There should be a category $3\text{CobTang}$ whose

- objects are some kind of surfaces with a finite set of marked points
- morphisms $M : S \to T$ are some kind of $3$-dimensional cobordisms $M$ between surfaces together with a choice of (framed) tangle in $M$ beginning at the marked points in $S$ and ending at the marked points in $T$.

Composition is given by gluing. Taking free $\mathbb{Z}[q, q^{-1}]$-modules on the morphisms in this category and quotienting by the skein relations defining the Kauffman bracket should give a category (the "Kauffman skein bracket category") $K$ in which

- the Temperley-Lieb category,
- skein algebras of surfaces, and
- skein modules of $3$-manifolds

all sit as subcategories. The first bullet is the subcategory where $S, T$ are both, say, the open disc with some marked points and $M$ is a cylinder. The second bullet is the subcategory where $S, T$ are both a fixed surface with no marked points and $M$ is a cylinder. The third bullet is the subcategory where $S, T$ have no marked points.

There is also a forgetful functor $3\text{CobTang} \to 3\text{Cob}$ given by forgetting the marked points and the tangle, and this gives a span of categories $3\text{Cob} \leftarrow 3 \text{CobTang} \rightarrow K$.

(Where) does this construction appear in the literature?

It seems related to a TQFT except that we only get a span and not a functor. In addition, the Temperley-Lieb category quantizes the representation theory of $\text{SU}(2)$ while skein algebras quantize character varieties. So:

What does $K$ quantize?