categories: Operads

[S.J.Vickers@open.ac.uk]

There's some discussion on the Universal Algebra list at present on operads.

I'm not very familiar with them. What I understand from the discussion is
they capture single sorted algebraic theories with respect to a symmetric
monoidal product ox. For each natural number n an object of n-ary operators
O_n is given, and an algebra A has operations O_n ox A^(n) -> A where A^(n)
is A ox ... ox A n times.

If you do this sort of thing with respect to categorical product, then it
already contains the information of the Lawvere theory category (for
single-sorted finitary algebraic theories), since hom(m,n) is hom(m,1)^n and
you take hom(m,1) to be O_m. But with a monoidal product this doesn't work.
It seemed to me that for proper generality the operad ought to have objects
O_mn (m, n natural numbers) representing the object of operations from A^(m)
to A^(n). Is there a name for that?

Steve Vickers
Department of Pure Maths
Faculty of Maths and Computing
The Open University
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