Consider the category whose objects are non-negative integers that are represented as dots along a line, and whose morphisms are generated by $X$---positive crossing, $\bar{X}$ --- negative crossing, $\cup$, $\cap$, $Y$ --- co-multiplication, $\Lambda$ --- multiplication, and of course, $|$--- the identity morphism. The relations in the category are the usual braid relations, adjointness for $\cup$ and $\cap$, distant $X$s and $Y$s commute, associativity, co-associativity, $(Y \otimes |) \circ X = (| \otimes X) \circ (X \otimes |) \circ (| \otimes Y)$ and variations, and relations involving $\cap$ and $Y$ to relate this to $\Lambda$. In brief, the relations in $\cup$, $\cap$, $Y$, and $\Lambda$ satisfy the Frobenius algebra axioms; the relations for $X$ satisfy the braid relations, and the obvious relations for crossings and trivalent vertices hold.

Is this structure is a free braided Frobenius category? I might not have the adjectives in the correct order. It seems that the structure should be the most free that satisfies braiding, Frobenius, and the intermingling of the two structures.

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    $\begingroup$ Typically ribbon categories are better behaved in this way than braided rigid monoidal categories (see Shum's theorem). I'd worry that the same issue would come up here. $\endgroup$ – Noah Snyder Jul 24 '11 at 6:17
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    $\begingroup$ I am trying to track down a copy of Shum's paper now. It is in the nether zone of science direct and print journals at my school, but can you elaborate about the issue a bit more, Noah? $\endgroup$ – Scott Carter Jul 25 '11 at 23:08
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    $\begingroup$ Sorry I didn't see your reply until now. Shum's theorem says that the category of ribbon tangles is the universal ribbon category. My impression was that there was some technical reason why ribbon tangles were better behaved from this point of view than ordinary tangles, but I'm not totally sure on this point. $\endgroup$ – Noah Snyder Oct 25 '11 at 13:21

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