category theory

Universal algebra nullary operations

May 10, 2020
category theory
universal algebra

Recall universal algebra approach to definining algebraic structures. Instead of explicitly stating existence of certain elements – for exapmle, by designating a unit element in a group or associating an inverse – we replace them with a collection of functions. With this in mind, if \(A\) is a group, a unit element can be replaced with 0-arity function that chooses an element of \(A\) . An inverse is a 1-arity function \(A \rightarrow A\) . ...

Partitions and Selectors

April 22, 2020
category theory
functor categories

Look at this interesting stuff Let I be a finite set, X be another set I -> X is a selector function X -> I is a partition function for example, Let I = {0, 1}, X = N f: {0, 1} -> N selects two natural numbers g: N -> {0, 1} partitions N into two subsets now, what happenes if we categorify this notion I becomes a discrete category – elements as objects, only identity morphisms X becomes a category of all sets ...