Collegare la Matematica: Teoria di Galois Categorica
Uno sguardo ai legami tra la teoria di Galois e la teoria delle categorie.
― 6 leggere min
Indice
- Cos'è la Teoria di Galois?
- Categorie e la loro Importanza
- Teoria Categorica di Galois
- Quasicategorie e il loro Ruolo
- Il Risultato Principale
- Sistemi di fattorizzazione
- Applicazioni della Teoria Categorica di Galois
- 1. Topologia Algebrica
- 2. Teoria dei Numeri
- 3. Teoria dei Fasci
- Direzioni Future
- 1. Generalizzare Risultati Noti
- 2. Indagare Strutture Di Dimensione Superiore
- 3. Stabilire Nuove Connessioni
- Conclusione
- Fonte originale
- Link di riferimento
La Teoria di Galois categorica è un modo per collegare diversi campi della matematica. Questo articolo mira a scomporre queste connessioni e spiegarle in parole semplici. Parleremo delle basi della teoria di Galois, di come si relaziona alle Categorie e delle implicazioni delle scoperte recenti.
Cos'è la Teoria di Galois?
La teoria di Galois prende il nome da Évariste Galois, che ha studiato le relazioni tra equazioni polinomiali e teoria dei gruppi. In parole semplici, ci aiuta a capire come le radici delle equazioni polinomiali interagiscono tra di loro. Questa teoria ci dice che se abbiamo un'equazione polinomiale, possiamo indagare le sue radici e le loro simmetrie attraverso il concetto di gruppo.
Un gruppo è un insieme di elementi che possono essere combinati sotto una certa operazione, come addizione o moltiplicazione. Galois ha mostrato che esaminando le simmetrie delle radici di un polinomiale, potremmo capire la risolvibilità del polinomiale stesso. Questo collegamento tra polinomi e teoria dei gruppi è ciò che rende potente la teoria di Galois.
Categorie e la loro Importanza
Le categorie sono un modo per organizzare oggetti matematici e le relazioni tra di essi. Permettono ai matematici di studiare strutture a un livello più alto. In parole semplici, una categoria consiste di oggetti e morfismi (o frecce) che collegano questi oggetti.
Ad esempio, pensa alle categorie come a una rete di città (oggetti) collegate da strade (morfismi). Ogni strada rappresenta un percorso possibile tra le città, proprio come i morfismi rappresentano relazioni tra oggetti.
Capire le categorie aiuta in molti campi della matematica, inclusi algebra, topologia e geometria. Fornisce un quadro per astrarre e analizzare diversi concetti matematici.
Teoria Categorica di Galois
L'idea della teoria categorica di Galois combina i concetti di teoria di Galois e teoria delle categorie. Si guarda a come i principi della teoria di Galois possono essere espressi usando la teoria delle categorie. Questo collegamento non è solo affascinante, ma offre anche nuovi modi di pensare ai risultati classici.
Nella teoria categorica di Galois, ci concentriamo su come oggetti e morfismi si relazionano all'idea di simmetria. Anziché guardare solo ai polinomi e alle loro radici, possiamo esaminare strutture più ampie e le loro relazioni.
Quasicategorie e il loro Ruolo
Una quasicategoria è un concetto più avanzato nella teoria delle categorie. Ci permette di esprimere relazioni in un modo che cattura strutture più complesse rispetto alle categorie tradizionali. Le quasicategorie possono rappresentare spazi che hanno un certo grado di flessibilità, il che è utile quando si trattano oggetti di dimensione superiore.
Nel contesto della teoria categorica di Galois, le quasicategorie forniscono un modo per esplorare connessioni tra vari concetti matematici in modo più flessibile. Ci permettono di costruire un ponte tra le idee classiche della teoria di Galois e la teoria moderna delle categorie.
Il Risultato Principale
Uno dei contributi chiave dei recenti lavori nella teoria categorica di Galois è lo sviluppo di un analogo quasicategoriale del teorema di Galois classico. Questa nuova prospettiva consente ai matematici di approfondire la loro comprensione delle relazioni tra diverse strutture matematiche.
