In some points, I’ve tried to fill in the most sketchy and un-articulated points with some simile of what I ended up actually saying.

Combinatorial species started out as a theory to deal with enumerative combinatorics, by providing a toolset & calculus for formal power series. (see Bergeron-Labelle-Leroux and Joyal)

As it turns out, not only is species useful for manipulating generating functions, btu it provides this with a categorical approach that may be transplanted into other areas.

For the benefit of the entire audience, I shall introduce some definitions.

Definition: A category C is a collection of objects and arrows with each arrow assigned a source and target object, such that

1. Each object has its own identity arrow 1.
2. Chains of arrows are associatively composable, with 1 the identity of this composition.

Examples: Sets, Finite sets, k-Vector spaces, left and right R-Modules, Graphs, Groups, Abelian groups.

: finite sets with only bijections as morphisms.

Examples:

• Category of a monoid.
• Category generated by a graph
• Category of a group (groupoid version of the category of a monoid)
• Category of a poset
• Category of Haskell types and functions.

Definition: A functor F is a map of categories, in other words, a pair of maps Fo on objects and Fa on arrows, such that the identity arrow maps to identity arrows and compositions of maps map to compositions of their images.

Examples:
Constant functor: sends every object to A, every map to 1.

Identity functor: sends objects and arrows to themselves.

Underlying set functor, free monoid/vector space/module/… functors.

We are now equipped to define Species:
Definition: A species of structures is a functor .

The idea is a set gets mapped to the set of all structures labelled by the elements in the original set. A bijection on labels maps to the transport of structures along the relabeling.

Examples:
: rooted trees, labels on vertices
: simple graphs, labels on vertices
: connected simple graphs, labels on vertices.
: trees, labels on vertices.
: directed graphs, labels on vertices
: subsets, .
: endofunctions
: involutions
: permutations
: cycles
: linear orders
: sets:
: elements:
: singletons: if and otherwise
: empty set: , otherwise
: empty species: .

In enumerative combinatorics, the power of species resides in the association to each species a number of generating functions:

The generating series of a species F is the exponential formal power series

where we use the convention , and .
Thus:

There are a number other in use for combinatorics, but for my purposes, this is the one I’ll focus on.

Operations on species

The real power, though, emerges when we start combining species, and carry over the combinations to actions on the corresponding power series.

Addition

The number of ways to form either, say, a graph or a linear order on a set of labels is the sum of the numbers of ways to form either in isolation.

This corresponds cleanly to the coproduct in Set, and we may write F+G for the species

(where the second + is the coproduct — i.e. disjoint union — of sets)

In the power series, we get .

Examples: We may define the species of all non-empty sets by

This kind of functional equations is where the theory of species starts to really shine.

Multiplication

A tricoloring of a set is a subdivision of the set into three disjoint subsets covering the original set. The number of tricolorings of size n is

A permutation fixes some set of points. The permutation restricted to the non-fixed points is a derangement. Total number of permutations on n elements is

In both of these cases, the total generating series is a product of the component series, and we end up defining

So .

Thus, tricolorings are and permutations are , where the set is that of fixed points, and the derangement captures the actual action of the permutation.

Composition

Endofunctions of sets decompose in their actions on the points as cycles or directed trees leading in to these cycles.

Since a collection of disjoint cycles corresponds to a permutation, we can consider such endofunctions to be permutations decorated with rooted trees attached to points of the permutations; or even permutations of rooted trees.

To form such a structure on a set U, we’d first partition U into subsets, put the structure of a rooted tree on each subset, and then the structure of a permutation on the set of these subsets.

Thus, the number of such structures on n elements is

This corresponds to the power series , and we write, in general, for the species of F-structures of G-structures on subsets.

Examples:
A = X·E(A)
L = 1 + X·L
B = 1 + X·B·B

Pointing

Picking out a single point in a structure on n points can be done in precisely n ways.

Thus the corresponding generating function will be

for f-structures with a single label distinguished.

Since we’re working with exponential power series, we may notice that

and thus that derivatives are shifts.
Furthermore,
so that the generating function for F-structures with a single distinguished are

In species, this process is called pointing.

In functional programming, Conor McBride related this construction to Huet’s Zipper datatypes.

As it turns out, many of the constructions for species make eminent sense outside the category . In fact, species in Hask are known to programming language researchers as container datatypes and the whole calculus translates relatively cleanly.

Functional equations translate to the standard new data type definitions in Haskell.

