Dirichlet polynomials form a topos

David I. Spivak and David Jaz Myers – GJM, Volume 8, Issue 2 (2023), 73-87.

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One can think of power series or polynomials in one variable, such as P(y)=2 y^3+ y+5    as covariant functors from the category   Set  of sets to itself; these are known as polynomial functors. In this paper, we will explore the contravariant analogue of polynomial functors which we call Dirichlet functors, since they are to Dirichlet series as polynomial functors are to power series. 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. We also characterize Dirichlet functors as those contravariant functors that commute with connected limits, dualizing the well-known similar characterization of polynomial functors. Finally, we generalize these results to contravariant functors of homotopy types, allowing us to categorify Dirichlet series with their usual negative exponents.

Milestones:

Received: Mars 28, 2023
Accepted: May 06, 2023
Revised: November 19, 2023

Authors:

David I. Spivak
Topos Institute,
2140 Shattuck Ave. # 610, Berkeley, CA 94704, USA.
David Jaz Myers
Department of Mathematics,
New York University - Abu Dhabi PO Box 129188, Abu Dhabi, UAE.

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