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A universal property of the monoidal 2category of cospans of ordinals and surjections
 Theory and Applications of Categories
, 2007
"... Abstract. We prove that the monoidal 2category of cospans of ordinals and surjections is the universal monoidal category with an object X with a semigroup and a cosemigroup structures, where the two structures satisfy a certain 2dimensional separable algebra condition. 1. ..."
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Abstract. We prove that the monoidal 2category of cospans of ordinals and surjections is the universal monoidal category with an object X with a semigroup and a cosemigroup structures, where the two structures satisfy a certain 2dimensional separable algebra condition. 1.
A universal property of the monoidal 2category of cospans of finite linear orders and surjections
, 706
"... We prove that the monoidal 2category of cospans of finite linear orders and surjections is the universal monoidal category with an object X with a semigroup and a cosemigroup structures, where the two structures satisfy a certain 2dimensional separable algebra condition. 1 ..."
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We prove that the monoidal 2category of cospans of finite linear orders and surjections is the universal monoidal category with an object X with a semigroup and a cosemigroup structures, where the two structures satisfy a certain 2dimensional separable algebra condition. 1
unknown title
, 2009
"... Cospans and spans of graphs: a categorical algebra for the sequential and parallel composition of discrete systems ..."
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Cospans and spans of graphs: a categorical algebra for the sequential and parallel composition of discrete systems
unknown title
, 801
"... Abstract. It is well known that strong monoidal functors preserve duals. In this short note we show that a slightly weaker version of functor, which we call “Frobenius monoidal”, is sufficient. The idea of this note became apparent from Prop. 2.8 in the paper of R. Rosebrugh, N. Sabadini, and R.F.C. ..."
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Abstract. It is well known that strong monoidal functors preserve duals. In this short note we show that a slightly weaker version of functor, which we call “Frobenius monoidal”, is sufficient. The idea of this note became apparent from Prop. 2.8 in the paper of R. Rosebrugh, N. Sabadini, and R.F.C. Walters [4]. Throughout suppose that A and B are strict 1 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 for all A, B, C ∈ A. ir = (1 ⊗ r)(i ⊗ 1) : F(A ⊗ B) ⊗ FC ir = (r ⊗ 1)(1 ⊗ i) : FA ⊗ F(B ⊗ C) � � FA ⊗ F(B ⊗ C) � � F(A ⊗ B) ⊗ FC, 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 [5]. A dual situation in A is a tuple (A, B, e, n), where A and B are objects of A and e: A ⊗ B � � I n: I � � B ⊗ A are morphisms in A, called evaluation and coevaluation respectively, satisfying the “triangle identities”: 1⊗n n⊗1
The parallel composition of processes
, 904
"... Abstract. We suggest that the canonical parallel operation of processes is composition in a wellsupported compact closed category of spans of reflexive graphs. We present the parallel operations of classical process algebras as derived operations arising from monoid objects in such a ..."
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Abstract. We suggest that the canonical parallel operation of processes is composition in a wellsupported compact closed category of spans of reflexive graphs. We present the parallel operations of classical process algebras as derived operations arising from monoid objects in such a
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
FROBENIUS OBJECTS IN CARTESIAN BICATEGORIES
"... Abstract. Maps (left adjoint arrows) between Frobenius objects in a cartesian bicategory B are precisely comonoid homomorphisms and, for A Frobenius and any T in B, Map(B)(T, A) is a groupoid. 1. ..."
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Abstract. Maps (left adjoint arrows) between Frobenius objects in a cartesian bicategory B are precisely comonoid homomorphisms and, for A Frobenius and any T in B, Map(B)(T, A) is a groupoid. 1.