I'm a newcomer to operads so apologies if this is a naive question.


The standard picture of an operad is of a collection of $n$-ary operations, thought of as objects with $n$ upward-pointing legs (inputs) and one downward-pointing leg (output), which can be composed in a sensible way. "Algebras over operads" then give rise to all sorts of algebras in the usual sense (associative, $A\_\infty$ etc).


My understanding is that a PROP is basically the same thing but with operations allowed to have more than one output. Algebras over PROPs can also accommodate coproduct-like operations, and hence also include useful things like coalgebras and Hopf algebras.


Historically though, operads came later than PROPs. So why was it useful to single out the notion of an operad, as opposed to a general PROP?



In what sense are operads "better" than PROPs?
I was friends with Frank Adams and Saunders Mac Lane, who invented PROPs in one of the world's most extensive unpublished collaborations. Saunders once showed me a box full of their correspondence. One reason they never published is that they lacked a way of showing the PROPs they were interested in acted on the things they wanted to have actions. Operads are of course equivalent to a special kind of PROP, and the specialization made it very much easier to find operad actions. The connection with monads was intrinsic to the definition of operads (I convinced Mac Lane to switch from "triples'' to "monads" in *Categories for the Working Mathematician* in large part in order to make the names operad and monad to mesh) and operads allowed tons of explicit computations (in the homology of iterated loop spaces in particular) that would not have not come naturally if at all from PROPs. Of course, there are interesting examples of PROPs of the more general sort, ones that do not come from an operad, but they are irrelevant to the original work with iterated loop spaces.