Il risultato principale suggerisce che possiamo esaminare il funzionamento interno di diverse categorie e come si relazionano tra loro attraverso i loro morfismi. Questo approccio apre a numerose possibilità per ulteriori esplorazioni nella teoria delle categorie superiori e nelle sue applicazioni.
Sistemi di fattorizzazione
In matematica, un sistema di fattorizzazione è un metodo per scomporre i morfismi in due classi, permettendoci di analizzare le loro proprietà in modo più efficace. Ci sono due tipi principali di morfismi: quelli che sono "essenzialmente suriettivi" e quelli che sono "completamente fedeli".
- Essenzialmente Suriettivi: Questo significa che un dato morfismo può raggiungere tutti gli oggetti nella categoria obiettivo.
- Completamente Fedeli: Questo significa che il morfismo preserva le relazioni tra gli oggetti in modo preciso.
Studiare questi sistemi di fattorizzazione ci permette di ottenere spunti su come diverse strutture matematiche interagiscono e come i risultati di tipo Galois possano essere generalizzati.
Applicazioni della Teoria Categorica di Galois
Le implicazioni della teoria categorica di Galois si estendono oltre la matematica astratta. Applicando questi concetti a vari campi, i matematici possono risolvere problemi pratici e scoprire nuovi spunti. Ecco alcune potenziali applicazioni:
1. Topologia Algebrica
Nella topologia algebrica, la teoria categorica di Galois può aiutarci a capire spazi complessi e le loro simmetrie. Analizzando le relazioni tra diversi spazi topologici, i matematici possono scoprire proprietà fondamentali ed esplorare nuovi risultati.
2. Teoria dei Numeri
Nella teoria dei numeri, la teoria di Galois gioca un ruolo cruciale nella comprensione del comportamento dei numeri e delle loro relazioni. Applicando principi categorici, i ricercatori possono indagare connessioni più profonde tra strutture algebriche e proprietà aritmetiche.
3. Teoria dei Fasci
La teoria dei fasci è un metodo per studiare sistematicamente il comportamento globale di dati locali. La teoria categorica di Galois può fornire nuovi strumenti per organizzare e relazionare i fasci, sbloccando nuove possibilità per la ricerca in questo campo.
Direzioni Future
Mentre i ricercatori continuano a esplorare la teoria categorica di Galois, ci sono diverse strade promettenti per future indagini. Ecco alcune aree dove ulteriori esplorazioni potrebbero portare a significativi progressi:
1. Generalizzare Risultati Noti
Concentrandosi sulle relazioni tra diverse strutture matematiche, i ricercatori possono estendere i risultati classici di Galois a contesti più complessi. Questo potrebbe portare a nuove intuizioni e applicazioni in varie aree della matematica.
2. Indagare Strutture Di Dimensione Superiore
La flessibilità delle quasicategorie consente di studiare strutture matematiche di dimensione superiore. Questa esplorazione potrebbe svelare nuove connessioni e risultati che erano precedentemente nascosti all'interno di quadri tradizionali.
3. Stabilire Nuove Connessioni
Esaminando le relazioni tra diversi campi della matematica, i ricercatori possono identificare nuove connessioni che possono portare a intuizioni interdisciplinari. La teoria categorica di Galois funge da strumento prezioso per colmare le lacune tra varie aree di studio.
Conclusione
La teoria categorica di Galois rappresenta un'intersezione emozionante tra la teoria di Galois e la teoria delle categorie. Esaminando le relazioni tra strutture matematiche e le loro simmetrie, i ricercatori possono ottenere nuove intuizioni sui risultati classici, aprendo anche porte per future esplorazioni.
Con l'evolversi della matematica, i concetti presentati in questo articolo offrono una base per studi futuri. L'interazione tra categorie e teoria di Galois è un'area ricca per ulteriori indagini, e le potenziali applicazioni sono vaste.
