This is probably straightforward, but I'm having trouble writing down a precise statement. "Everyone knows" that the cobordism category $\text{2Cob}$ (all manifolds compact and oriented) is the free symmetric monoidal category on a commutative Frobenius object. What is the analogous statement for $\text{1Cob}$?


It looks something like the free symmetric monoidal category on an object with a (left and right) dual, but I'm not sure if I'm interpreting the orientation on points correctly.



Fill in the blanks: "1Cob is the free ____ category on a ____"
Qiaochu, yes, but you don't need to say "left and right dual" because a left dual is a right dual in a symmetric monoidal category. It would be enough to say "with a left dual", or say it as Theo did. 


An equivalent way of describing it is "the free [compact closed category](http://en.wikipedia.org/wiki/Compact_closed_category) generated by a single object". For some information on free compact closed categories, there is an old paper by Kelly and LaPlaza, Coherence for compact closed categories, Journal of Pure and Applied Algebra 19 (1980), pp. 193–213. The description in terms of 1-cobordisms is implicit in that paper. 


