked point, and all values $I(b_j,c)$ quoted on this page are computed in that fixed picture, whose data are fixed there once and for all.

Fix $m\ge 1$, let $A_m$ be the Khovanov–Seidel type A algebra, and let $C_m=K^b({\mathrm{proj}}^{gr}{A}_m)$ be the bounded homotopy category of finite graded projective left $A_m$-modules of The bounded projective homotopy category C_m and the two shifts, with its distinguished triangles and its homological shift $[1]$. The graded Grothendieck group of $A_m$ is the triangulated Grothendieck group $G(A_m):=K_0(C_m)=K_0^{\mathrm{t}\mathrm{r}\mathrm{i}}(K^b({\mathrm{proj}}^{gr}{A}_m))$ of Grothendieck group of an essentially small triangulated category applied to the small coded presentation of The triangulated K_0 of the Khovanov–Seidel projective category: the free abelian group on the set of isomorphism classes of objects of $C_m$ modulo the subgroup generated by the relations $[Y]=[X]+[Z]$ for every distinguished triangle $X\to Y\to Z\to X[1]$ of $C_m$. Write $[X]$ for the class of $X$.

Class rules. The definition is the triangulated one, so:

• the cone triangle $X\oplus Y\to Y\to X[1]\to X[1]\oplus Y[1]$ of a biproduct gives $[X\oplus Y]=[X]+[Y]$;
• the rotated triangle $X\to 0\to X[1]\to X[1]$ gives $[X[1]]=-[X]$, the statement of Homological and internal shifts on K_0(C_m);
• the internal shift $\{1\}$, which is an exact automorphism of $C_m$ distinct from $[1]$, makes $G(A_m)$ into a module over $\mathbb{Z}[q,q^{-1}]$ with $[X\{r\}]=q^r[X]$, again by Homological and internal shifts on K_0(C_m).

The two shifts are never identified: $[1]$ is homological and contributes a sign $-1$, $\{1\}$ is internal and contributes a unit $q$.

Comparison with perfect complexes. The composite $\mathrm{\Theta }\text{\hspace{0.17em}⁣}:C_m=K^b({\mathrm{proj}}^{gr}{A}_m)⟶D^b(A_m\text{-mod})$ of inclusion and localization is exact, full, faithful and essentially surjective by The bounded projective comparison for the derived category; the graded clause of Triangle K0 of perfect complexes equals split K0 of finite projectives identifies the target's triangulated Grothendieck group with $K_0^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}$ of the finite graded projectives. Consequently $G(A_m)$ is canonically isomorphic to the Grothendieck group of the graded perfect complexes of Perfect complexes over a ring and its graded version on the bounded projective model, and $[X]$ corresponds to the Euler characteristic ${\sum }_n(-1{)}^n[P^n]$ of any bounded finite graded projective complex representing $X$.

Scope and warnings. This is emphatically the triangulated $K_0$ of projective complexes and not the short-exact $G_0$ of the abelian category of all finitely generated graded $A_m$-modules, and no identification of the two is claimed; $G_0$ may a priori be a quotient of $G(A_m)$. No structure theorem for finite-dimensional algebras over a field is applied to the integral algebra $A_m$. The freeness of $G(A_m)$ on the shifted vertex-projective classes is not asserted here: it is the content of The graded Grothendieck group is free on the vertex-projective classes, which uses the classification of finite graded projectives and the comparison above.

Notation used below. Since the internal shift is an automorphism, write $q^r[P_i]:=[P_i\{r\}]$, so that $G(A_m)$ is generated as a $\mathbb{Z}[q,q^{-1}]$-module by the classes of the vertex projectives $P_0,\dots ,P_m$; the next lemma on this page shows these generate freely.