Con la ricerca in corso, la teoria categorica di Galois ha il potere di approfondire la nostra comprensione della matematica e dei suoi molti rami. Il viaggio in questo affascinante regno invita matematici ed appassionati a esplorare le intricate relazioni che plasmano il nostro mondo matematico.
Titolo: The Unreasonable Efficacy of the Lifting Condition in Higher Categorical Galois Theory I: a Quasi-categorical Galois Theorem
Estratto: In (Borceux-Janelidze 2001) they prove a Categorical Galois Theorem for ordinary categories, and establish the main result of (Joyal-Tierney 1984), along with the classical Galois theory of Rings, as instances of this more general result. The main result of the present work refines this to a Quasicategorical Galois Theorem, by drawing heavily on the foundation laid in (Lurie 2024). More importantly, the argument used to prove the result is intended to highlight a deep connection between factorization systems (specifically the lex modalities of (Anel-Biedermann-Finster-Joyal 2021)), higher-categorical Galois Theorems, and Galois theories internal to higher toposes. This is the first part in a series of works, intended merely to motivate the lens and prove Theorem 3.4. In future work, we will delve into a generalization of the argument, and offer tools for producing applications.
Autori: Joseph Rennie
Ultimo aggiornamento: 2024-09-05 00:00:00
Lingua: English
URL di origine: https://arxiv.org/abs/2409.03347
Fonte PDF: https://arxiv.org/pdf/2409.03347
Licenza: https://creativecommons.org/licenses/by/4.0/
Modifiche: Questa sintesi è stata creata con l'assistenza di AI e potrebbe presentare delle imprecisioni. Per informazioni accurate, consultare i documenti originali collegati qui.
Si ringrazia arxiv per l'utilizzo della sua interoperabilità ad accesso aperto.
Link di riferimento
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- https://q.uiver.app/#q=WzAsMyxbMCwxLCJ4Il0sWzEsMCwieSJdLFsyLDEsInoiXSxbMCwyLCJoIiwxXSxbMCwxLCJmIiwxXSxbMSwyLCJnIiwxLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZG90dGVkIn19fV0sWzMsMSwiXFxjaXJjbGVhcnJvd3JpZ2h0X1xcYWxwaGEiLDEseyJzaG9ydGVuIjp7InNvdXJjZSI6MjB9LCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJub25lIn0sImhlYWQiOnsibmFtZSI6Im5vbmUifX19XV0=
- https://q.uiver.app/#q=WzAsMyxbMCwxLCJ4Il0sWzEsMCwieSJdLFsyLDEsInoiXSxbMCwyLCJoIiwxLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZG90dGVkIn19fV0sWzAsMSwiZiIsMV0sWzEsMiwiZyIsMV0sWzMsMSwiXFxjaXJjbGVhcnJvd3JpZ2h0X1xcYWxwaGEiLDEseyJzaG9ydGVuIjp7InNvdXJjZSI6MjB9LCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJub25lIn0sImhlYWQiOnsibmFtZSI6Im5vbmUifX19XV0=
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- https://stacks.math.columbia.edu/tag/023F