Examples:
L = 1 + X·L

data List a = Nil | Cons a (List a)


B = 1 + X·B·B

data BinaryTree a = Leaf | Node (BinaryTree a) a (BinaryTree a)


A = X·E(A)

data RootedTree a = Node a (Set (RootedTree a))

usually, we simulate the Set here by a List. If we need for our rooted trees to be planar, we can in fact impose a Linear Order structure instead, and get something like
Ap = X·L(Ap)

The species interpretation of corresponding to leaving a hole in the structure carries over cleanly, so that is the type of T-with-a-hole.

Two derivatives of lists

We can deal with in two different ways:

1.
DL = D(1+X·L) = D1 + D(X·L) = 0 + DX·L+X·DL
and thus
DL-X·DL = 1·L
so (1-X)·DL = L
and thus
DL = L·1/(1-X)

Now, from L=1+X·L follows by a similar cavalier use of subtraction and division — which, by the way, in species theory is captured by the idea of a virtual species, and dealt with relatively cleanly — that
L = 1+X·L
so
L-X·L = 1
and thus
(1-X)·L = 1
so
L = 1/(1-X)

Thus, we can conclude that
DL = L·1/(1-X) = L·L
and thus, a list with a hole is a pair of lists: the stuff before the hole and the stuff after the hole.

2.
We could, instead of using implicit differentiation, as in 1, attack the derivation we had of
L = 1/(1-X) = 1+X+X·X+X·X·X+…

Indeed,
DL = D(1/(1-X)) = D(1+X+X·X+…) = 0+1+2X+3X·X+…
which we can observe factors as
= (1+X+X·X+X·X·X+…)·(1+X+X·X+X·X·X+…) = L·L

Or we can just use the division rule
DL = D(1/(1-X)) = D(1/u)·D(1-X) = -1/(u·u)·(-1) = 1/[(1-X)·(1-X)] = L·L

Gröbner bases for operads

All of this becomes relevant to the implementation of Buchberger’s algorithm on shuffle operads (see Dotsenko—Khoroshkin and Dotsenko—Vejdemo-Johansson) in the step where the S-polynomials (and thereby also the reductions) are defined. With a common multiple defined, we need some way to extend the modifications that take the initial term to the common multiple to the rest of that term.

For this, it turns out, that the derivative of the tree datatype used provides theoretical guarantees that only partially filled in trees of the right size, with holes the right size, can be introduced; and also provides an easy and relatively efficient algorithm for contructing the hole-y trees and later filling in the holes.

Recall how you would run the Gaussian algorithm on a matrix. You’d take the leftmost upmost non-zero entry, divide its row by its value, and then use that row to eliminate anything in the corresponding column.

Once we have the matrix on echelon form, we can then do lots of things with it. Importantly, we can use substitution using the leading terms to do equation system solving.

The starting point for the theory of Gröbner bases was that the same method could be used – with some modification – to produce something from a bunch of polynomials that ends up being as useful as a row-reduced echelon form.

Basically, the central theorem-definition of Gröbner bases (by Buchberger, named for his thesis advisor Gröbner), says that a Gröbner basis for some ideal in some polynomial ring is a bunch of generators for that ideal such that if we do polynomial division of any element of the polynomial in the ring with the generators, one after the other, until no more divisions could be performed, then what we get out of it all is uniquely determined.

This need not necessarily be the case, even to begin with – we could get weird loops and various kinds of bad behaviour that screws up the concept of dividing, with remainder, by a whole bunch of polynomials. Having a Gröbner basis means this is no longer a problem.

The important bit of the theorem is that we can GET a Gröbner basis by just iteratively trying out combinations of the generators, trying to find overlaps of the leading terms, and generating “bad examples”. Any bad example that doesn’t get completely reduced to 0 by division by the generators is also needed to complete the Gröbner basis, and by adjoining it we grow our generating set, but not the things it generates. And the method works by growing the generating set until anything we can build out of two of the generators really will reduce completely to 0.

And, the central theorem says, if THIS works, then reduction works in general, and we have a tool as good for solving systems of polynomial equations as the echelon form is for linear equation systems.

Losing commutativity

The next interesting step is to look at non-commutative polynomial rings. Here, we suddenly have a deep theoretic issue popping up. We know that the word problem for monoids – i.e. whether a specific string can be reduced with rewriting rules to an empty string – is unsolvable in general. It hooks up to Turing machines and the limits of theoretical computer science, but in essence we can encode problems as Gröbner basis computations for non-commutative algebras that we know cannot possibly be solved in finite time.

So we cannot hope for the situation to be as good as for commutative algebras. However, there is a theorem floating around – the Diamond lemma by Bergman – that says that if we DO get a Gröbner basis computation that halts in finite time, then all the good things that we had in the commutative case – such as reductions modulo the generators being well defined – hold for the things we’ve computed. In other words, the only thing that could go wrong would be that the computation of the Gröbner basis doesn’t finish in finite time.

