Non-unital Morita equivalence in a bicategorical setting
Date
2017-06-28
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Abstract
Morita teooria sai alguse Jaapani matemaatiku Kiiti Morita töös moodulite duaalsuse alal, mis avaldati aastal 1958. Morita defineeris ekvivalentsiseose ringide klassil ringide moodulite kategooriate ekvivalentsuse kaudu. Ta tõestas ka mõned alustulemused selle ekvivalentsiseose kohta ning tänapäeval kutsutakse seda seost Morita ekvivalentsiks ning neid tulemusi Morita teoreemideks.
Ringide korral on Morita ekvivalentsil mitmeid samaväärseid vorme. Morita ekvivalentsi mõistet on nüüdseks rakendatud mitmesugustele teistele struktuuridele. Seda on tehtud kasutades erinevaid tingimusi, mis ringide korral kokku langevad, kuid muude struktuuride korral seda teha ei pruugi. Mõni tingimus pole isegi teistsuguses kontekstis sõnastatav.
Mitmed inimesed on pannud tähele, et suur osa Morita teooriast on seotud loomuliku bikategoorse struktuuriga moodulitel. Tänapäeval on see fakt võrdlemisi tuntud. Bikategooriatest võib mõelda kui moodulite tensorkorrutise abstraktsioonist, mis jätab alles ainult tensorkorrutise bifunktoriaalsuse, assotsiatiivsuse ja ühikute olemasolu.
Kuigi on teada, et Morita ekvivalentsil on bikategoorses keskkonnas hea teooria, on mitmeid olukordi, kuhu meil bikategoorne keskkond ei rakendu. Selline on olukord näiteks ühikuta ringide korral, mis pole üldsegi eksootiline näide.
Selles töös nõrgendatakse bikategooria eeldusi, lastes bikategooria ühikutel olla lõtv. Paistab, et see on õige lähenemine, et uurida abstraktset Morita teooriat ühikuta struktuuride, näiteks ühikuta ringide ning poolrühmade korral. Töös uuritakse, kuidas ekvivalentsuse definitsioonid erinevatel eeldustel kokku langevad või erinevaid mõisteid annavad.
Töö lõpus näidatakse, kuidas saab selliseid üldiseid lõtvu bikategoorseid struktuure koostada üldise ühikuta multiplikatiivsete struktuuride korral – seda siis poolrühmobjektide jaoks monoidkategoorias. Kuigi enamasti ei ole see konstruktsioon uus, illustreerib see töö eesmärki.
Morita theory originates from the work of Kiiti Morita, a Japanese mathematician, who introduced the concept in a paper of his in 1958, where he studied the theory of dualities of modules. Morita defined an equivalence relation on rings in terms of the equivalence of categories of modules. He proved some fundamental properties of the relation and nowadays the relation is known as Morita equivalence and the properties are known as Morita theorems. In the case of rings, Morita equivalence has many equivalent formulations. The notion of Morita equivalence has since then been used for various different structures, using differing formulations inspired by the theory's original context, the case of rings. Often, not all of the different formulations of Morita equivalence coincide for some given structure, some of the conditions might not even be applicable for certain structures. It has been observed by many people, and in these days it is quite well known, that at lot of aspects of Morita theory have to do with the natural bicategorical structure on modules. Bicategories can be thought of as abstractions of the concept of the tensor product of modules, only retaining the bifunctorial, associative and unital structure of the tensor product. While it is known that Morita equivalence has a nice theory in the bicategorical setting, the bicategorical setting is not always applicable for the study of the Morita theory of certain structures. Such is the case even in the remarkably non-exotic case of rings without an identity element. In this work the notion of a bicategory is relaxed a bit, allowing the unital structure of the bicategory to be lax. This setting seems to be the right way to abstract the Morita theory of non-unital structures like rings without unit element or semigroups. We study how different formulations of the equivalence relation diverge and converge under different assumptions. At the end of the thesis it is shown how one can construct such lax bicategorical structures by studying the notion of a non-unital multiplicative structure in general, in the context of semigroup objects in monoidal categories. This construction is for the most part not new, but it illustrates the objective
Morita theory originates from the work of Kiiti Morita, a Japanese mathematician, who introduced the concept in a paper of his in 1958, where he studied the theory of dualities of modules. Morita defined an equivalence relation on rings in terms of the equivalence of categories of modules. He proved some fundamental properties of the relation and nowadays the relation is known as Morita equivalence and the properties are known as Morita theorems. In the case of rings, Morita equivalence has many equivalent formulations. The notion of Morita equivalence has since then been used for various different structures, using differing formulations inspired by the theory's original context, the case of rings. Often, not all of the different formulations of Morita equivalence coincide for some given structure, some of the conditions might not even be applicable for certain structures. It has been observed by many people, and in these days it is quite well known, that at lot of aspects of Morita theory have to do with the natural bicategorical structure on modules. Bicategories can be thought of as abstractions of the concept of the tensor product of modules, only retaining the bifunctorial, associative and unital structure of the tensor product. While it is known that Morita equivalence has a nice theory in the bicategorical setting, the bicategorical setting is not always applicable for the study of the Morita theory of certain structures. Such is the case even in the remarkably non-exotic case of rings without an identity element. In this work the notion of a bicategory is relaxed a bit, allowing the unital structure of the bicategory to be lax. This setting seems to be the right way to abstract the Morita theory of non-unital structures like rings without unit element or semigroups. We study how different formulations of the equivalence relation diverge and converge under different assumptions. At the end of the thesis it is shown how one can construct such lax bicategorical structures by studying the notion of a non-unital multiplicative structure in general, in the context of semigroup objects in monoidal categories. This construction is for the most part not new, but it illustrates the objective
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Keywords
poolrühmad (mat.), homoloogiline algebra, kategooriateooria, semigroups (math.), homological algebra, category theory