transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
symmetric monoidal (∞,1)-category of spectra
(also nonabelian homological algebra)
Context
Basic definitions
Stable homotopy theory notions
Constructions
Lemmas
Homology theories
Theorems
constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
Talk about the decimal number system for integers and decimal fractions, and then infinite sequences of decimals as a terminal coalgebra for an endofunctor.
Define a set of digits , and the free monoid on with unit , quotiented by an equivalence relation. Then define a function on such that is a natural numbers object.
Let be the free monoid on the closed interval , which represents the list of digits in the decimal numeral representation of the natural numbers, with length function . There is a surjection defined as
representing the numerical value of the list of digits.
The algorithm is as follows: Let and be lists of digits, with the dividend and the divisor. If , then and . Otherwise, we iterate for times before stopping:
For each iteration , let be the quotient extracted from the algorithm at iteration , let be the dividend at iteration , let be the remainder at iteration , and let be the next digit of the quotient, with the restriction that and
We set initial conditions to be
The successive conditions are defined inductively as
work in progress…
See below, but define in terms of an initial algebra instead of a terminal coalgebra.
Consider the category of intervals , i.e., linearly ordered sets with identified elements and , and let
be the endofunctor which takes an interval to , the the interval obtained by taking ten copies of and identifying the of the -th copy with the of the -th copy, for . The real interval becomes a coalgebra if we identify with and consider the multiplication-by-10 map as giving a coalgebra structure.
The interval is terminal in the category of such coalgebras.
Given any coalgebra structure , any value lands either in the -th tenth (the -th in ) for , or at the precise spot between them, where the in the -th copy is glued to the in the -th for . Intuitively, one could think of a coalgebra structure as giving an automaton where on input there is output of the form , where is one of 19 states, “0”, “1”, “2”, “3”, “4”, “5”, “6”, “7”, “8”, “9”, “either 0 or 1”, “either 1 or 2”, “either 2 or 3”, “either 3 or 4”, “either 4 or 5”, “either 5 or 6”, “either 6 or 7”, “either 7 or 8”, and “either 8 or 9”. By iteration, this generates a behavior stream . Let “0”, “1”, “2”, “3”, “4”, “5”, “6”, “7”, “8”, and “9” be decimal digits , the form a decimal expansion to give a number between 0 and 1, and therefore we have an interval map which sends to that number. Of course, if is ever one of the “either or ” states, for we have a choice to resolve it as either or and continue the stream, but these streams are identified, and this corresponds to the identifications of decimal expansions
as real numbers. In this way, we produce a unique well-defined interval map , so that is the terminal coalgebra.
Let be the set of integers, the initial set with an element , a linear order , a monotone such that for all , and an antitone such that and . Let be a set with a structure preserving function and a function into the monotone poset such that and for all . The set of real numbers is the initial such system. By uncurrying the function , one gets a function . An element is called an infinite decimal representation, where the comma used to represent pairs in set theory or type theory is literally the decimal separator commonly seen in non-English speaking countries.
The arithmetic operations and topological properties on can be defined by the properties of the function algebra of and currying.
Localisation of the rig of natural numbers at 10 , finite decimals as canonical representatives of , and then group completion of the additive monoid to .
Sequence algebra and Cauchy sequences
Let be the cyclic group consisting of 10 elements, and let be a chain complex of abelian groups consisting of a sequence of indexed by . The indices are called place values, and -cochains are called digits.
A 10-adic number is a cochain such that for all or for all for . A 10-adic integer is a cochain such that for all or for all . A real number is a cochain such that for all and for . A decimal rational is a real 10-adic number, and an integer is a real 10-adic integer.
The cochain complex defined in the previous section has a structure of an abelian group, making it into a 10-adic solenoid.
A cyclic group has a canonical cyclic order . We define the cyclic order on such that is false for all .
For each , there exists a cocycle called the digitwise carry function at place value , defined such that for all , if is false, and if is true.
We define the addition without carry on the cochain complex as the addition of all digits using the abelian group operation, and we define the carry as the digitwise carry of all digits. Then, addition is defined recursively as .
The cochains consisting of all s for all are additive identity elements of the addition operation defined above. As such, they are algebraically equal to the same chain zero, the chain consisting of all zeroes. We define negation such that for all chains , the digits in are . As a result, the chain complex itself is an abelian group.
Let one denote the cochain with all digits equal to zero except at place value , where the digit is equal to . The cochain with all digits equal to zero except at place value is called the -th power of ten and is denoted as .
Let us define an -action such that and for all and . This represents the -fold sum of a cochain .
One could also establish a ring structure on and construct a multiplication operation on the chain complex such that the chain complex itself with the defined abelian group structure and multiplication should be equivalent to the quotient ring of the Laurent series .
Wikipedia, Decimal
Wikipedia, Decimal representation
Last revised on May 13, 2022 at 00:36:10. See the history of this page for a list of all contributions to it.