delooping of the bicyclic monoid

 algebra
Notation BCBC Objects a single object ∗* Morphisms the elements of the bicyclic monoid C≔⟨p,q:pq=1⟩C \coloneqq \langle p,q : pq = 1 \rangle Related BDBD, BNB\IN, BGBG
Every monoid MM induces a category BMBM with a single object ∗*, morphisms given by the elements of MM, and composition given by the monoid operation. Some of the properties of this category depend on the specific monoid.In this entry, we take the monoid CC freely generated by p,qp,q subject to the relation pq=1pq=1, known as the bicyclic monoid and often denoted by C(p,q)C(p,q). Every element of CC can be uniquely written as qnpmq^n p^m for n,m≥0n,m \geq 0.This category is a one-object version of the walking splitting, but its properties are completely different.

Satisfied properties

Assigned properties

Deduced properties

Unsatisfied properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

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Special objects

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Special morphisms