Operads

Now, what I really wanted to talk about was operads.

Here, an operad is a collection {O(n) : n?1} of vector spaces with permutations attached to them. Hence, for each n, there is a map S_n ← Hom(O(n),O(n)), or in other words, we can apply any permutation of n things to any element in the component O(n) and get something back out of it.

Furthermore, an operad O = {O(n)} has defined on it structure operations. These behave like the composition of multilinear functions – so we have a way of plugging one function into another:
f(a1,a2,…,g(ai,…,aj),…,an)
and this composition is associative, so the order we figure out some sequence of compositions doesn’t matter.

These gadgets show up in modern approaches to universal algebra like questions, and also all over the place in topology; some of the earliest instances were from homotopy theory.

Recently, Dotsenko and Khoroshkin released a paper on Gröbner bases for operads. In the paper, they figure out a way to find the kind of canonical and ordered basis that you need in order to mimic the whole workflow of Gröbner bases above; and specified how one should go about producing a Diamond Lemma for operads. It turns out that Gröbner bases would be useful to prove operads to be Koszul – something I won’t discuss here, but which is important in the operad theory context.

Basically, instead of working in a polynomial ring on some set of variables, we start out with out variables in one of these graded sets {V(n)}. Then we can build rooted trees, whose internal vertices of degree n+1 are labeled by elements from V(n).

The vector space spanned by all such trees forms an operad where the composition operation works by taking a tree and attaching the root to the leaf numbered by whatever position we need to compose at.

The resulting construction is the free operad on the generating set and takes the role that the polynomial ring had in the previous examples. And what Dotsenko and Khoroshkin pointed out was that if we restrict the kinds of actions we allow from permutations on these trees somewhat, we end up with something that has exactly one representative that fits in the restricted context for each tree that occurs as a basis element of the free operad.

So we can impose some sort of ordering on these trees, and use them to mimic the Diamond lemma.

Indeed, what we end up doing is forming the overlaps between leading terms by finding trees that parts of the trees from the leading terms from pairs of operad elements can embed into, and using the resulting procedure to build the same kind of bad cases we need to test for Buchberger’s algorithm.

And again, it turns out that while we may not always get an answer within finite time, if we’re lucky then the answer we DO get has all the properties we could dream of. And not only that – all the previous kinds of Gröbner bases embed as special cases of doing it this way.

I heard of this, and got my hands on the paper, when I first arrived in CIRM in Luminy, outside Marseille, for 2 weeks of operad theory with a master’s course and a conference on the subject. And I got so excited – once upon a time this was essentially my proposal for PhD thesis project – that I decided to sit down and code the whole thing up right away!

And code away I did. Once the two weeks were gone, with the valuable help from Vladimir Dotsenko and Eric Hoffbeck, I had ended up with a working implementation in Haskell of the whole paradigm. It’s now available from the HackageDB, and runs at least in GHC 6.8 and 6.10:

And we see here that with the particular set of generators given, namely using m2 as the variable, and for a generating set, we pick
m2(m2(x1,x2),x3) + m2(x1,m2(x2,x3))
and
m2(m2(x1,x3),x2) – m2(x1,m2(x2,x3))

we get a Gröbner basis almost instantly containing additionally the generator
m2(x1,m2(x2,m2(x3,x4)))

However, as such, this is not enough to solve the issue. For one thing, we can set Q’=P and P’=Q, and things are the way he asks for.

Now, I wonder whether we can solve this by using PROPs (or possibly di-operads or something like that). Let’s represent a document as a list of some sort of tokens. We’ll set the set of all lists of length , and we’ll set to denote operations that take a list of length n and returns a list of length m.

One operation here is obvious – the identity operation. So we’ll take that into the mix. And we’ll also want to have some manner of composing operations. So, set to be the operation that applies the second argument to the elements i,i+1,…,i+m of the list outputted by the first argument.

This way, we can make patch trees – using the identities to fill out when a patch doesn’t influence everything; and have composition of operations represent composition of patches.

Now, the commutativity that Brent asks for would manifest as an additional relation – on top of those inherent in the definition of a PROP – to rebuild trees. One obvious one pops out immediately – as long as the trees don’t overlap, in other words, as long as the subtree we want to reorganize is contractible (in the topological sense), we can commute patches freely.

When trees -do- overlap, however, the undo operation Brent asks for is much more tricky. Consider the following sequence of edits:

[] -> [a] -> [b] -> [cbd]

Now, undo the insertion of [a].

