## Born Free January 12, 2009

Posted by dcorfield in Uncategorized.

Continuing our discussion of free entities, from Fiore and Leinster’s A simple description of Thompson’s group F we read

…many entities of interest can be described as free categories with structure. For instance, the braided monoidal category freely generated by one object is the sequence $(B_n)_{n \geq 0}$ of Artin braid groups; the monoidal category freely generated by a monoid consists of the finite ordinals; the symmetric monoidal category freely generated by a commutative monoid consists of the finite cardinals; the symmetric monoidal category freely generated by a commutative Frobenius algebra consists of 1-dimensional smooth oriented manifolds and diffeomorphism classes of 2-dimensional cobordisms.

You can find definitions of many terms used in higher category theory at the exciting new wiki nLab. For example, see braided monoidal category.

Fiore and Leinster continue,

In this vein, our result is that the monoidal category freely generated by an object $A$ and an isomorphism $A \otimes A \longrightarrow A$ is equivalent to the groupoid $1 + F$, where $1$ is the trivial group and $+$ is coproduct of groupoids.

and for the related Thompson’s group V,

just replace ‘monoidal category’ by ‘symmetric monoidal category’, or equally ‘finite-product category’.