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Operads are a generic structure giving a more precise definition of these terms. An operad is an abstract set of operations of various arities (an ugly word to precise the number or arguments taken by an operation: a ternary operation is said to have arity three), subject to relations between them or their compositions. For example, the operad of vector spaces consists of two basic operations: sum and product by scalars (which are actually infinitely many operations), which are tied by distributivity, commutativity and associativity among other relations. An example of operation of this monad is (x, y, z) → 2x + 3y + z, which is a ternary operation. An operad T defines a natural associated monad, which associates to a set X the set

Is not an “operad” a particular case of a “signature”/“theory” in mathematical logic? Why invent a new word?

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No. They aren't as powerful as generic first-order theories, or even universal algebra. On the other hand, they are defined using the language of category theory and can be studied using those tools. (Comment because I don't actually know the details.) –  Zhen Lin Mar 21 '11 at 8:51
    
Sorry, I mean “a particular case of a signature/theory”. –  beroal Mar 21 '11 at 9:08
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Although they have non-empty intersection, none is a special case of the other.

The analogue of an operad in universal algebra is the clone of the theory; but this is a minor difference. The more substantial difference is that operads are defined in such a way that they can have models in any monoidal category. This allows for example, to speak of monoids in the category of linear spaces with monoidal structure given by tensor product a.k.a. (associative, unital) algebras. On the other hand, this implies some restrictions; for example there is no operad for groups. Why? Well, the inverse axiom is

$x x^{-1}= 1, x^{-1} x= 1$

but to express this axiom you need to "double" the variable $x$. How do you that categorially? You need the diagonal map $X\to X\times X$ and for this map to exist, the monoidal structure must come from products. The tensor product of linear spaces is an obvious example of a monoidal structure that does not come from products.

Hope it helps, regards, G. Rodrigues

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Yeah, I know that. BTW, to describe groups, we need $X\to I$ in addition to $X\to X\otimes X$. So, an “operad” is a generalization of a “Lawvere theory”, replacing a category with all categorical products with a monoidal category? Is an “operad” different from a “PRO”, “PROP”, “PROB”? –  beroal Mar 24 '11 at 8:47
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Those generalizations you mention are exactly PROs, PROPs and PROBs. An operad is a further generalization. The idea is that while one can use tensor products to say what you mean by morphisms of "two variables" (in vector spaces, for example, a bilinear map is just a linear map out of a tensor product), you could directly define category-like things where the domains of morphisms are allowed to be a sequence of objects (and there might be no way to define a tensor product to reduce the domain to a single object). Those are called multicategories and one with a single object is an operad. –  Omar Antolín-Camarena Mar 27 '11 at 0:44
    
(To be precisee, symmetric operads generalize PROPs, non-symmetric ones generalize PROs, and I don't recall hearing about braided operads. Also, bewqare that when people say operad, they almost always mean symmetric operad.) –  Omar Antolín-Camarena Mar 27 '11 at 1:43
    
Arch, I forgot one other difference between PROPs and (symmetric) operads: in an operad all the operations have n inputs and one output, in a PROP you can have operations with any number of outputs (often this is unnecessary and I think people now use operads for several of the original uses of PROPs). So operads and PROPS are actually incomparable in generality. –  Omar Antolín-Camarena Mar 30 '11 at 15:55
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