ScalingStacks

Recall the following definition, from e.g. [4].

If $\mathcal{C}$ as in Definition 4.9 is not just enriched in $K$-modules, but $M$-graded $K$-modules for some monoid $M$, then $\mathrm{HH}_{0}(\mathcal{C})$ inherits the structure of an $M$-graded $K$-module. The following is now an immediate consequence of Theorem 4.7 and the Definitions 4.5 and 4.9.

We now recall some facts about the zeroth Hochschild homology, which we will use to show that the 1-handle formula may compute vector spaces which are not locally finite-dimensional.

In a slight reformulation of the functoriality results from [8], the tangle invariant underlying the $\mathfrak{gl}_{N}$ link homology over $\mathbbm{k}$ can be described as a 2-functor:

We now briefly explain the relevant algebraic structures here.

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  As in [27, Definition 6.1] one defines a category $\mathbf{TD}$ of tangle diagrams, whose objects are finite words in the alphabet $\{\uparrow,\downarrow\}$ (which encode possible sequences of oriented boundary points for tangles) and whose morphisms are finite words in generating morphisms $\{\mathrm{cup}_{i},\mathrm{cap}_{i},\mathrm{crossing}_{i},\mathrm{crossing}_{i}^{-1}\}$ (where the index $i$ specifies the strands participating in the generator), that are admissible in the sense that the composite describes a tangle diagram. The composition is concatenation of words. For details see [27, Definition 6.1].

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  $\boldsymbol{\mathrm{Tang}}$ is a $2$-category whose objects and $1$-morphisms are as in $\mathbf{TD}$. The $2$-morphisms are the framed, oriented tangle cobordisms in $[0,1]^{4}$ between standard lifts of tangle diagrams to actual tangles in $[0,1]^{3}$, considered up to isotopy rel boundary.

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  $\boldsymbol{\mathrm{Foam}}_{N}$ is a (monoidal) $2$-category, enriched at the level of $2$-morphism spaces in $\mathbbm{k}$-vector spaces and equipped with grading shift functors on $1$-morphisms. It has the same objects²² 2 More generally, one can consider labelled oriented points as objects in $\boldsymbol{\mathrm{Foam}}_{N}$, but we will not need labels other than $1$. as $\boldsymbol{\mathrm{Tang}}$. The $1$-morphisms are (formal direct sums of grading shifts of) $\mathfrak{gl}_{N}$ webs embedded in $[0,1]^{2}$ and the $2$-morphisms are (matrices with entries given by) $\mathbbm{k}$-linear combinations of $\mathfrak{gl}_{N}$ foams embedded in $[0,1]^{3}$, modulo certain local relations. For details see [8].

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  $\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}}$ is the (monoidal) $2$-category that is obtained from $\boldsymbol{\mathrm{Foam}}_{N}$ by replacing its $\mathbbm{k}$-linear $\operatorname{Hom}$-categories by the corresponding dg categories. This means it has the same objects, but the $1$-morphisms are now chain complexes formed from $1$-morphisms in $\boldsymbol{\mathrm{Foam}}_{N}$, where the differentials are given by $2$-morphisms in $\boldsymbol{\mathrm{Foam}}_{N}$. The $2$-morphisms spaces are chain complexes of homologically homogeneous and quantum grading-preserving maps, spanned by $2$-morphisms from $\boldsymbol{\mathrm{Foam}}_{N}$ (not necessarily chain maps). The differential on $2$-morphisms is the usual supercommutator with respect to the differential on the source- and target complexes. With respect to this differential the zero cycles are exactly the classical chain maps. There is also an enriched $2$-hom in $\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}}$, which is assembled from $2$-homs between objects shifted in quantum grading.

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  $H^{\bullet}(\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}})$ is the cohomology category of $\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}}$. It has the same objects and $1$-morphisms, but the $2$-morphism spaces are now graded $\mathbbm{k}$-modules obtained by taking cohomology. The zeroth cohomology $H^{0}(\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}})$ is also called the homotopy category; its $2$-morphisms are chain maps up to homotopy.

  In the following we will also consider enriched $2$-homs. For objects $s,t$ and $1$-morphisms $A,B\colon s\to t$ we define the bigraded $\mathbbm{k}$-modules:

  (21)

  $$H^{\bullet}(\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}})^{*}(A,B):=\bigoplus_{k\in{\mathbb{Z}}}\operatorname{Hom}_{H^{\bullet}(\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}})}(A\{k\},B)$$

  Here one grading, the quantum grading, is given by the displayed direct sum, while the other grading, the homological grading, is already internal to $H^{\bullet}(\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}})$. Using the grading shift automorphisms, these enriched $2$-homs admit composition maps and thus assemble into a bigraded $\mathbbm{k}$-linear enriched morphism category $H^{\bullet}(\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}})^{*}(s,t)$ whose objects are the $1$-morphisms from $s$ to $t$.

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  The functor $\left\llbracket-\right\rrbracket$ is the identity on objects. On $1$-morphisms it sends a tangle diagram to a chain complex of webs and foams in the way that is usual for $\mathfrak{gl}_{N}$ link homology, and $2$-morphisms, i.e. isotopy classes of tangle cobordisms are sent to the corresponding homotopy classes of chain maps as specified in the functoriality proof in [8].

We recall from [27, Section 6] that the tangle invariant corresponding to the $\mathfrak{gl}_{N}$ link homology can be organized into a braided monoidal $2$-category. Here we give a similar construction of this category $\boldsymbol{\mathrm{T}}_{N}$ (which was denoted $\mathbf{KhR}_{N}$ in[27]) by replacing the top morphism layer of $\boldsymbol{\mathrm{Tang}}$:

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  objects are sequences of tangle endpoints, as in $\mathbf{TD}$ and $\boldsymbol{\mathrm{Tang}}$,

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  1-morphisms consist of Morse data for tangles, as in $\mathbf{TD}$ and $\boldsymbol{\mathrm{Tang}}$,

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  2-morphisms between tangles $S$ and $T$ with equal source and target objects are the bigraded $\mathbbm{k}$-modules computed as the enriched 2-hom $H^{\bullet}(\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}})^{*}(\left\llbracket S\right\rrbracket,\left\llbracket T\right\rrbracket)$ from (21) between the $\mathfrak{gl}_{N}$ chain complexes of the tangles.

As an important special case, one gets for a framed, oriented link $L$:

Moreover, if $T$ and $S$ are framed, oriented tangles with endpoints identified, so that we can form the link $T\cup\overline{S}$, then we set $2p=|\partial S|=|\partial T|$ and have:

Given a $3$-ball $B^{3}$ with a set $P_{p}$ of $2p$ framed, co-oriented points in the boundary, together with a suitable identification of $(B^{3},P_{p})$ with $([0,1]^{3},s\cup t)$, we associate to it the morphism category $\boldsymbol{\mathrm{T}}_{N}(s,t)$, whose objects are tangles from $s$ to $t$. By construction, $\boldsymbol{\mathrm{T}}_{N}(s,t)$ is equivalent to $\mathcal{S}_{0}^{N}(B^{3};P_{p})$ from Definition 4.5. Moreover, $\boldsymbol{\mathrm{T}}_{N}(s,t)$ can be considered as a full subcategory of the bigraded enriched morphism category $H^{\bullet}(\boldsymbol{\mathrm{Foam}}_{N}^{\mathrm{dg}})^{*}(s,t)$.