categorical aspects of generating functions

8/8/2019 Categorical Aspects of Generating Functions

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.Journal of Algebra 240, 4082 2001

doi:10.1006rjabr.2000.8602, available online at http:rrwww.idealibrary.com on

.Categorical Aspects of Generating Functions I :

Exponential Formulas and

KrullSchmidt Categories

Tomoyuki Yoshida

Department of Mathematics, Hokkaido Uni ersity, Kita-10, Nishi-8,Sapporo 060-0810, Japan

E-mail: [email protected]

Communicated by Walter Feit

Received October 12, 1999

DEDICATED TO THE MEMORY OF PROFESSOR MICHIO SUZUKI

In this paper, we study formal power series with exponents in a category. Forexample, the generating function of a category EE with finite hom sets is defined by

. X < .


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CATEGORICAL ASPECTS OF GENERATING FUNCTIONS 41

.C Operations on categories and functors.

.D Polynomials and power series with exponents in locally finitetoposes.

.E Polynomials and power series with coefficients in a Mackeyfunctor with multiplicative induction.

In the present paper we study topics A and B, in the next paper we willstudy topic C, and in the third paper we will study topics D and E. Thepurpose of topics D and E, which also are the final purpose of this paper,is to build the external theory of polynomials and power series. Here, forexample, the external integer ring with respect to a category Set G of finitefG-sets and G-maps is nothing but the Burnside ring of the finite group G,that is, the Grothendieck ring of Set G with respect to disjoint union andf

w xcartesian product 15, 51 . On the other hand, the internal integer ringwith respect to Set G is the rational integer ring with trivial G-action whichis defined by an internal Peano axiom and internal Grothendieck construc-tion. Of course, the external integer ring with respect to Set and thefinternal integer ring inside Set are both isomorphic to the rational integer

w xring Z; see MacLane and Moerdijks book 31, Sect. V.5 .

1.2. Species and EE Structures

The most famous and successful abstract theory of generating functionsw x w xis Joyals theory of species 26, 7 and its generalizations 6, 36 , which

affect also the present paper. Let Bij be the category of finite sets andfbijections, and let Set be the category of finite sets and maps. Then af

species A is now a functor from Bij to Set . An element of the imagef fw xA E of E under A is called an A-structure on the label set E. Further-

more, its generating function is defined by

w xA nnA t [ t . .

n!ns0

w x w x 4Here A n denotes the value at the n-point set n [ 1 , 2 , . . . , n .On the other hand, for a faithful functor F: EESS with finite fibers

< y1 .


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TOMOYUKI YOSHIDA42

.a bijective correspondence Example 2.10

G 66w xStr Set r n r( Hom G , S . . .f n

Furthermore, if EEs RTree is the category of rooted trees and F: RTree Set is the forgetful functor which assigns a rooted tree to its vertexfset, then an isomorphism class of RTree-structures on N bijectivelycorresponds to a labeled rooted tree on N.

THEOREM 3.6. Let S: GG Set be a faithful functor with finite fibersffrom a groupoid GG. Then the assignment

6

A : N Str EErN r( , Ng Bij , .S f

is a species. This construction gi es a bijecti e correspondence between aspecies and a faithful functor with finite fibers from a groupoid to Set .f

For example, the functor corresponding to the species of graphs is thefunctor from the groupoid of graphs and graph isomorphisms to Setfwhich assigns its vertex set to a graph. Here note that the same species is

also obtained from the category of graphs and all graph homomorphisms.Theorem 3.6 states that there exists an equivalence

6 : GpdrSet Bij , Set . f f f f f f

between the category of species and the category of faithful functors withfinite fibers from groupoids to Set . This equivalence is reminiscent of anf

w xequivalence familiar in sheaf theory 31, II.6; 25, 0.2, Chap. 3 ,

op( 6 : EtalerX Sheave X : OO X , Set , . .

where EtalerX is the category of local homeomorphisms over the space .X; see also 2.3.b .

The concept of faithful functors with finite fibers has some advantagesin comparison with that of species as follows:

1. The source category of the functor corresponding to a species viaTheorem 3.6 is a groupoid like Bij . However, this restriction is notfessential. Using faithful functors from any category with finite fibersinstead of species, we can generalize the theory of species. Particularly, we

w xcan take a locally finite topos as the source category 25, 31 .

2. Considering a faithful functor S: EESS instead of a species, wecan separate the roles that the source category EE and the functor S play.For example, the exponential formula essentially depends on the property

.that EE is a strict KS category see Section 5.5 ; for the functor S we needto assume only its additivity.


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CATEGORICAL ASPECTS OF GENERATING FUNCTIONS 43

3. When we want to substitute a natural number a for the variable t . nin the generating function A t of polynomial type, the value of t at t sa

w xshould be the number of mappings from an n-point set n to an a-point

w x < w x w x.

categorical aspects of generating functions

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