ScalingStacks

We very briefly recall the key points of the definition of a disklike 4-category, from §6 of [MW12]. A disklike $n$-category ${\mathcal{C}}$ consists of:

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  for each $0\leq k\leq n$, a functor

  $${\mathcal{C}}^{k}:\{\text{$k$-balls and diffeomorphisms}\}\to{\mathsf{Set}}$$

  (and we interpret ${\mathcal{C}}^{k}(X)$ as the set of $k$-morphisms with shape $X$),

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  for each $k-1$-ball $Y$ in the boundary of a $k$-ball $X$, a restriction map ${\mathcal{C}}^{k}(X)\to{\mathcal{C}}^{k-1}(Y)$ (to be more careful, these restriction maps only need to be defined on sufficiently large subsets of ${\mathcal{C}}^{k}(X)$, for example to allow for transversality issues),

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  for each $k$-ball $X$ presented as the gluing of two $k$-balls $X_{1}$ and $X_{2}$ along a common $k-1$-ball $Y$ in their boundaries, a gluing map

  $${\mathcal{C}}^{k}(X_{1})\times_{{\mathcal{C}}^{k-1}(Y)}{\mathcal{C}}^{k}(X_{2})\to{\mathcal{C}}^{k}(X),$$

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  such that these gluing operations are compatible with the action of diffeomorphisms, and associative on the nose,

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  and that two diffeomorphisms of $n$-balls which are isotopic rel boundary act identically,

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  along with some data and axioms concerning identities which we omit here.

(As a reminder, the surprising feature of this definition is that while gluing is required to be strictly associative, this definition actually models fully weak $n$-categories. The key point is that we do not choose canonical models for the shape of a $k$-morphism, and it is up to ‘the end user’ to pick reparametrisations of glued balls back to any standard model balls that they prefer. It is these reparametrisations that are responsible for introducing all the difficult structural isomorphisms of most definitions. This is analogous to the idea of a Moore loop space, which has a strictly associative composition, versus an ordinary loop space, which has a complicated higher associator structure described by Stasheff polyhedra.)

As explained in [MW12], one of the primary examples of a disklike $n$-category is string diagrams for a pivotal traditional $n$-category. This string diagram construction works just as well for a lasagna algebra (which is essentially a pivotal 4-category with trivial 0- and 1-morphisms). Specifically, starting from the lasagna algebra $\mathrm{KhR}_{N}$, we define a disklike $4$-category $\mathsf{KhR}_{N}$ as follows:

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  For $X$ a 0-ball, we define $\mathsf{KhR}_{N}^{0}(X)$ to be a single-element set.

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  For $X$ a 1-ball, we define $\mathsf{KhR}_{N}^{1}(X)$ to be a single-element set.

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  For $X$ a 2-ball, we define $\mathsf{KhR}_{N}^{2}(X)$ to be the set of all configurations of finitely many framed oriented points in $X$.

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  For $X$ a 3-ball, we define $\mathsf{KhR}_{N}^{3}(X)$ to be the set of all framed oriented tangles (not up to isotopy) properly embedded in $X$. If $c$ is a finite configuration of oriented points in $\partial X$, we define $\mathsf{KhR}_{N}^{3}(X;c)$ to be the set of all oriented tangles which restrict to $c$ on $\partial X$.

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  For $X$ a 4-ball, and $L$ a link in $\partial X$, we define $\mathsf{KhR}_{N}^{4}(X;L)$ to be the bigraded abelian group $\mathcal{S}^{N}_{0}(X;L)$ defined above, that is, all lasagna fillings of $X$ which restrict to $L$ on the boundary, modulo relations described above. Recall that by Example 5.6 we know $\mathsf{KhR}_{N}^{4}(X;L)\cong\mathrm{KhR}_{N}(\partial X,L)$.

We define $\mathsf{KhR}_{N}^{4}(X;L)$ to be lasagna fillings modulo relations rather than simply defining it to be $\mathrm{KhR}_{N}(\partial X,L)$ in order to make it easier to define composition below.

We will henceforth drop superscripts and write $\mathsf{KhR}_{N}(X)$ instead of $\mathsf{KhR}_{N}^{k}(X)$.

In dimensions 0 through 3, it is clear that $\mathsf{KhR}_{N}(X)$ is functorial with respect to diffeomorphisms. In dimension 4, it is clear the diffeomorphisms act on lasagna fillings; what remains is to show that the relations we impose are compatible with the action of diffeomorphisms. Specifically, for a diffeomorphism $f$ we must show that if $\mathrm{KhR}_{N}(F)=\mathrm{KhR}_{N}(F^{\prime})$ then $\mathrm{KhR}_{N}(f(F))=\mathrm{KhR}_{N}(f(F^{\prime}))$. This follows from the fact that any diffeomorphism of a 4-ball (rel boundary) is isotopic to the identity away from a small 4-ball in the interior. We can arrange that this small 4-ball is disjoint from $\Sigma$ and the $B_{i}$. The argument is similar to (but simpler than) the argument given in Lemma 4.7.

We must now define gluing (composition) of morphisms. In dimensions 0 through 3 the morphisms are purely geometric and the gluing is defined to be the obvious geometric gluing of submanifolds. In dimension 4, there is again an obvious geometric gluing map of lasagna fillings. We must show that this gluing map is compatible with the relations we impose on fillings. This follows from the operad composition property proved in the previous section.

Finally, the (omitted above) axioms about identities require that we check that 4-ball diffeomorphisms which are supported away from the surface $\Sigma$ act trivially. The diffeomorphism action on lasagna fillings is just moving submanifolds around (and, if the internal balls move, applying the $S^{3}$-functoriality action from the first piece of data for a lasagna algebra to the labels), so a diffeomorphism supported away from the surface and the internal balls does not change a lasagna filling.