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.



Is there something like "Noncommutative geometry internal to a category"?
I'll leave open how this might connect to "internal noncommutative geometry", but you can say something about internal $\*$-algebras.


You can formulate $\*$-algebras, anti-linear involution and all, internal to [*dagger* monoidal categories](https://en.wikipedia.org/wiki/Dagger_symmetric_monoidal_category), such as the category of Hilbert spaces. In fact, it turns out you can characterize finite-dimensional C\*-algebras precisely as $\*$-algebras internal to the category of finite-dimensional Hilbert spaces satisfying the so-called *Frobenius law*, see [this paper](http://arxiv.org/abs/0805.0432). 


This doesn't quite generalize to arbitrary dimension, because the carrier object still has to be a Hilbert space. So the best you can hope for is to characterize C\*-algebras whose underlying Banach space is in fact a Hilbert space. But with this caveat, that is possible: these algebras are called *H${}^\*$-algebras*, and are $\*$-algebras internal to the category of Hilbert spaces that satisfy an axiom close to the Frobenius law, see [this paper](http://arxiv.org/abs/1011.6123). 


What's interesting is that you can also do this in other dagger monoidal categories. For example, in the [category of sets and relations](https://en.wikipedia.org/wiki/Category_of_relations), the internal $\*$-algebras as above (Frobenius algebras and H\*-algebras) correspond precisely to *groupoids*! See [this paper](http://arxiv.org/abs/1112.1284).


