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Title: A categorical view of algebraic theories
Author: Ferrà Marcús, Aina
Director/Tutor: Casacuberta, Carles
Keywords: Categories (Matemàtica)
Treballs de fi de grau
Teoria de models
Programació funcional (Informàtica)
Grups abelians
Categories (Mathematics)
Bachelor's thesis
Model theory
Functional programming (Computer science)
Abelian groups
Issue Date: 27-Jun-2018
Abstract: [en] Classically, algebraic structures such as groups, rings, and many others were jointly studied with the language of universal algebra. It was later found that certain tools from category theory, called monads, are especially suitable to encode the whole amount of information contained in algebraic theories. In this work we discuss monads, and, in particular, some monads that are relevant in functional programming in Computer Science. We give a proof of the equivalence between the category of algebraic theories (formalized as Lawvere theories) and the category of finitary monads on the category of sets. We also prove that there is an equivalence between the category of algebras over a monad and the category of models of the associated Lawvere theory. Finally, we apply this equivalence of categories to give a new proof of the fact that all localizations on the category of abelian groups can be uniquely lifted to $R$-modules for every ring $R$.
Note: Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: Carles Casacuberta
Appears in Collections:Treballs Finals de Grau (TFG) - Matemàtiques

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