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.

  1. What is a tropical chain complex?

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

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

  4. Same questions for cyclic homology.

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    $\begingroup$ Durov in his thesis arxiv.org/abs/0704.2030 develops (cf. chapter 10) nonabelian homological algebra for "generalized algebraic geometry" which, as special cases, includes a variant of the geometry over a field of one element, tropical geometry and an algebraic version of Arakelov geometry. $\endgroup$ – Zoran Skoda Feb 10 '11 at 14:01
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    $\begingroup$ Zoran, since this question is more or less of a "yes or no" type, your comment would make a fine answer. $\endgroup$ – S. Carnahan Feb 10 '11 at 15:58
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    $\begingroup$ An interesting couple of question along these lines is: 1. can one interpret the Baker-Norine rank of a linear system on a metric graph as the rank of a tropical module? 2. are there higher cohomology analogs of the Baker-Norine rank. The Baker-Norine rank makes sense from the point of view of deformation theory, but there should be a way of doing tropical homological algebra to make it natural. $\endgroup$ – Eric Katz Feb 15 '11 at 19:55

Your question has two parts - the main question, existence of tropical homotopical algebra, seems to be nicely resolved by Zoran's answer. The questions about Hochschild homology though are rather context independently true - I am not familiar with the Durov story, but assuming he provides a symmetric monoidal model category (and hence symmetric monoidal $\infty$-category) that should suffice. Given an algebra object $A$ in any symmetric monoidal $\infty$-category, we can define its Hochschild homology and cohomology, as self-tor and ext as bimodules (see e.g. here - note that what I'm referring to as the homology and cohomology corresponds to the chain level versions of these notions, not the cohomology groups). The former has an $S^1$ action, with invariants giving cyclic homology, and can be calculated by a version of the cyclic bar construction. Regarding the Gerstenhaber structure on Hochschild COhomology, again the answer is yes by a general form of the Deligne conjecture (see Lurie's Higher Algebra - the part that until last week was called DAG VI..) - there's an $E_2$ structure on the Hochschild cohomology. In fact if your algebra $A$ is separable (dualizable as $A$-bimodule) one obtains the full structure of a framed two-dimensional topological field theory (via Lurie's Cobordism Hypothesis theorem), which includes all these operations and many more. (The Hochschild homology and cohomology are assigned to the circle with two different framings in this story.)

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