Sure, taken this way, we cheat a little. Maybe we want to restrict edit descriptions to explicit deletions and additions. So we would have

[] -> [a] -> [] -> [b] -> [cb] -> [cbd]

and here we could probably move things around a bit easier. I don’t quite see how to do it, right now, though.

If you do enough of a particular brand of homotopy theory, you’ll sooner or later encounter algebras that occur somewhat naturally, but which aren’t necessarily associative as such, but rather only associative up to homotopy. The first obvious example is that of a loop space, viewed as a group: associativity only holds after you impose equivalence between homotopic loops.

In fact, if we start with a based space — the normal situation in original homotopy of topological spaces — and start looking at the space of all loops based in the basepoint of the space. A loop is simply an image of the circle in the topological space, with the circle based in 0 and running up to 2π. Then we know we can compose loops, by just running through first one on half the time, and then the other on the second half the time. So from 0 to π we run through one loop, and then from π to 2π we run through the other.

However, we now get genuinely different loops — seen as functions — from the different ways to compose three loops. Let the loops be called f,g,h. Then f(gh) is a different loop from (fg)h, since in the first loop, f happens from 0 to π and in the second, f happens only from 0 to π/2. But we were doing homotopy! So we have some continuous map, named the homotopy, that goes from f(gh) to (fg)h. So then everything is alright after all. We don’t have strict associativity in the loop space, but we have associativity up to homotopy.

But now, if we look at the different ways to compose four loops, we get odd things happening. We’d have f(g(hi)) and ((fg)h)i as to extremal versions, and then two different ways to move between them. One way is to go
f(g(hi)) → (fg)(hi) → ((fg)h)i and the other one is to go
f(g(hi)) → f((gh)i) → (f(gh))i → ((fg)h)i

So we’re stuck in the same place again, but with a higher level set of problems. Not really a big problem: we’ll simply require there to be a homotopy between these two paths. So we slot in a continuous function that takes care of that.

But then, if we look at the ways to compose five loops, we end up getting homotopies that form a spherical shell, and the same problem that we can do things in different way. But we can always make sure these are homotopical as well. And so on. The different homotopies needed are called associahedra and were introduced by Stasheff. A topological space that admits all these is called an A_∞-space. Moreover, if a topological space admits an A_∞-structure, then it is homotopy equivalent to a loop space of something.

But hang on a second. I’m doing algebra, not topology. What good is this?

Well, we certainly have a concept of associativity for algebras. And we do, once we start juggling the right kind of objects, have a concept of homotopies in an algebraic setting. So, we’ll try to mimic all of this but with a suitable setting to be able to talk about algebras that are simultaneously chain complexes.

A-infinity algebras enter the stage

So, we’ll want an algebra over some field k. We’ll start of gently, introducing it first as a graded vectorspace ; and we’ll call the component of degree p. The category of graded vector spaces come equipped with one rather natural endofunctor: the suspension S. It works by . An n-ary operator of degree k on a graded vector space V is a family of maps .

We want to tune a family of maps corresponding to the composition and higher homotopies in the topological situation, as well as handling the differential; which we want there just because we’re doing homological algebra and see a graded space. So we’ll want a differential
of degree 1, and a multiplication of degree 0. The homotopy of the associativity of the multiplication will be some 3-ary map h such that , whereby the lefthand side has degree 0, and the righthand side has degree 1 more than the degree of h, since the differential has degree 1. Thus, for each of the higher homotopies, we’ll fall one degree step. All in all, the n-ary higher homotopy operator will have degree 2-n.

We note that (just check the degrees on both sides…), and so that this, maybe slightly artificial looking, degree condition on the higher homotopies can be handled rather neatly. Each n-ary homotopy is a map
of degree 2-n. That means, that we can just as well view it as a map
of some degree; since the S operator only changes the degrees within V. We can find the degree of this map by looking at where the degree 0 slice of ends up. This is, in reality the degree n slice of , and thus ends up in degree 2 of V, which is to say that it ends up in degree 1 of SV. So the degree 2-n map defined on V turns into a degree 1 map of SV; which is rather neat.

So, each interesting map — let’s call them all is a degree 1 map . This family of maps is a part of the structure definition of an A_∞-algebra. The rest of its definition is the set of properties we want these maps to fulfill. And what would those be?

