Seminario de Categorías

Expositor: Paolo Perrone (Oxford).

Resumen: Partial evaluations are a way to encode, in terms of monads, operations which have been computed only partially. For example, the sum "1+2+3+4" can be evaluated to "10", but also partially evaluated to "3+7", or to "6+4". Such structures can be defined for arbitrary algebras over arbitrary monads, and even 2-monads, and can be considered the 1-skeleton of a simplicial object called the bar construction. The higher simplices of the bar construction can be interpreted as ways to compose partial evaluations. Recent research has shown that, while for cartesian monads partial evaluations form a category, for weakly cartesian monads the compositional structure is more complex, and in particular it does not in general form any of the standard higher-categorical structures. Moreover, partial evaluations return known concepts of "partially evaluated operations" in the following contexts: - For the free cocompletion monad, where the operation is taking the colimit, partial evaluations correspond to left Kan extensions; - For probability monads, where the operation is taking the expected value, partial evaluations correspond to conditional expectations.

The research presented in this talk has been carried out jointly with Carmen Constantin, Tobias Fritz, Brandon Shapiro, and Walter Tholen.

The talk will be in English, but you are welcome to ask questions in Spanish if anything is not clear.

The following are references for the topics discussed in the talk:

arxiv.org/abs/1810.06037

arxiv.org/abs/2009.07302

arxiv.org/abs/2101.04531

https://www.youtube.com/watch?v=ynxfrlqr4I0&feature=youtu.be