I have heard that one can do algebraic geometry internal to symmetric monoidal categories. Topological quantum field theories also exist internal to symmetric monoidal categories, and the usual definition is recovered by inserting the category of vector spaces.
I wonder whether this can, or has, been done for noncommutative geometry. I'm interested in Connes' approach via spectral triples. I guess it is feasible to define what a $*$-algebra internal to a symmetric monoidal category is, but how does the story go on?
Edit: As has been pointed out in the comments, this seems to hard in all generality. Let us restrict to finite-dimensional $*$-algebras and Hilbert spaces in order not to worry about topology and functional analysis.