Dirichlet Polynomials form a Topos

Spivak, David I.; Myers, David Jaz

doi:10.48550/arxiv.2003.04827

Cited by 3 publications

(4 citation statements)

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“…Following [SM20], we can think of Dirichlet polynomials as functors FSet op → FSet, where FSet is the category of finite sets. Indeed, given a natural number n ∈ N, the exponential n y can be thought of as the Yoneda embedding of the set with n elements 2 , i.e.…”

Section: Length Width Widthmentioning

confidence: 99%

“…A brief outline of this paper is as follows: § 2: We recall the definitions of Dirichlet polynomials 1 and set-theoretic bundles, along with their rig structures, from [SM20]; we then study the equivalence between these two notions. § 3: We explain how empirical probability distributions correspond to set-theoretic bundles (and thus to Dirichlet polynomials).…”

Section: Introductionmentioning

confidence: 99%

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Dirichlet polynomials and entropy

Spivak,

Hosgood

2021

Preprint

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A Dirichlet polynomial d in one variable y is a function of the form d(y) = a n n y + • • • + a 2 2 y + a 1 1 y + a 0 0 y for some n, a 0 ,... , a n ∈ N. We will show how to think of a Dirichlet polynomial as a set-theoretic bundle, and thus as an empirical distribution. We can then consider the Shannon entropy H(d) of the corresponding probability distribution, and we define its length (or, classically, its perplexity) by L(d) = 2 H (d) . On the other hand, we will define a rig homomorphism h : Dir → Rect from the rig of Dirichlet polynomials to the so-called rectangle rig, whose underlying set is R 0 × R 0 and whose additive structure involves the weighted geometric mean; we write h(d) = (A(d),W(d)), and call the two components area and width (respectively).The main result of this paper is the following: the rectangle-area formula A(d) = L(d)W(d) holds for any Dirichlet polynomial d. In other words, the entropy of an empirical distribution can be calculated entirely in terms of the homomorphism h applied to its corresponding Dirichlet polynomial. We also show that similar results hold for the cross entropy.

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Section: Length Width Widthmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dirichlet polynomials and entropy

Spivak,

Hosgood

2021

Preprint

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“…Alternatively, on the multiplicative side, one might want infinite products and a complete distributivity law over infinite coproducts. This type of '2-rig' would be germane to the study of polynomial functors in the sense of [20,31,50,51,49], which have provided a unifying setting for studying numerous structures in applied category theory. Given the multiplicity of possible definitions of 2-rig, we believe it makes sense not to fix a single notion of 2-rig but to be flexible and contemplate a whole spectrum of possible theories, or 'doctrines' of 𝑫-rigs parametrized by 𝑫, a 2-monad on Cat (locally small categories) whose algebras will possess colimits of a certain shape.…”

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confidence: 99%

Differential 2-rigs

Loregian

Trimble

2023

Electron. Proc. Theor. Comput. Sci.

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We study the notion of a differential 2-rig, a category R with coproducts and a monoidal structure distributing over them, also equipped with an endofunctor 𝜕 : R → R that satisfies a categorified analogue of the Leibniz rule. This is intended as a tool to unify various applications of such categories to computer science, algebraic topology, and enumerative combinatorics. The theory of differential 2-rigs has a geometric flavour but boils down to a specialization of the theory of tensorial strengths on endofunctors; this builds a surprising connection between apparently disconnected fields. We build free 2-rigs on a signature, and we prove various initiality results: for example, a certain category of colored species is the free differential 2-rig on a single generator.3For the sake of completeness, we shall mention yet another approach to 'categorical differentiation' recently developed in [54] with applications to ZX calculus in mind; again, there seems to be no relation with our theory of differential 2-rigs, since derivations on their category Mat-𝑆 are not Leibniz on objects.

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“…The reason for the name Dirichlet is that if one replaces polynomials with Dirichlet series by reversing each summand y A to A y , the result is the usual product. For example (3 y + 2 y ) × (4 y + 0 y ) 12 y + 8 y + 2•0 y See[SM20] for more on the connection between Dirichlet series and polynomials.…”

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confidence: 99%

Poly: An abundant categorical setting for mode-dependent dynamics

Spivak¹

2020

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Dynamical systems-by which we mean machines that take time-varying input, change their state, and produce output-can be wired together to form more complex systems. Previous work has shown how to allow collections of machines to reconfigure their wiring diagram dynamically, based on their collective state. This notion was called "mode dependence", and while the framework was compositional (forming an operad of re-wiring diagrams and algebra of mode-dependent dynamical systems on it), the formulation itself was more "creative" than it was natural.In this paper we show that the theory of mode-dependent dynamical systems can be more naturally recast within the category Poly of polynomial functors. This category is almost superlatively abundant in its structure: for example, it has four interacting monoidal structures (+, ×, ⊗, •), two of which (×, ⊗) are monoidal closed, and the comonoids for • are precisely categories in the usual sense. We discuss how the various structures in Poly show up in the theory of dynamical systems. We also show that the usual coalgebraic formalism for dynamical systems takes place within Poly. Indeed one can see coalgebras as special dynamical systems-ones that do not record their history-formally analogous to contractible groupoids as special categories. Introduction to mode-dependence Plant Cy ABController By C System Cy A .(2)Observe that in each case the output type is the coefficient on y, and the input type is the exponent on y. In Section 3.3 we will see that the wiring diagram (1) itself, as well as the interacting dynamics, can be represented by morphisms involving these polynomials.

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Dirichlet Polynomials form a Topos

Cited by 3 publications

References 2 publications

Dirichlet polynomials and entropy

Dirichlet polynomials and entropy

Differential 2-rigs

Poly: An abundant categorical setting for mode-dependent dynamics

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