Has anyone out there thought about homological algebra over the tropical semifield $\mathbb{T}$? For example, I'm interested in the Hochschild homology and cyclic homology of tropical algebras, if this can be made to make sense.

[Edit] Perhaps I should add some motivation. Topologists and geometric group theorists have been interested in the moduli space of metric graphs for at least 25 years now, mainly because of its appearance as a classifying space for automorphism groups of free groups. This space turns out to have a second identity as the moduli space of tropical curves, and people in tropical geometry tell me that it should probably in fact carry the structure of a tropical orbifold.

This means that, in addition to homological invariants built from (sheaves of) continuous of PL functions on the space (containing essentially the information of rational homotopy), one can try to build and study homological invariants made from the tropical structure sheaf. I'm interested in what sort of geometric information these invariants might carry. Is there perhaps some new information hiding in here?

It is always exciting when one finds that an object one has known for many years turns out to have a hidden new structure.

Following Zoran's comment, it looks like Durov has constructed, among many other things, a model category structure on complexes of modules over $\mathbb{T}$. This means that, in principle, something like homological algebra can be done. But it's a different matter to explicitly develop homological algebra.

So, to expand on my original question, here are some explicit questions.

What is a tropical chain complex?

Is there an explicit tropical analogue of the usual Hochschild chain complex and does it compute the correct derived functor?

Ordinary Hochschild homology carries a Gerstenhaber algebra structure. Is there an analogous structure on a tropical Hochschild homology?

Same questions for cyclic homology.