ScalingStacks

Definition 4.5. For $p\in{\mathbb{N}}$ fix a configuration $P_{p}$ of $2p$ framed points in $S^{2}=\partial B^{3}$, partitioned into two halves with opposite co-orientations. We define a category $\mathcal{S}_{0}^{N}(B^{3};P_{p})$ enriched in bigraded $\mathbbm{k}$-vector spaces with:

• •

  objects: framed, oriented tangles $T$ in $(B^{3};P_{p})$ inducing the given orientation on $P_{p}$

• •

  morphisms given by

  (18)		$\displaystyle\operatorname{Hom}_{\mathcal{S}_{0}^{N}(B^{3};P_{p},\mathbbm{k})}(T_{1},T_{2})$	$\displaystyle:=\operatorname{KhR}_{N}(T_{2}\cup_{P_{p}}\overline{T_{1}},\mathbbm{k})\{p(N-1)\}$
  (19)			$\displaystyle=\mathcal{S}_{0}^{N}(B^{4};T_{2}\cup_{P_{p}}\overline{T_{1}},\mathbbm{k})\{p(N-1)\}$

with (grading-preserving) composition maps induced in the case of the right-hand side of (18) by the action of merging cobordisms, as described in [27, Section 6.1 (vertical composition of 2-morphisms)], and in the case of (19) induced by the gluing of lasagna fillings of balls.