First of all, we want it to be a chain complex when we’re done, so that we can use it to do homological stuff. So we’ll want . Next, we’ll want it to fulfill the Leibniz rule, so that it really does behave like a differential. So . And we want the associativity to hold – but only up to homotopy. That f and g are homotopic means that there is some chain map h such that f-g=hd+dh for the differentials on the chain complexes that f and g go between. Now, the differential on V we take to be our , and the differential on is induced from this as the sum . Now, the various versions of associating three elements under this multiplication we’ve defined are and . Both of these are chain maps (chain maps since the Leibniz rule holds). So we’ll want , where the first d is just and the second is . If we go on with all the higher associahedra, and choose our signs in a neat way, we’ll end up with the generic condition that

So that gives us some sort of abstract feel for what an A_∞-algebra is. Do we know any examples? Can we construct any?

Examples

First of all, any algebra is a differential graded algebra concentrated in degree 0. So if for all non-zero i, and is the only non-zero structure map, then we recover the normal associative algebras.

A differential graded algebra is an A_∞-algebra, associative and not only up to homotopy. This is the same as all higher homotopies vanishing, so it is the case where are the only non-zero maps.

Hochschild cohomology

Inspired by the success of simplicial complexes in algebraic topology, the inspiration pops up to try and introduce similar things in other theories. Thus, the idea of simplicial objects appears – which are defined as a functor from the category of finite ordered sets of integers with nondecreasing monotone functions as the arrows. In this setting, face maps and degeneracy maps get introduced: the face maps miss precisely one element, and the degeneracy maps duplicate precisely one element. This corresponds closely to our intuition of faces of simplices and degenerate simplices. With a simplicial objects theory, the differential in a chain complex ends up being just the sum of faces with appropriate signs. The whole theory varies far more than here indicated though.

Suppose now we want to construct a cohomology theory which includes this at its core. One method of arriving there is the theory of Hochschild cohomology. This is built basically the same way as any other cohomology theory, with the salient difference that the differentials and degeneracy maps are chosen differently. So to a bimodule M over a ring R, we set and .
The face maps on this complex is defined as



and the degeneracy maps just slot in a at the appropriate index.

We build a complex from this by just putting . The homology of the resulting chain complex is called the Hochschild homology . Dualizing everything, we get a cochain complex of multilinear maps , and the face maps



and the degeneracy maps again just inserting a 1 at the appropriate position. The homology of the chain complex we get from has homology the Hochschild cohomology .

Now, let’s take some associative algebra B, and look at the graded algebra with . Note to the attentive reader. This construction is very reminiscent of a graded version of what sigfpe is doing with practical synthetic differential geometry. We furthermore pick some multilinear map , and define to be normal multiplication in A, and =0 for all other i. Now, we can get a new A_∞ structure by setting for all and .

Suppose first that we picked our . Then we get, from the A_∞-conditions that
(sure, anyway…)
(no problem. is still 0)
, which we can expand to, and insert values to get


where x(yz)-(xy)z=0 since B is associative anyway, and by multilinearity, we note that c(x,y)z=c(x,yz) and xc(y,z)=c(xy,z), so everything cancels out. Thus this is fulfilled by the preconditions and does not bring any additional information.

Being a Hochschild 2-cocycle, however, only mandates that , which using the definitions means that dc(x,y,z)=xc(y,z)-c(xy,z)+c(x,yz)-c(x,y)z=0, which fits perfectly with our observation. Suppose now that we pick some other . Then, the only relations that survive, since all but two operations vanish, are those where and are combined. This occurs precisely for the relations of degrees N+1 and N(N-1). For the latter case, we combine with , which gives us something multiplied with , thus vanishing. For the first relation, we receive instead
, or if we translate it to something readable, involving an application to elements, it turns into

which looks very much like the Hochschild differential on cochains.

So, this is an A_∞-algebra precisely when the chosen map c is a Hochschild cocycle.

If you’ve read this far, I’m impressed. If you’ve understood what I’ve been saying, I’m even more impressed – and probably will run across you sooner or later at some conference or other.

First out in this mathematical expose, though, is André Henriques, talking about

An operad coming from representation theory

There is a way to connect to a finite Lie algebra first it’s universal enveloping algebra and quantum groups . From representations of , one path leads on over braided tensor products to braided tensor categories. Such categories are described by operads, which occur in the study of Gerstenhaber algebras and their homology.

By instead of studying representation, studying crystals, Henriques finds an interesting operad as the result of an analogous chain of associations. A crystal is built up in analogy to a representation; as follows:

Representation	Crystal
V a vector space with direct sum decomposition to weight spaces	B a finite set with disjoint union decomposition
Chevalley operators:	Raising and lowering operators:

Would it be possible to find bases for the representation that gets mapped to “itself” under the operators, then the work would be done here, and crystals would be the same as representations. This is, however, not possible.

