论文标题
dirichlet多项式形成topos
Dirichlet Polynomials form a Topos
论文作者
论文摘要
可以将一个变量中的功率序列或多项式考虑，例如$ p（x）= 2x^3+x+5 $，是类别$ \ mathsf {set} $ sets set of to to本身的函数；这些被称为多项式函数。用$ \ mathsf {poly} _ {\ mathsf {set}} $在$ \ mathsf {set set} $上及其之间的自然变换上的多项式函数类别。 $ p（x）$中发生的常数$ 0,1 $和操作$+，\ times $实际上是初始对象和终端对象，而$ \ mathsf {poly} _ {\ mathsf {set set}} $中的coproduct and product中 正如$ \ mathsf {set} $上的多项式函数是可以写入代表成分的总和一样，也可以表达任何dirichlet系列，例如，例如\ $ \ sum_ {n = 0}^\ infty n^x $，作为代表性的预示。 DIRICHLET多项式是一个有限的Dirichlet系列，即代表$ n^x $的有限总和。我们讨论了如何从捆绑中理解多项式函数及其dirichlet类似物，并继续证明dirichlet多项式类别是基本的topos。
One can think of power series or polynomials in one variable, such as $P(x)=2x^3+x+5$, as functors from the category $\mathsf{Set}$ of sets to itself; these are known as polynomial functors. Denote by $\mathsf{Poly}_{\mathsf{Set}}$ the category of polynomial functors on $\mathsf{Set}$ and natural transformations between them. The constants $0,1$ and operations $+,\times$ that occur in $P(x)$ are actually the initial and terminal objects and the coproduct and product in $\mathsf{Poly}_{\mathsf{Set}}$. Just as the polynomial functors on $\mathsf{Set}$ are the copresheaves that can be written as sums of representables, one can express any Dirichlet series, e.g.\ $\sum_{n=0}^\infty n^x$, as a coproduct of representable presheaves. A Dirichlet polynomial is a finite Dirichlet series, that is, a finite sum of representables $n^x$. We discuss how both polynomial functors and their Dirichlet analogues can be understood in terms of bundles, and go on to prove that the category of Dirichlet polynomials is an elementary topos.