- https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXGxHcm90aFxcb3AiXSxbMSwwLCJcXExvY1xcb3AiXSxbMiwwLCJcXGxHcm90aFxcb3AiXSxbMywwLCJcXENDYXQiXSxbMCwxLCJMIl0sWzEsMiwiXFxTaCJdLFswLDIsIjFfe1xcbEdyb3RoXFxvcH0iLDIseyJsYWJlbF9wb3NpdGlvbiI6ODAsImN1cnZlIjo1fV0sWzIsMywiXFxpb3RhIl0sWzEsNiwiXFx1bml0XFxvcCIsMix7ImxhYmVsX3Bvc2l0aW9uIjozMCwic2hvcnRlbiI6eyJ0YXJnZXQiOjIwfSwic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiYXJyb3doZWFkIn0sImhlYWQiOnsibmFtZSI6Im5vbmUifX19XV0=
- https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXGxTcGxpdF9cXGFscGhhKFxcbGFtYmRhKSJdLFsxLDAsIlxcRSJdLFsxLDEsIlxcU2goXFxMKSJdLFswLDEsIlxcU2goTChcXFNoKFxcTCkpKSJdLFsxLDJdLFszLDIsIlxcc2ltIl0sWzAsMV0sWzAsM10sWzAsMiwiIiwxLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV1d
- https://tex.stackexchange.com/questions/577605/footnote-appearing-on-the-next-page-when-theres-too-much-space-on-a-page?noredirect=1&lq=1
- https://math.arizona.edu/~aprl/publications/mathclap/
- https://q.uiver.app/?q=WzAsNCxbMCwxLCJGXzFee1xcSEh9Il0sWzEsMSwiRl8yXlxcSEgiXSxbMSwwLCJGXzJeXFxHRyJdLFswLDAsIlxcU3BsaXRfXFxhbHBoYShcXGdhbW1hKSJdLFsyLDFdLFswLDFdLFszLDJdLFszLDBdLFszLDEsIiIsMSx7InN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dXQ==
- https://q.uiver.app/?q=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
- https://q.uiver.app/?q=WzAsNSxbMCwwLCJcXFNwbGl0X1xcYWxwaGEoXFxsYW1iZGEpIl0sWzEsMCwiXFx2TUFQe1xcZ3JwZEVtYmVkZGluZ3tcXEdHX1N9LEZfMX0iXSxbMiwwLCJcXHZNQVB7XFxncnBkRW1iZWRkaW5ne1xcR0dfXFxsYW1iZGF9LEZfMX0iXSxbMSwxLCJcXHZNQVB7XFxncnBkRW1iZWRkaW5ne1xcR0dfU30sRl8xfSJdLFswLDEsIlxcdk1BUHtcXGdycGRFbWJlZGRpbmd7XFxHR19TfSxGXzF9Il0sWzEsMiwiKDEpIiwwLHsibGV2ZWwiOjIsInN0eWxlIjp7ImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbNCwzXSxbMSwzXSxbMCw0XSxbMCwxXSxbMCwzLCIiLDAseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0=
- https://q.uiver.app/?q=WzAsNCxbMCwwLCJGXzFee1xcR0dfXFxsYW1iZGF9Il0sWzEsMCwiRl8yXntcXEdHX1xcbGFtYmRhfSJdLFswLDEsIkZfMV57XFxHR19TfSJdLFsxLDEsIkZfMl57XFxHR19TfSJdLFswLDFdLFsxLDNdLFswLDJdLFsyLDNdXQ==
- https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXEhvbV97XFxzU2V0fShcXERlbHRhXm5cXHRpbWVzIFxcTWFwX1xcQyhDLFIpLFxcRCkiXSxbMCwxLCJcXEhvbV97XFxzU2V0fShcXERlbHRhXm5cXHRpbWVzIFxcTWFwX1xcQyhDLFMpLFxcRCkiXSxbMSwwLCJcXEhvbV97XFxzU2V0fShcXERlbHRhXm5cXHRpbWVzIFxcTWFwX1xcQyhDLFIpLFxcRCcpIl0sWzEsMSwiXFxIb21fe1xcc1NldH0oXFxEZWx0YV5uXFx0aW1lcyBcXE1hcF9cXEMoQyxTKSxcXEQnKSJdLFsyLDNdLFsxLDNdLFswLDFdLFswLDJdXQ==
- https://arxiv.org/search/advanced?advanced=&terms-0-operator=AND&terms-0-term=Galois+Theory&terms-0-field=title&terms-1-operator=OR&terms-1-term=Galois+Group&terms-1-field=title&classification-physics_archives=all&classification-include_cross_list=include&date-year=&date-filter_by=date_range&date-from_date=2010&date-to_date=2022-10-17&date-date_type=submitted_date&abstracts=show&size=50&order=-announced_date_first
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- https://q.uiver.app/#q=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