Which isn’t to say that they don’t have anything to do with each other. There is a bijection of isomorphism classes between representations and crystals over a Lie algebra

The analogy to the braid group, when studying the categorical properties of the crystals, is the cactus group – which also is the fundamental group of the manifold whose points correspond to isomorphism classes of real algebraic curves such that

1. each component is homeomorphic to
2. the components are glued together along a tree
3. all singularities are at gluepoints between components
4. each component has at least three points – either crossing points or marked points

These end up being governed by an operad; which in turn has 2-Gerstenhaber algebras as their homology.

Scherer, later, held a talk on

Relative homotopy cyclic homology

in which he seems to want to recast the +-construction of Quillen in operadic terms.

Next morning, the funky stuff starts. First out is Eugenia Cheng

Operads and multicategories

The talk was expository, early, brilliant and very lucid. In my humble opinion the best of the whole conference.

Cheng set out to illustrate the theory and current state of research of multicategories; and did this by displaying multicategories as a simultaneous generalisation of operads and categories. Operads describe operations with several inputs and one input. This will be generalized to encompass both many different objects and inputs in some interesting configuration. (This requires nifty pictures which I don’t really have any decent way of reproducing here. Poke me if you want the pictures that go with the text…)

Categorically, we’re motivated by the possibility of getting composition of higher cells (0-cells = objects, 1-cells = morphisms/arrows, 2-cells = natural transformations, …. an arrow between two n-1-cells is an n-cell)

Topologically, this gives us a way to deal with composition in loop spaces; or to even deal with “path spaces” with many possible objects, but compatibility requirements on compositions of paths.

So. To get her idea through, Cheng starts by giving the definition of category as she sees it:
A category C is a collection ob C of objects, for any pair of objects, a set of arrows , for any object a canonical arrow and a composition ; with axioms that make this work as expected.

A multicategory C, as defined by Lambek in 1969, is a collection of objects ob C, for any sequence of inputs and an output, b, a set of arrows, with a canonical arrow in C(a;a) and a composition of arrows.

A non-symmetric operad in the category of sets is a multicategory with one object. The hom-sets can be indexed by their number of inputs, and compotision needs no source/target matching.

This way of looking at categories with a single object is quite useful:

The topological varieties go on beyond the pair listed here, but the idea is clear.

In some terminology, multicategories are called “coloured operads”. Cheng points out that in that case, it would be consistent to call categories “coloured monoids” and groupoids “coloured groups” et.c.; which would be … poetic.

Some examples

A category is a multicategory where every arrow is unary.

A monoidal category has an underlying multicategory; with arrows in given by the arrows for the monoidal operation.

For many monoidal thingies, the monoidality is not needed, but merely a multicategory conditions. As the categorists like to hunt down minimal required conditions, this is a very relevant observation.

Now, this multicategory game could just as well be seen as using a monad to capture the input. More specifically, the “free monoid”-monad (i.e. the composition of the adjoint pair of a forgetful and a free functor) gives rise to these strings of inputs that occured up previously. By replacing by any other monad T, and expanding the obvious (for some value of obvious) axioms and propositions, you get a sane theory for T-multicategories.

By expanding on this idea, various combinations of base categories and monads in these give rise to a new, cool and exciting sequence of new categories.

Tom Leinster in Glasgow has expanded on the idea of PROPs by introducing monads in a similar manner.

After this talk, Mark Weber gave a highly technical talk on

Applications of 2-categorical algebra to the theory of operads

This talk tried, it seems, to generalize operads to higher category levels. I didn’t really understand much. At all.

Finally, Peter May gave

An oldfashioned elementary talk

Peter May started by lamenting that:

I had good results in topology to talk about – but not all here are topologists.
I had nice category theory to talk about – but not all here are categorists.
I knew some nice things about operads in algebraic geometry – but definitely not all here are algebraic geometers

So he ended up talking about constructions of Steenrod operations in various situations, based on a construction through the Eilenberg-Zilber operad. Among other things, it seems that for good algebras A, has Steenrod operations. What this means, and whether it’s of any use outside of topology, is unknown to me and quite interesting. Or so it seems.

However, we have had more talks. First out, yesterday evening, was Pascal Lambrechts

Coformality of the little ball operad and rational homotopy types of spaces of long knots

The theme of interest for this talk was long knots; i.e. embeddings of into such that outside some finite region in the middle, the embedding agrees with the trivial embedding . The space of all such is denote , and the item of study is more precisely the rational homology and rational homotopy of the fiber of the inclusion of into the space of all immersions of into .

