Exploring Voronov's Contributions to Operads: A Comprehensive Guide

[Kea]

Er. Thanks, Greg. Do you think maybe we could have just one link to all
these nice notes? Voronov's homepage is

http://www.math.umn.edu/~voronov/

Cheers
Kea

 

[Kea]

lecture 6

In lecture 6, Voronov defines an operad as a collection of sets indexed by $n$. What this really means is that it is a monoid, ie. based on a single object, in a category of functors from some category P into Set. The category P is just a disjoint collection of symmetric groups as one object categories.

Now there is a nicer category than P which is also related to the ordinals. That is, the category S whose objects are labelled by $n$, sets of $n$ elements, and whose morphisms are the maps between these sets. The really wonderful thing is that a (set valued) operad based on this category is something called a Lawvere theory! These show up everywhere. One is given a sequence of sets $S_{n}$ of n-ary operations. In conjunction with a set of Axioms one has a Lawvere theory. For example, the theory of a commutative ring with unit has two elements in $S_{0}$, namely 0 and 1, one operation in $S_{1}$, namely the negation, and the binary operations of addition and multiplication.

Theories can have models, ie. interpretations in any category (well, we need products). For example, the models of the theory of groups in Set turns out to be the same thing as the category of groups. Anyway, it turns out that Lawvere theories have universal models given, for instance, by the Yoneda embedding!

This is nicely explained in an old article of Kelly's, On the Operads of J.P. May http://www.tac.mta.ca/tac/reprints/articles/13/

Sorry if I got a bit carried away ... Greg started it.

 

[Greg Bernhardt]

Kea, originally I had these in our tutorial section, but we thought they'd be better served in here. Enjoy!