Fix $m\ge 1$ and let $R_i,R_i^{-1}$ be the twist complexes of The twist complexes R_i and R_i^{-1}, built from ${\beta }_i,{\gamma }_i$ of The Khovanov–Seidel bimodule maps β_i and γ_i and acting on $C_m=K^b({\mathrm{proj}}^{gr}{A}_m)$. Fix a word $\sigma ={\tau }_1{\tau }_2\cdots {\tau }_k,{\tau }_{\mathrm{\ell }}\in \{{\sigma }_i^{\pm 1}:1\le i\le m\},$ in the Artin generators and their inverses, and put $R_{{\sigma }_i}:=R_i,R_{{\sigma }_i^{-1}}:=R_i^{-1},R_1:=A_m,$ the last being the diagonal bimodule $A_m$ concentrated in homological degree $0$. The complex of the word $\sigma$ is the iterated signed totalization $R_{\sigma }:=R_{{\tau }_1}{\otimes }_{A_m}{R}_{{\tau }_2}{\otimes }_{A_m}\cdots {\otimes }_{A_m}{R}_{{\tau }_k}$ of Signed totalization of graded A_m-bimodule actions, and $R_{\sigma }$ also denotes the endofunctor $R_{\sigma }\text{\hspace{0.17em}⁣}:C_m\to C_m,M\mapsto R_{\sigma }{\otimes }_{A_m}M\mathrm{.}$

Claims. (i) $R_{\sigma }$ is a bounded complex of graded $(A_m,A_m)$-bimodules each of whose terms is finitely generated graded projective as a left $A_m$-module and as a right $A_m$-module; (ii) consequently the functor $R_{\sigma }{\otimes }_{A_m}-$ is an exact additive endofunctor of $C_m$ carrying distinguished triangles to distinguished triangles and agreeing with the derived tensor product through the identity replacements, by Bounded two-sided projective bimodule complexes act on C_m. These two claims are verified below. The definition fixes the complex attached to the chosen word; that different words for the same braid give isomorphic functors is the content of the weak action theorem below and is not asserted here.

Fix $m\ge 1$ and let $A_m$ be the Khovanov–Seidel type A algebra of Khovanov–Seidel type A algebra, with its internal grading $\mathrm{deg}{e}_i=0,\mathrm{deg}(i\mathrm{\mid }i+1)=0,\mathrm{deg}(i+1\mathrm{\mid }i)=1$ on vertices and arrows, extended to paths additively, and with vertex idempotents $e_0,\dots ,e_m$ and unit $1=e_0+\cdots +e_m$. Let $J\subseteq A_m$ be the two-sided ideal generated by the classes of all arrows $(i\mathrm{\mid }i\pm 1)$ (The ideal generated by a subset and principal ideals); it is the smallest two-sided ideal containing every arrow, and it is computed in the proof below.

Claims. $J$ is homogeneous for the internal degree; $J$ is spanned as a $\mathbb{Z}$-module by the classes of all paths of length at least one, and $J^2$ is spanned by the returns $(i\mathrm{\mid }i-1\mathrm{\mid }i)$, so that $J^3=0$; and the quotient ring of The quotient ring $R\mathrm{/}I$ with $(r+I)(s+I)=rs+I$ satisfies $A_m\mathrm{/}J\cong {\mathbb{Z}}^{m+1}$ through the vertex idempotents, the isomorphism sending the class of $e_i$ to the $i$-th standard basis vector. In particular the quotient module $A_m\mathrm{/}J\cong {\mathbb{Z}}^{m+1}$ is concentrated on the vertex idempotents.

Let $G$ be a group with unit $1$ (Left group actions, transitive actions, and faithful actions) and $Q$ a category (Category, object, morphism, domain, codomain, identity, composition, and hom-collection). A weak action of $G$ on $Q$ is a choice of a functor $F_g\text{\hspace{0.17em}⁣}:Q\to Q(g\in G)$ for every element of $G$ (Covariant functor, identity functor, composite functor, and contravariant functor) such that

1. $F_1$ is the identity functor of $Q$, and
2. for all $f,g\in G$ the functors $F_{fg}$ and $F_f{F}_g$ are isomorphic, that is, there exists a natural isomorphism between them (Natural isomorphism).

No isomorphisms $F_{fg}\cong F_f{F}_g$ are chosen, they are not required to be compatible with the associativity of $G$, and no pentagon is imposed: the definition records only that such isomorphisms exist. This is Definition 2.6 of the source, stated there for a group and a category with the same comments.

