We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A = F-Vect, where F is a field. An object X ∈ A with two-sided dual X gives rise to a Frobenius algebra in A, and under weak additional conditions we prove a converse: There exists a bicategory E with ObjE = {A, B} such that EndE(A) ⊗ ≃ A and such that there are J, J: B ⇋ A producing the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity,...) of A carry over to the bicategory E. We define weak monoidal Morita equivalence of tensor categories, denoted A ≈ B, and establish a correspondence between Frobenius algebras in A and tensor categories B ≈ A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence A ≈ B has remarkable consequences: A and B have equivalent (as braided tensor categories) quantum doubles (‘centers’) and (if A, B are semisimple spherical or ∗-categories) have equal dimensions and give rise the same state sum invariant of closed oriented 3-manifolds as recently defined by Barrett and Westbury. An instructive example is provided by finite dimensional semisimple and cosemisimple Hopf algebras, for which we prove H − mod ≈ ˆH − mod. The present formalism permits a fairly complete analysis of the center of a semisimple spherical category, which is the subject of the companion paper math.CT/0111205. 1

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Abstract. Mednykh [Me78] proved that for any finite group G and any orientable surface S, there is a formula for #Hom(π1(S), G) in terms of the Euler characteristic of S and the dimensions of the irreducible representations of G. A similar formula in the nonorientable case was proved by Frobenius and Schur [FS06]. Both of these proofs use character theory and an explicit presentation for π1. These results have been reproven using quantum field theory ([FQ93], [MY05], and others). Here we present a greatly simplified proof of these results which uses only elementary topology and combinatorics. The main tool is an elementary invariant of surfaces attached to a semisimple algebra called a lattice topological quantum field theory. 1.

I dedicate this paper to a man who throughout his career has exemplified the power of conceptual thought in math: Bob Moody. In 1978, John McKay made an intriguing observation: 196 884 ≈ 196 883. Monstrous Moonshine is the collection of questions (and a few answers) inspired by this observation. In this paper we provide a few snapshots of what we call the underlying theory. But first we digress with a quick and elementary review. By a lattice in C we mean a discrete subgroup of C under addition. We can always express this (nonuniquely) as the set of points Zw + Zz def = Λ{w, z}. We dismiss as too degenerate the lattice Λ = {0}. Call two lattices Λ, Λ ′ similar if they fall into each other once the plane C is rescaled and rotated about the origin — i.e. Λ ′ = αΛ for some nonzero α ∈ C. In Figure 1 we draw (parts of) two similar lattices. For another example, consider the degenerate case where w and z are linearly dependent over R: then in fact w and z are linearly dependent over Z (otherwise discreteness would be lost) and any such lattice is similar to Z ⊂ C. 01

Abstract. The interface between classical physics and quantum physics is explained from the point of view of Quantum Information Theory (Feynman Processes), based on the qubit model. The interpretation depends on a hefty sacrifice: the classical determinism or the arrow of time. As a benefit, the wave-particle duality naturally emerges from the qubit model, as the root of creation and annihilation of possibilities (quantum logic). A few key experiments are briefly reviewed from the above perspective: quantum erasure, delayed-choice and wave-particle correlation. The CPT-Theorem is interpreted in the framework of categories with duality and a timeless interpretation of the Feynman Processes is proposed. A connection between the fine-structure constant and algebraic number theory is