ScalingStacks

Remark 6.2. The definitions of the 2-morphism spaces of $\boldsymbol{\mathrm{KhR}}_{N}$ and the composition operations are motivated by the isomorphisms

$$\boldsymbol{\mathrm{KhR}}_{N}(f,g)\cong\boldsymbol{\mathrm{HCh}}^{b}(\boldsymbol{\mathrm{Foam}}_{N})(\left\llbracket f\right\rrbracket,\left\llbracket g\right\rrbracket),$$

where the latter denotes the cohomology category of the dg category of bounded chain complexes over $\boldsymbol{\mathrm{Foam}}_{N}$ (compare with [CMW09, Proposition 3.1]). Under these isomorphisms, the horizontal composition corresponds to stacking tangles, the tensor product corresponds to placing tangles side by side, and the vertical composition corresponds to composing homotopy classes of chain maps. In the following, we will take this space saving point of view when describing 2-morphisms. For example, we will say that the 2-morphism in (6.1) is induced by the movie of tangle diagrams:

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