arXiv subjects
Publications and source records attributed to Shai Haran.
This work explores new arithmetic and combinatorial structures arising from the interplay between Farey-type graphs, Fibonacci expansions, and operadic constructions. We introduce Fibonadic numbers, defined as an inverse limit under the Zeckendorf shift, equipped with a metric, order, and commutative rig structure.
The usual dictionary between geometry and commutative algebra is not appropriate for Arithmetic geometry because addition is a singular operation at the "Real prime". We replace Rings, with addition and multiplication, by Props (=strict symmetric monoidal category generated by one object), or by Bioperad (=two closed symmetric operads acting on each other): to a ring we associate the prop of all matrices over it, with matrix multiplication and block direct sums as the basic operations, or the bioperad consisting of all raw and column vectors over it. We define the "commutative" props and bioperads, and using them we develop a generalized algebraic geometry, following Grothendieck footsteps closely. This new geometry is appropriate for Arithmetic (and potentially also for Physics).
The usual language of algebraic geometry is not appropriate for Arithmetical geometry: addition is singular at the real prime. We developed two languages that overcome this problem: one replace rings by the collection of "vectors" or by bi-operads and another based on "matrices" or props. These are the two languages of [Har17], but we omit the involutions which brings considerable simplifications. Once one understands the delicate commutativity condition one can proceed following Grothendieck footsteps exactly. The square matrices, when viewed up to conjugation, give us new commutative rings with Frobenius endomorphisms.
We study the paths in the Farey graph going from (1,0) to (0,1) , or the finite subtrees of the Stern-Brocot tree . We show they form an operad and isolate the "coronas", which are especially spiked paths. We show that {(x,y), gcd(x,y)=1 , x+y < R } is a corona.
We define the concept of a bi-operad. We develop the homotopy theory of "Bital-Sets" and of infinite-bi-operads. We develop a geometry of generalized schemes based on the spectra of distributive monochromatic bi-operads.
For a commutative ring $A$, we have the category of (bounded-below) chain complexes of $A$-modules $Ch_{+}(A\mymod)$, a closed symmetric monoidal category with a compatible stable Quillen model structure. The associated homotopy category is the derived category $\mathbbm{D}(A\mymod)$, where one inverts all the quasi-isomorphisms, and it has the good description as the chain complexes made up of projective $A$-module in each dimension, and chain maps taken up to chain homotopy. We give here the analogous theory for a (commutative) generalized ring in the sense of \cite{MR3605614}. We refer to the new concept as ``$\aset$''. For an ordinary commutative ring $A$, an $A$-set is just an $A$-module in the usual meaning, and our construction will be equivalent to $\mathbbm{D}(A\mymod)$. For the initial object of the category of generalized rings $\mathbb{F}$ ``the field with one element'', we obtain the category of symmetric spectra, and the associated stable homotopy category with its smash product (an $\mathbb{F}$-set is just a pointed set, i.e. a set $X$ with a distinguish element $O_X\in X$). Thus the analogous theories of stable homotopy and of chain complexes of modules over a commutative ring appear as two sides of the same coin, and moreover, they appear in a context where they interact (via the forgetful functor and its left adjoint - the base change functor). For the ``real integers'' $A=\Z_{\R}$, the $\Z_{\R}$-sets include the symmetric convex subsets of $\R$-vector spaces. We also give the global theory of the derived category of $\bigo_X$-sets, for a generalized scheme $X$, in a way that is based on the local projective model structure.
We give (two) non additive languages for geometry via simple generalisations of commutative rings.
We shall describe a simple generalization of commutative rings. The category GR of such "rings", contains the ordinary commutative rings (fully faithfully), but also the "integers" and "residue field" at a real or complex place of a field ; the "field with one element" (the initial object of GR ); the "arithmetical surface" ( the sum in the category GR of the integers with them self: Z(x)Z ) . We shall show that this geometry "see" the real and complex places of a number field (there is an Ostrowski theorem that the valuation sub-GR of a number field correspond to the finite and infinite primes; there is a compactification of the spectrum of the integers ). One can develop algebraic geometry using GR following Grothendieck paradigm . Quillen's homotopical algebra replaces homological algebra . There is an analogue of Quillen's cotangent bundle - in particular there is a theory of non-additive derivations, with modules of Kahler differentials which satisfy all the usual exact sequences - we compute explicitly the differentials of the integers Z over the initial object F1 . Finally we associate with any topological compact valuation GR a meromorphic function - its "zeta" : for the p-adic integers we get the p-local factor of zeta, for the "real integers" we get the gamma factor.