symmetric monoidal (∞,1)-category of spectra
The idea is that there should be some kind of “complete local ring” corresponding to the archimedean valuation on , by analogy with the (genuine!) complete local rings corresponding to the (non-archimedean) -adic valuations on : the p-adic integers. However, the naïve approach taking
fails because this is not a subring of !
Nikolai Durov’s definition of is inspired by classical Arakelov geometry and starts with the observation that any -lattice in a finite-dimensional -vector space defines a maximal compact subgroup submonoid of , and if and only if and are similar lattices; accordingly, a -lattice up-to-similarity should correspond to maximal compact submonoids of for a finite-dimensional -vector space , i.e. the monoid of -linear endomorphisms of that are short with respect to some norm on . Thus, Durov defines a -lattice to be a finite-dimensional normed -vector space, and a morphism of lattices is defined to be a short -linear map. This gives a category with finite limits and colimits. (Note, however, that it is not an additive category!)
Now, notice that every flat -module is the filtered colimit of its finite-dimensional -submodules (which are necessarily free because is a local ring), and in fact the category of flat -modules is equivalent to the category of ind-objects of . So we may define a flat -module to be an ind-object of . One may also give the following explicit description: a flat -module is a (possibly infinite-dimensional) -vector space together with a symmetric convex body , and a morphism is an -linear map that restricts to a map . This defines a category .
Finally, noting that the forgetful functor taking a flat -module to its underlying symmetric convex body has a left adjoint , Durov defines a (not necessarily flat) -module to be a module for the induced monad . The comparison functor embeds as a full subcategory of .
Let be the functor sending a set to the set
i.e. the solid regular cross-polytope with -many vertices. The action of on maps of sets is the obvious one. Let be the natural transformation given by insertion of generators, and let be the natural transformation given by “evaluation” of “octahedral” combinations:
The monad is a monad with arities: the category of arities may be taken to be .
The -module structure on a set is entirely determined by the map given by . Conversely, a set together with an element and a map satisfying certain equations is a -module. A map commuting with and is a homomorphism of -modules, thus the theory of -modules is a finitary algebraic theory, with all that this implies.
The category has the following properties:
It is a complete category (with the forgetful functor creating all limits).
It is a cocomplete category (with the forgetful functor creating all filtered colimits).
It has a zero object.
Every normed -vector space induces a flat -module where and . (In the other direction, every flat -module induces a seminorm on the underlying -vector space.)
A finitely-generated flat -module is the symmetric convex hull of a finite set of vectors.