


Categories, allegories



Description: 
A binary relation from a set A to a set B is a subset of the cartesian product AxB. Moreover, a function f: A>B can be identified with a particular type of relation, namely the relation consisting of pairs (x,f(x)). Consequently, the usual category of sets can be regarded as a subcategory of the category Rel(Set) whose objects are sets and morphisms are relations between them. In this talk, we are going to present a categorical generalization of this situation due to P. Freyd and A. Scedrov, and explore its applications in category theory. Not every category C is suitable for this: we will first characterize those categories in which we can still do relational calculus, that is, where we can still define its category of relations Rel(C). More interestingly, we will see that ‘categories of relations' (equivalently, their axiomatized version, ‘allegories') provide an alternative and quite natural framework for understanding certain sorts of categories, e.g. regular categories, coherent categories, Heyting categories, and toposes. As a consequence, a nice corresponde will arise between ‘categorical structure' and ‘allegorical structure'. Time permitting, we shall briefly discuss why a very large and important class of toposes can be built from fairly simple categories of relations.

Date: 
20181114

Start Time: 
15:00 
Speaker: 
Igor Arrieta (UCUP PhD programme student)

Institution: 
UCUP PhD programme

Place: 
Sala 2.5, DMat UC

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