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Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory
 Mem. Amer. Math. Soc
"... The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2category theory, such as versions of algebra, limit, colimit, and adjun ..."
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Cited by 19 (8 self)
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The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2category theory, such as versions of algebra, limit, colimit, and adjunction, are necessary for this
Linearlyused continuations in an enriched effect calculus
 In preparation
, 2009
"... Abstract. The enriched effect calculus is an extension of Moggi’s computational metalanguage with a selection of primitives from linear logic. In this paper, we present an extended case study within the enriched effect calculus: the linear usage of continuations. We show that established callbyval ..."
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Cited by 6 (4 self)
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Abstract. The enriched effect calculus is an extension of Moggi’s computational metalanguage with a selection of primitives from linear logic. In this paper, we present an extended case study within the enriched effect calculus: the linear usage of continuations. We show that established callbyvalue and callby name linearlyused CPS translations are uniformly captured by a single generic translation of the enriched effect calculus into itself. As a main syntactic theorem, we prove that the generic translation is involutive up to isomorphism. As corollaries, we obtain full completeness results for the original callbyvalue and callbyname translations. The main syntactic theorem is proved using a categorytheoretic semantics for the enriched effect calculus. We show that models are closed under a natural dual model construction. The canonical linearlyused CPS translation then arises as the unique (up to isomorphism) map from the syntactic initial model to its own dual. This map is an equivalence of models. Thus the initial model is selfdual. 1
of A and
, 2008
"... Abstract. It is well known that strong monoidal functors preserve duals. In this short note we show that a weaker version of functor, which we call “Frobenius monoidal”, is sufficient. Further properties of Frobenius monoidal functors are developed. The idea of this note became apparent from Proposi ..."
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Abstract. It is well known that strong monoidal functors preserve duals. In this short note we show that a weaker version of functor, which we call “Frobenius monoidal”, is sufficient. Further properties of Frobenius monoidal functors are developed. The idea of this note became apparent from Proposition 2.8 in the paper of R. Rosebrugh, N. Sabadini, and R.F.C. Walters [5]. Throughout suppose that A and B are strict1 monoidal categories. Definition 1. A Frobenius monoidal functor is a functor F: A ��B which is monoidal (F, r, r0) and comonoidal (F, i, i0), and satisfies the compatibility conditions ir =(1⊗r)(i ⊗ 1) : F (A ⊗ B) ⊗ FC ��FA ⊗ F (B ⊗ C) ir =(r ⊗ 1)(1 ⊗ i):FA ⊗ F (B ⊗ C) ��F (A ⊗ B) ⊗ FC, for all A, B, C ∈ A. The compact case ( ⊗ = ⊕) of Cockett and Seely’s linearly distributive functors [2] are precisely Frobenius monoidal functors, and Frobenius monoidal functors with ri = 1 have been called split monoidal by Szlachányi in [6]. A dual situation in A is a tuple (A, B, e, n), where A and B are objects
of A and
, 2008
"... Abstract. It is well known that strong monoidal functors preserve duals. In this short note we show that a weaker version of functor, which we call “Frobenius monoidal”, is sufficient. Further properties of Frobenius monoidal functors are developed. The idea of this note became apparent from Proposi ..."
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Abstract. It is well known that strong monoidal functors preserve duals. In this short note we show that a weaker version of functor, which we call “Frobenius monoidal”, is sufficient. Further properties of Frobenius monoidal functors are developed. The idea of this note became apparent from Proposition 2.8 in the paper of R. Rosebrugh, N. Sabadini, and R.F.C. Walters [5]. Throughout suppose that A and B are strict1 monoidal categories. Definition 1. A Frobenius monoidal functor is a functor F: A ��B which is monoidal (F, r, r0) and comonoidal (F, i, i0), and satisfies the compatibility conditions ir =(1⊗r)(i ⊗ 1) : F (A ⊗ B) ⊗ FC ��FA ⊗ F (B ⊗ C) ir =(r ⊗ 1)(1 ⊗ i):FA ⊗ F (B ⊗ C) ��F (A ⊗ B) ⊗ FC, for all A, B, C ∈ A. The compact case ( ⊗ = ⊕) of Cockett and Seely’s linearly distributive functors [2] are precisely Frobenius monoidal functors, and Frobenius monoidal functors with ri = 1 have been called split monoidal by Szlachányi in [6]. A dual situation in A is a tuple (A, B, e, n), where A and B are objects
Example
, 2004
"... Recall the ordinary notion of Frobenius algebra over a field k. Step 2 Lift the concept from linear algebra to a general monoidal category and justify this with examples and theorems. Step 3 Lift the concept up a dimension so that monoidal categories themselves can be examples. 1 Frobenius algebras ..."
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Recall the ordinary notion of Frobenius algebra over a field k. Step 2 Lift the concept from linear algebra to a general monoidal category and justify this with examples and theorems. Step 3 Lift the concept up a dimension so that monoidal categories themselves can be examples. 1 Frobenius algebras An algebra A over a field k is called Frobenius when it is finite dimensional and equipped with a linear function e:A æÆ æ k such that: e ( ab) = 0 for all a ŒA implies b = 0.