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COMMUTATIVE MONADS AS A THEORY OF DISTRIBUTIONS
"... Abstract. It is shown how the theory of commutative monads provides an axiomatic framework for several aspects of distribution theory in a broad sense, including probability distributions, physical extensive quantities, and Schwartz distributions of compact support. Among the particular aspects cons ..."
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Abstract. It is shown how the theory of commutative monads provides an axiomatic framework for several aspects of distribution theory in a broad sense, including probability distributions, physical extensive quantities, and Schwartz distributions of compact support. Among the particular aspects considered here are the notions of convolution, density, expectation, and conditional probability.
MFPS 2008 Semimodule enrichment
"... A category with biproducts is enriched over (commutative) additive monoids. A category with tensor products is enriched over scalar multiplication actions. A symmetric monoidal category with biproducts is enriched over semimodules. We show that these extensions of enrichment (e.g. from homsets to h ..."
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A category with biproducts is enriched over (commutative) additive monoids. A category with tensor products is enriched over scalar multiplication actions. A symmetric monoidal category with biproducts is enriched over semimodules. We show that these extensions of enrichment (e.g. from homsets to homsemimodules) are functorial, and use them to make precise the intuition that “compact objects are finitedimensional” in standard cases. Keywords: Semimodules, enriched categories, biproducts, scalar multiplication, compact objects.
Universal Properties of Impure Programming Languages
"... We investigate impure, callbyvalue programming languages. Our first language only has variables and letbinding. Its equational theory is a variant of Lambek’s theory of multicategories that omits the commutativity axiom. We demonstrate that type constructions for impure languages — products, sums ..."
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We investigate impure, callbyvalue programming languages. Our first language only has variables and letbinding. Its equational theory is a variant of Lambek’s theory of multicategories that omits the commutativity axiom. We demonstrate that type constructions for impure languages — products, sums and functions — can be characterized by universal properties in the setting of ‘premulticategories’, multicategories where the commutativity law may fail. This leads us to new, universal characterizations of two earlier equational theories of impure programming languages: the premonoidal categories of Power and Robinson, and the monadbased models of Moggi. Our analysis thus puts these earlier abstract ideas on a canonical foundation, bringing them to a new, syntactic level. F.3.2 [Semantics of Pro
Coherence for Monoidal Monads and Comonads
, 907
"... The goal of this paper is to prove coherence results with respect to relational graphs for monoidal monads and comonads, i.e. monads and comonads in a monoidal category such that the endofunctor of the monad or comonad is a monoidal functor (this means that it preserves the monoidal structure up to ..."
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The goal of this paper is to prove coherence results with respect to relational graphs for monoidal monads and comonads, i.e. monads and comonads in a monoidal category such that the endofunctor of the monad or comonad is a monoidal functor (this means that it preserves the monoidal structure up to a natural transformation that need not be an isomorphism). These results are proved first in the absence of symmetry in the monoidal structure, and then with this symmetry. The monoidal structure is also allowed to be given with finite products or finite coproducts. Monoidal comonads with finite products axiomatize a plausible notion of identity of deductions in a fragment of the modal logic S4.
Coherence for Monoidal Endofunctors
, 907
"... The goal of this paper is to prove coherence results with respect to relational graphs for monoidal endofunctors, i.e. endofunctors of a monoidal category that preserve the monoidal structure up to a natural transformation that need not be an isomorphism. These results are proved first in the absenc ..."
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The goal of this paper is to prove coherence results with respect to relational graphs for monoidal endofunctors, i.e. endofunctors of a monoidal category that preserve the monoidal structure up to a natural transformation that need not be an isomorphism. These results are proved first in the absence of symmetry in the monoidal structure, and then with this symmetry. In the later parts of the paper the coherence results are extended to monoidal endofunctors in monoidal categories that have diagonal or codiagonal natural transformations, or where the monoidal structure is given by finite products or coproducts. Monoidal endofunctors are interesting because they stand behind monoidal monads and comonads, for which coherence will be proved in a sequel to this paper.
TENSORS, MONADS AND ACTIONS Dedicated to the memory of Pawel Waszkiewicz
"... We exhibit sufficient conditions for a monoidal monad T on a monoidal ..."
Vol. XXIII. 19.72 113 Strong Functors and Monoidal Monads By
"... In [4] we proved that a commutative monad on a symmetric monoidal closed category carries the structure of a symmetric monoidal monad ([4], Theorem 3.2). We here prove the converse, so that, taken together, we have: there is a 11 correspondence between commutative monads and symmetric monoidal mona ..."
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In [4] we proved that a commutative monad on a symmetric monoidal closed category carries the structure of a symmetric monoidal monad ([4], Theorem 3.2). We here prove the converse, so that, taken together, we have: there is a 11 correspondence between commutative monads and symmetric monoidal monads (Theorem 2.3 below). The main computational work needed consists in constructing an equivalence between possible strengths 8tA,B: A c ~ B+ A T ~ B T on a functor, and possible "tensorial stren~hs " on T t"X,B: X ( ~ BT> (X ( ~ B) T; T is assumed to be a functor between categories tensored over a monoidal closed category 3~'. The equivalence is stated in Theorem 1.3. (There is a similar theorem for the notion of eotensorial strength Ax,B: (Xt ~ B) T+ Xr B T, which we do not include in this note.) As an application of the theory here, we construct strength on certain functors related to the power set monad. If ~r is a 3~category, we use t ~ to denote the homfunctor ~r x ~r as well as to denote the homfunctor of 3r ~ itself. 1. Making a functor strong. Let ~r and ~ be categories tensored over the symmetric monoidal closed ~r [3]. Let T: ~0> ~0 be a functor between the underlying categories. To a family of maps (1.1) 8tA,A,: Ac~A'> A Tc~A ' T we associate a family of maps (1.2) t"X,A: X (D A T> ( X @ A) T by commutativity of (1.3) ua