## Categorical aspects of polar decomposition (2010)

### BibTeX

@MISC{Heunen10categoricalaspects,

author = {Chris Heunen},

title = {Categorical aspects of polar decomposition},

year = {2010}

}

### OpenURL

### Abstract

Polar decomposition unquestionably provides a notion of factorization in the category of Hilbert spaces. But it does not fit existing categorical notions, mainly because its factors are not closed under composition. We observe that the factors are images of functors. This leads us to consider notions of factorization that emphasize reconstruction of the composite

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Citation Context ...rase this in terms of stratisfied factorization systems, instead of weak factorization systems. 7.1 Recall that the category Cat of categories is cocomplete. Colimits in it can be computed as follows =-=[12]-=-: 1. calculate the colimit of underlying graphs; 2. take the free category on the resulting graph; 3. quotient out the smallest congruence making the graph homomorphisms of the colimit cone into funct... |

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Citation Context ...dly, polar decomposition provides a novel example of a factorization that is of interest in its own right categorically, since factorization has been a topic of quite intense study in category theory =-=[11, 8, 13, 17, 23]-=-, not in the least because it is a fundamental part of abstract homotopy theory [21]. Although we are currently unaware of similar phenomena in other categories, we tentatively propose notions that si... |

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Citation Context ...dly, polar decomposition provides a novel example of a factorization that is of interest in its own right categorically, since factorization has been a topic of quite intense study in category theory =-=[11, 8, 13, 17, 23]-=-, not in the least because it is a fundamental part of abstract homotopy theory [21]. Although we are currently unaware of similar phenomena in other categories, we tentatively propose notions that si... |

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Citation Context ...em for categories of the form Hilb C skel. This can have interesting consequences for (topological or unitary conformal) quantum field theories, which can be formulated as functors into Hilbskel (see =-=[3]-=- and [24], respectively): apparently, such quantum field theories can be reconstructed from a functor into Pos and a functor into PInj. 7.12 Connecting to the analogy with prime factorization of natur... |

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Citation Context ...ve operators to act on, as in the following definition. 3.2 Definition The category Pos has as objects measure spaces, i.e. triples (S, Σ, µ) of a set S, a σ-algebra Σ on S, and a measure µ on (S, σ) =-=[10]-=-. An endomorphism on (S, Σ, µ) is an equivalence class of measurable functions f : S → R >0 whose range is compact, where two such functions are identified when they differ only on a negligible set. E... |

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Citation Context ...∑ y∈Y y∈Y x∈f −1 2 (y) y∈Y x∈f −1 2 (y) = ∑ |ϕ(f1(x))| 2 ≤ ∑ |ϕ(x)| 2 < ∞. x∈F x∈X |ϕ(f1(x))| 2 That this breaks down for functions f in general, instead of (partial) injections, was first noticed in =-=[4]-=-. Functoriality of ℓ 2 is easy to verify. 4.13 The following calculation shows that the ℓ2 functor preserves daggers. For a partial injection (X f1 F � f2 � 2 2 Y ), ϕ ∈ ℓ (X) and ψ ∈ ℓ (Y ): 〈(ℓ 2 f)... |

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Citation Context ...dly, polar decomposition provides a novel example of a factorization that is of interest in its own right categorically, since factorization has been a topic of quite intense study in category theory =-=[11, 8, 13, 17, 23]-=-, not in the least because it is a fundamental part of abstract homotopy theory [21]. Although we are currently unaware of similar phenomena in other categories, we tentatively propose notions that si... |

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Citation Context |

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Citation Context ...gory Hilb: (i) has (co)equalizers; (ii) does not have infinite (co)products; (iii) does not have directed (co)limits. Proof Part (i) holds because Hilb is enriched over abelian groups and has kernels =-=[15]-=-. For (ii), consider the following counterexample. Define an Nindexed family Xn = C of objects of Hilb. Suppose the family (Xn) had a coproduct X with coprojections κn : Xn → X. Define fn : Xn → C by ... |

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Citation Context ...n to the fact that the functor ℓ 2 is part of an equivalence between PInj and a category of Hilbert spaces with a chosen orthonormal basis whose morphisms are partial injections of those chosen bases =-=[1]-=-. 4.17 Proposition A morphism i: X → Y in Hilbskel is a partial isometry if and only if i = ℓ 2 f for a morphism f in PInj. Proof Clearly a map of the form ℓ 2 f is a partial isometry. Conversely, sup... |

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Citation Context ... right categorically, since factorization has been a topic of quite intense study in category theory [11, 8, 13, 17, 23], not in the least because it is a fundamental part of abstract homotopy theory =-=[21]-=-. Although we are currently unaware of similar phenomena in other categories, we tentatively propose notions that simultaneously fit polar decomposition and established categorical notions of factoriz... |

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Citation Context ...1 , j = , 1 0 2 ( ) 1 1 1 1 and take k the partial isometry got by factorizing . 1 1 1 0 It should be noted that other compositions of partial isometries, that do make them into a category, are known =-=[16]-=-. However, we follow a similar strategy as in Section 3, and ‘split out objects’ as follows. 4.2 Definition A partial injection is a partial function that is injective, wherever it is defined. More pr... |

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Citation Context .... We refrain from doing so here, because that is not so interesting from the viewpoint of reconstructing a category from factors. 7.10 The situation is not quite that of an amalgamation of categories =-=[20]-=-, and the functors L 2 and ℓ 2 are not embeddings. Nevertheless, it would be interesting to see for what refactorization systems the category C coincides with the pushout category P, i.e. when the squ... |

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Citation Context ...ategories of the form Hilb C skel. This can have interesting consequences for (topological or unitary conformal) quantum field theories, which can be formulated as functors into Hilbskel (see [3] and =-=[24]-=-, respectively): apparently, such quantum field theories can be reconstructed from a functor into Pos and a functor into PInj. 7.12 Connecting to the analogy with prime factorization of natural number... |