Algebraic theories, monads, and arities

Details Algebraic theories, monads, and arities Algebraic theories, monads, and arities

2011-10-14

arxiv.org/abs/1110.3294v1

Mathematics

Category Theory

65 pages, this is a survey made for a Master thesis

Scientific paper

Monads are of interest both in semantics and in higher dimensional algebra. It turns out that the idea behind usual notion finitary monads (whose values on all sets can be computed from their values on finite sets) extends to a more general class of monads called monads with arities, so that not only algebraic theories can be computed from a proper set of arities, but also more general structures like n-categories, the computing process being realized using Kan extensions. This Master thesis compiles the required material in order to understand this question of arities and reconstruction of monads, and tries to give some examples of relevant interest from both semantics and higher category theory. A discussion on the promising field of operads is then provided as appendix.

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