Categorical Aspects of Generating Functions (I): Exponential Formulas and Krull–Schmidt Categories
✍ Scribed by Tomoyuki Yoshida
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 260 KB
- Volume
- 240
- Category
- Article
- ISSN
- 0021-8693
No coin nor oath required. For personal study only.
✦ Synopsis
In this paper, we study formal power series with exponents in a category. For example, the generating function of a category E E with finite hom sets is defined by Ž .
X < Ž .< E E t s Ýt r Aut X , where the summation is taken over all isomorphism classes of objects of E E. We can use such power series to enumerate the number of Ž . E E-structures along a faithful functor Theorem 4.6 . Our theory is closely related to Ž . the theory of species Joyal, 1981 . A species can be identified with a faithful Ž . functor from a groupoid to the category of finite sets Theorem 3.6 . We use mainly the concept of faithful functors with finite fibers instead of species, so that we can separate the roles categories and functors play. For example, the exponential Ž . Ž Ž .Ž .. formula E E t s exp Con E E t means the unique coproduct decomposition prop-Ž . erty Theorem 5.8 . In the final section, we give some applications of our theory to rather classical enumerations.