Jason (jcreed) wrote,

A nice natural example of a bicategory (non-strict 2-category) I didn't know about before:

Objects are sets.

Morphisms from A to B are "state machines from A to B", that is, a set X together with functions i : A → X and s : X → X + B. The set X is the set of nonfinal states. The function i says how to initialize the state given an input from A. The function s says how to take a step; you might get back another nonfinal state, or you might get an output in B.

A 2-cell from (X, i, s) to (Y, j, t) is a simulation of one state machine by the other, that is, a function h : X → Y such that hi = j and th = (h+B)s.

Like the bicategory of spans, a universal construction that's associative only up to isomorphism (in the state machine case, coproduct, in the bicategory-of-spans, the pullback) means that composition of morphisms is only associative up to isomorphism. Unlike the bicategory of spans, (and this to my mind makes it a more satisfyingly general example of a bicategory) not every morphism has a dual.
Tags: bicategories, categories, math
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