Andrej Bauer wrote:
The category of finite sets and functions may be characterized (up to equivalence) as the category with finite coproducts freely generated from one object. Is there a similar nice characterization for the category of finite sets and _injective_ functions?
Best regards,
Andrej
There are two universal characterisations of the category of finite sets and injections: I - It is the free symmetric monoidal category on one generator, with an initial unit. II - It is the free symmetric monoidal category on one generator G, with a monoidal indeterminate x: I -> G. In both cases, the symmetric monoidal structure is given by finite coproducts. The general notion of 'monoidal indeterminates' requires some spelling out, but should be clear enough in this simple case. II implies I above; it follows from a general construction of monoidal indeterminates subject to a naturality constraint (which is trivial in this case). Regards, Claudio Hermida