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I created the article properad, essentially a brief description of the definition together with a reference.
Dimitri: Thanks. That reads a bit more like an ‘Idea’ than a definition (We probably need the full details, but I do not have them to hand.)
What are the ’bisymmetric sequences’?
This talks about monoids in the monoidal category of $\mathbb{S}$-bimodules.
Bisymmetric sequences are simply functors Σ×Σ→C, just like symmetric sequences are functors Σ→C. S-modules mean something completely different (the EKMM model for spectra) and there is a well-established terminology of symmetric sequences for operads, which I don’t want to change.
Still, probably better to add some detail to the entry of what the sequences consist in. Or refer to operad for comparison.
The ’prop’ mentioned is not PROP, it seems from Vallette’s article
A prop is the support of the operations acting on algebraic structures. That’s-why we denote it with small letters.
Is that worth spelling out?
Prop is the same thing as PROP, of course, only spelled in a modern way. (MacLane originally wrote PROP, but these days almost everybody writes prop, to make it more readable.)
MacLane only talked about props in the cartesian monoidal category of sets, whereas Vallette talks about props in (Vect_k,⊗). In general, props are defined in any symmetric monoidal category, see, for example, Fresse, “Props in model categories and homotopy invariance of structures”, http://arxiv.org/abs/0812.2738.
It’s strange that nLab elected to use the archaic spelling, and doesn’t even have a redirect for the modern spelling. Perhaps we should rename the page PROP to prop?
Re 7, My guess would be that the people who created and edited the page were unaware of what you say the modern spelling is. I’ve never seen it spelled lowercase before.
Re 9: Recent papers on props, e.g., ones by Fresse, Vallette, etc., use the lowercase spelling.
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