Vassiliev approached this by constructing a spectral sequence converging to the homology for high enough d, and where the first term has a combinatorial description in terms of chord diagrams. The idea, building on some preparatory work by Dev Sinha, is to replace the long knot with a sequence of very many distinct points on the knot. Thus you end up in the configuration space of many knots, which can be put in the context of being (a compactification of) configuration spaces; which in turn form a cosimplicial space. These spaces, in fact, are (more or less) the Fulton-MacPherson operad, and using this correspondence, Lambrechts is able to show that the spectral sequence by Vassiliev collapses.

Paolo Salvatore had a speech on mainly the same theme; however, I haven’t really gotten any substantial notes out of that speech.

Today, however, started with Dev Sinha

The duality between Lie and Comm revisited

Introducing a function called a Configuration pairing; which takes a directed graph on [n] and a rooted, half-planar tree with leafs in [n] to an integer; by sending to 0 if the map that takes an edge to the lowest vertex of the path in the tree between the corresponding leafs is not bijective; and sending to (-1)^k if there are k edges such that the corresponding path runs right to left otherwise, we can get a function on the pairs that vanishes on anti-symmetry and Jacobian identity of the trees; and on anti-symmetry, loops and the Arnold identity – meaning vanishes – on the graph half; we get a perfect pairing between the slices Lie(n) generated by the trees and the slices Eil(n) generated by the graphs. Lie(n) has a basis consisting of tall trees (i.e. (((((((1,i1),i2),….,in)) and Eil(n) a basis of long graphs (i.e. paths beginning with 1). The pairing matches a tree and a graph if they have the same indices in the same order.

Using this, and involving the binomial operad, he then manages by introduction of co-Lie-brackets by removing an edge of the graph and taking antisymmetric differences to make the Eil operad describe Lie co-algebras; and thus gives a way to construct a linear duality between DG-Lie-algebras and DG-Lie-Coalgebras which does not go the way over bar and cobar constructions.

After Sinha, Martin Markl spoke on

Cohomology operations and the Deligne conjecture

He characterizes the renaissance of operad theory as being all about calculating Kontsevich graph (co)homology in various settings. More specifically, he looks on the operad of “natural operations” on multilinear functions – i.e. anything given by free compositions and pointwise multiplications. This turns out to be a rather big object, and many conjectures and questions were asked and discussed. In the ideal case, Markl wishes, the structure of the this operad will give information about the theory it’s based in.

Finally, Jonathan Scott had a speech on

Co-rings of operads

In which he gives a way of viewing an operad as a monoid, whose left modules are the algebras and right modules are the co-algebras of the operad.

That is, basically, all there is to report at this point. For all things, just poke me if you’re interested in more details.

With the morning thus came, again, the pain in the legs. However, I’m told it’ll be better if I keep on skiing.

The mathematics in this report will come sooner than in the last; mainly because the lectures start at 8.30 and not at 17.00.

First out is Bênoit Fresse with

Little cubes operad actions on the bar construction of algebras

The reduced bar construcion of augmented associative algebras is given by fixing a field , and for an augmented associative algebra giving a chain complex such that where denotes the augmentation ideal of with the differential

If the product of is commutative, then the shuffle product of tensors provides with the structure of a differential commutative algebra. In the talk, Fresse starts looking at the algebraic structure of for algebras with a homotopy commutative algebra:

Results

1. If is an algebra, then can be equipped with the structure of an algebra.
2. If is an algebra, then can be equipped with the structure of an algebra.

1 was obtained by Smirnov, by Justin Smith, …
2 was obtained for n=2 by an explicit construction by Hans Baues.
For the cochain algebra of a space similar structure results have been obtained by Kadeishvili-Saneblidze and by Hess-Parent-Scott.

1 can be motivated in arXiv:math.AT/0601085.

2 is a consequence of the assertion that the endomorphism prop of the bar construction is equivalent to the prop of co-associative and -multiplicative bialgebras.

Next up is Clemens Berger, talking about

On the combinatorial structure of E_n operads

May et.al. studied the n-fold loop spaces of maps of the n-sphere to the space, by considering the k-fold n-fold loop space, i.e. maps . It turns out that it isn’t necessary to study the entire space of maps but it’s enough to look at the suboperad of “little n-disks” .

Theorem (Boardmann-Voft, May, Segal): Each n-fold loop space is canonically a [tex[\mathcal D_n[/tex] algebra.

gives rise to an operad. gives rise to an operad. If we can find a similar intrinsic characterisation of operads for other n, it would yield a combinatorial structure of an n-fold loop space.

Fiedorowicz has been able to obtain such a classification for n=2 (with symmetric props replaced by braid groups).