Normalized coherent actions. In the identity-unit normalization, the action is coherent, or a genuine $2$-action, when isomorphisms ${\mu }_{f,g}\text{\hspace{0.17em}⁣}:F_f{F}_g⟶F_{fg}(f,g\in G)$ are chosen so that ${\mu }_{1,g}={\mathrm{i}\mathrm{d}}_{F_g}$, ${\mu }_{f,1}={\mathrm{i}\mathrm{d}}_{F_f}$ and the two composites $F_f{F}_g{F}_h\stackrel{{\mu }_{f,g}{F}_h}{\to }{F}_{fg}{F}_h\stackrel{{\mu }_{fg,h}}{\to }{F}_{fgh},F_f{F}_g{F}_h\stackrel{F_f{\mu }_{g,h}}{\to }{F}_f{F}_{gh}\stackrel{{\mu }_{f,gh}}{\to }{F}_{fgh}$ agree, the associativity (pentagon) condition. This is the normalized special case of a strong monoidal action. In the general definition, the unit constraint is a chosen natural isomorphism $u:F_1\Rightarrow {\mathrm{Id}}_Q$; when $F_1={\mathrm{Id}}_Q$, the unit triangles read ${\mu }_{f,1}=F_{f}u$ and ${\mu }_{1,g}=u{F}_g$, and need not be identity maps. No strictification to the normalized case is asserted. A coherent action with $F_1={\mathrm{Id}}_Q$ is in particular a weak action after forgetting its chosen compositors. For a general coherent action with only $u:F_1\Rightarrow {\mathrm{Id}}_Q$, first replace the identity component of the functor assignment by ${\mathrm{Id}}_Q$; the unit isomorphism and the compositors then supply the pairwise isomorphisms required by the weak definition. This replacement asserts no strictification of the coherence data.

Standing convention. The whole page uses "weak" in the sense of this definition and never silently substitutes a coherent action: whenever a compositor or pentagon argument would be needed, the weakness of the available data is stated.

Terminology. The functors $F_g$ are the components of the weak action, the assignment $g\mapsto F_g$ is its functor assignment, and a weak action is determined by the functor assignment with its prescribed identity component together with the existence of the pairwise isomorphisms in condition 2. We do not distinguish two weak actions that are naturally isomorphic componentwise.

Arbitrarily chosen pairwise isomorphisms of a weak action need not satisfy coherence: the companion-page counterexample Weak actions do not supply pentagon coherence data ↗ exhibits a weak action with chosen pairwise isomorphisms violating the pentagon. That example also admits identity compositors satisfying coherence, so it does not assert that no coherent choice exists.

Fix $m\ge 1$ and let $R_i$ and $R_i^{-1}$ be the twist complexes of graded $(A_m,A_m)$-bimodules of The twist complexes R_i and R_i^{-1}, with $R_i=[U_i\stackrel{{\beta }_i}{\to }{A}_m]$ having $U_i$ in homological degree $-1$ and $A_m$ in degree $0$. If $\mathrm{\mid }i-j\mathrm{\mid }>1$ then there is an isomorphism of complexes of graded $(A_m,A_m)$-bimodules $R_i{\otimes }_{A_m}{R}_j\cong R_j{\otimes }_{A_m}{R}_i,$ and consequently an isomorphism of endofunctors of $C_m$ $R_i{R}_j\cong R_j{R}_i\mathrm{.}$

Fix $m\ge 1$ and let $R_i=[U_i\stackrel{{\beta }_i}{\to }{A}_m]$ and $R_i^{-1}=[A_m\stackrel{{\gamma }_i}{\to }{U}_i\{-1\}]$ be the positive and negative twist complexes of The twist complexes R_i and R_i^{-1}, with $U_i$ in homological degree $-1$ and $A_m$ in degree $0$ in the first complex and $A_m$ in degree $0$ and $U_i\{-1\}$ in degree $1$ in the second. Then for every $1\le i\le m$ there are homotopy equivalences of complexes of graded $(A_m,A_m)$-bimodules $R_i{\otimes }_{A_m}{R}_i^{-1}\simeq A_m\simeq R_i^{-1}{\otimes }_{A_m}{R}_i,$ where $A_m$ denotes the diagonal bimodule concentrated in homological degree $0$; they become isomorphisms in $C_m$ and induce isomorphisms of endofunctors $R_i{R}_i^{-1}\cong {\mathrm{Id}}_{C_m}\cong R_i^{-1}{R}_i\mathrm{.}$ In particular $R_i^{-1}$ is a two-sided inverse of $R_i$ on $C_m$, and both are equivalences of $C_m$.