## Involutive categories and monoids, with a GNS-correspondence (2010)

Venue: | In Quantum Physics and Logic (QPL |

Citations: | 4 - 2 self |

### BibTeX

@INPROCEEDINGS{Jacobs10involutivecategories,

author = {Bart Jacobs},

title = {Involutive categories and monoids, with a GNS-correspondence},

booktitle = {In Quantum Physics and Logic (QPL},

year = {2010}

}

### OpenURL

### Abstract

This paper develops the basics of the theory of involutive categories and shows that such categories provide the natural setting in which to describe involutive monoids. It is shown how categories of Eilenberg-Moore algebras of involutive monads are involutive, with conjugation for modules and vector spaces as special case. The core of the so-called Gelfand-Naimark-Segal (GNS) construction is identified as a bijective correspondence between states on involutive monoids and inner products. This correspondence exists in arbritrary involutive symmetric monoidal categories. 1

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Citation Context ...gates. Lemma 3.3 Let C be an involutive category; SC(C) inherits all limits and colimits that exist in C, and the forgetful functor SC(C) → C preserves them. For the record we note the following (see =-=[18, 7]-=- for background). Lemma 3.4 The mapping C ↦→ SC(C) is a 2-functor ICat → ICat, and even a 2comonad.� �� �� � � � � �� � �� Involutive Categories and Monoids, with a GNS-correspondence 5 4 Involutive ... |

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Citation Context ...ly, for a Boolean algebra B, negation ¬ yields a self-conjugate ¬: B ∼ = → B in the category of Boolean algebras. There are similar self-conjugates via orthosupplements (−) ⊥ in orthomodular lattices =-=[13]-=- and effect algebras [9]. In Cat a self-conjugate is given by a self-dual category C op ∼ = C. Recall the conjugation on vector spaces. Suppose V ∈ VectC has a basis (vi)i∈I. Then we can define a self... |

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Citation Context ... B, negation ¬ yields a self-conjugate ¬: B ∼ = → B in the category of Boolean algebras. There are similar self-conjugates via orthosupplements (−) ⊥ in orthomodular lattices [13] and effect algebras =-=[9]-=-. In Cat a self-conjugate is given by a self-dual category C op ∼ = C. Recall the conjugation on vector spaces. Suppose V ∈ VectC has a basis (vi)i∈I. Then we can define a self-conjugate V ∼ ( ∑ = → V... |

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Citation Context ...lso be defined on a category. It then consists of an endofunctor C → C, which is typically written as X ↦→ X. It should satisfy X ∼ = X. Involutive categories occur in the literature, for instance in =-=[6, 1]-=-, but have not been studied very extensively. This paper will develop the basic elements of such a theory of involutive categories. Its main technical contribution is a bijective correspondence betwee... |

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Citation Context ...uld like to show that these categories of algebras of an involutive monoidal monad are also involutive monoidal categories. The monoidal structure is given by the standard construction of Anders Kock =-=[15, 14]-=-. Tensors of algebras exist in case certain colimits exist. This is always the case with monads on sets, due to a result of Linton’s, see [5, § 9.3, Prop. 4]. Theorem 6.2 Suppose T is an involutive mo... |

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Citation Context ...id in such categories of algebras. Pre C∗-algebras (without norm) are such monoids. Once this setting has been established we take a special look at the famous GelfandNaimark-Segal (GNS) construction =-=[3]-=-. It relates C∗-algebras and Hilbert spaces, and shows in particular how a state A → C on a C∗-algebra gives rise to an inner product on A. Using conjugation as involution, the latter can be described... |

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Citation Context ...uld like to show that these categories of algebras of an involutive monoidal monad are also involutive monoidal categories. The monoidal structure is given by the standard construction of Anders Kock =-=[15, 14]-=-. Tensors of algebras exist in case certain colimits exist. This is always the case with monads on sets, due to a result of Linton’s, see [5, § 9.3, Prop. 4]. Theorem 6.2 Suppose T is an involutive mo... |

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Citation Context ...states and inner products, using the language of involutive categories and monoids.876 Found Phys (2012) 42:874–895 In what follows we assume familiarity with basic category theory, see for instance =-=[4, 10, 24]-=-. 2 Involutive Categories Definition 1 A category C will be called involutive if it comes with a ‘involution’ functor C → C, written as X ↦→ X, and a natural isomorphism ι: id ⇒ (−) with components ιX... |

