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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.
Contents CALCULATING COLIMITS COMPOSITIONALLY
, 712
"... ABSTRACT. We show how finite limits and colimits can be calculated compositionally using the algebras of spans and cospans, and give as an application a proof of the Kleene ..."
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Cited by 2 (2 self)
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ABSTRACT. We show how finite limits and colimits can be calculated compositionally using the algebras of spans and cospans, and give as an application a proof of the Kleene
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.
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
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
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
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
TANGLED CIRCUITS
"... Abstract. We consider commutative Frobenius algebras in braided strict monoidal categories in the study of the circuits and communicating systems which occur in Computer Science, including circuits in which the wires are tangled. We indicate also some possible novel geometric interest in such algebr ..."
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Abstract. We consider commutative Frobenius algebras in braided strict monoidal categories in the study of the circuits and communicating systems which occur in Computer Science, including circuits in which the wires are tangled. We indicate also some possible novel geometric interest in such algebras. For example, we show how Armstrong’s description ([1, 2]) of knot colourings and knot groups fit into this context. 1.