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Citation Context ...ry, the notion of involutive monoid requires an appropriate notion of involutive monoidal category. This is in line with the “microcosm principle”, formulated by Baez and Dolan [4], and elaborated in =-=[12, 11, 10]-=-: it involves “outer” structure (like monoidal structure 1 I → C ⊗ ← C × C on a category C) that enables the definition of “inner” structure (like a monoid I 0 → M + ← M ⊗ M in C). We briefly illustra... |

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Citation Context ...ry, the notion of involutive monoid requires an appropriate notion of involutive monoidal category. This is in line with the “microcosm principle”, formulated by Baez and Dolan [4], and elaborated in =-=[12, 11, 10]-=-: it involves “outer” structure (like monoidal structure 1 I → C ⊗ ← C × C on a category C) that enables the definition of “inner” structure (like a monoid I 0 → M + ← M ⊗ M in C). We briefly illustra... |

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Citation Context ...ry, the notion of involutive monoid requires an appropriate notion of involutive monoidal category. This is in line with the “microcosm principle”, formulated by Baez and Dolan [4], and elaborated in =-=[12, 11, 10]-=-: it involves “outer” structure (like monoidal structure 1 I → C ⊗ ← C × C on a category C) that enables the definition of “inner” structure (like a monoid I 0 → M + ← M ⊗ M in C). We briefly illustra... |

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Citation Context ...∈ Q in a + ib) form an involutive semiring, albeit not a complete one. The multiset monad MS : Sets → Sets associated with S is defined on a set X as: MS(X) = {ϕ: X → S | supp(ϕ) is finite}, see e.g. =-=[8]-=-. The category of algebras of this monad is the category ModS of modules over S. This monad is monoidal / commutative, because S is commutative. It is involutive, with involution ν : MS(X) → MS(X) giv... |

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categories and star operations
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Citation Context ...o be defined on a category C. It then consists of an endofunctor C → C, which is typically written as X ↦→ X. It should satisfy X ∼ = X. Involutive categories occur in the literature, for instance in =-=[1, 7, 13]-=-, but have not been studied very extensively. This paper will develop the basic elements of such a theory of involutive categories. Its main technical contribution is a bijective correspondence B. Jac... |

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Citation Context ...o be defined on a category C. It then consists of an endofunctor C → C, which is typically written as X ↦→ X. It should satisfy X ∼ = X. Involutive categories occur in the literature, for instance in =-=[1, 7, 13]-=-, but have not been studied very extensively. This paper will develop the basic elements of such a theory of involutive categories. Its main technical contribution is a bijective correspondence B. Jac... |

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Citation Context ...ting maps X → S Y and Y → S X factor as Kleisli maps X → MS(Y ) and Y → MS(X). This yields a dagger category with (dagger) tensors and biproducts, whose set of scalars is S itself. See [19], and also =-=[9, 18]-=-, for more details. 7 Inner Product Spaces and the GNS-Construction In this final section we consider inner product spaces V , with an inner product operation 〈− | −〉: V ⊗ V → X. The input type V ⊗ V ... |

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Citation Context ...t the two resulting maps X → S Y and Y → S X factor as Kleisli maps X → MS(Y ) and Y → MS(X). This yields a dagger category with (dagger) tensors and biproducts, whose set of scalars is S itself. See =-=[19]-=-, and also [9, 18], for more details. 7 Inner Product Spaces and the GNS-Construction In this final section we consider inner product spaces V , with an inner product operation 〈− | −〉: V ⊗ V → X. The... |

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categories and star operations. Algebras and Representation Theory
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Citation Context ...lso be defined on a category. It then consists of an endofunctor C → C, which is typically written as X ↦→ X. It should satisfy X ∼ = X. Involutive categories occur in the literature, for instance in =-=[6, 1]-=-, but have not been studied very extensively. This paper will develop the basic elements of such a theory of involutive categories. Its main technical contribution is a bijective correspondence betwee... |

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(Show Context)
Citation Context ...ting maps X → S Y and Y → S X factor as Kleisli maps X → MS(Y ) and Y → MS(X). This yields a dagger category with (dagger) tensors and biproducts, whose set of scalars is S itself. See [19], and also =-=[9, 18]-=-, for more details. 7 Inner Product Spaces and the GNS-Construction In this final section we consider inner product spaces V , with an inner product operation 〈− | −〉: V ⊗ V → X. The input type V ⊗ V ... |