The next talk was held by Muriel Livernet on

The associative operad and the weak order on the symmetric groups

Livernet started out by defining symmetric and nonsymmetric operads, and gave a pair of adjoint functors – the forgetful functor from symmetric to non-symmetric operads, and tensoring with the group algebra to symmetrize a non-symmetric operad.

At this point, parts of the lighting in the room fell down on the audience. No serious consequences.

The weak Bruhat order can be defined as ordering permutations if where is the inversion set of the permutation, i.e. the set of pairs i,j such that i.

Generating a new basis for the associative operad, using the Möbius inversion on the relevant Bruhat order, the composition rule ends up being a summation over a specific interval in the Bruhat order.

Interesting combinatorial question asked: How many permutations are there (of a fixed length) such that no proper substring of the permutation (written as a permuted string) is an entire interval. For instance, (2,4,1,3) has no proper substring that fills out an interval, whereas (2,4,3,1) has the substring 2,4,3 which fills out [2,4].

Shouldn’t this question be easily answered by simply counting all permutation that DO have a subinterval? It’s easy enough to count your way through the intervals – choose a length and a startingpoint. There are length! ways to embed that interval, and easily enough controlled number of points to insert the interval. The remaining parts are also easily distributed. However, double counting will occur – which messes up the counting quite a bit. Maybe using inclusion-exclusion will lead the way? For, for instance [2,4] to be counted a particular time, you’d want [2,4] to be embedded without contact to 3 and 5, as well as the remaining two sections to be interval-free. Maybe it’s doable with some sort of recursion? There are, even then, cases of permutations with several intervals embedded, for instance (2,4,3,6,9,8,1,7) which contains [2,4] and [8,9].

As I’m writing this, they’re still out there – I went back when the pain in my legs caused tears in my eyes for just turning on the skis. The techniques were solid as concrete. The muscles not so much. It took half an hour in the sauna to get to the point where I actually was able to walk again.

Among the more amusing things that happened was that when I was stopping to wait up for Arne and Miles in a narrow forest path, I ended up standing too close to the edge, which subsequently gave up and dropped me down into a few meters of powder just under a fern tree. Getting out of there was awkward – to begin with my legs were in the wrong angle to get out of the ski bindings; and once Miles helped me out, the only reason I wasn’t buried in snow to my shoulders was that I packed it as I stood on it, and the carrying point ended up being roughly waistdeep.

Go on. You try it. Get up from waistdeep loose powder snow. Straight up, a meter or so, onto that hardened shell that you once were skiing on. It’s an extremely amusing and rather hard exercise.

Now, for the mathematics. The (only) talk today was by Bruno Vallette, on

Manin products and Koszul duality

For associative quadratic algebras, given as a quotient of the free tensor algebra on a space by , Manin defines two different products, by for the standard twisting homomorphism; and . The main and most relevant result here is that the two products are Koszul dual to each other, i.e. .

Using the concept of lax 2-monoidal categories, it is possible to generalize this to basically any possible interesting category – with the same particular construction viable for Algebras, Nonsymmetric and symmetric operads, dioperads, coloured operads, properads, PROPs (and probably ½PROPs as well).

Notes on this will be forthcoming, and I’ll post here once I get a relevant URL.

Except for that, though, the travel went fine, and after being treated to some immensely beautiful views (glittering lake of Genéve with rows and rows of snowcovered grapevines in front, anyone?) and reminded of just how much I miss the deep-snow winters, I got up on this mountain in southwestern (very much frenchspeaking) Switzerland to the Hotel du Golf. The receptionist told me, straight off, that a number of my colleagues had already arrived, and then I went to eat (Crêpes – expensive and not even correctly delivered…) and started wrestling the connector dance. Y’see, half of the connectors used in civilized Europe work here. The other half don’t. And those who do work, only do work if they’re impeccably straight. So I had to work for quite a while to actually, y’know, get my laptop, my mp3player, my loudspeakers (I sleep with music, mmmkay? Headphones are NOT very nice to sleep in, mmmkay?) and my cell phone all connected. I think I disconnected 75% of the room lights in the process.

Just before leaving, i.e. yesterday, I discovered that the colleague of Gooseania, Hadi Zare is also coming here to this workshop. I plan on saying Hi.
Most of my raison d’être for this trip has evaporated since friday though. It was supposed to be a threefold thing: Skiing, preparing for an Operadic PhD and looking for application spots to get that PhD position. Since I now HAVE a position (Jena, Group Cohomology) and it’s not in operads, the latter two kinda fell into oblivion. But I’m looking forward to skiing!!

I will do a 30-minute stunt online each day, trying to cram it full of updating here, reading my mail, webcomics, livejournals, blogs, whatever, calling my fiancee on Skype and generally hanging out on IRC.

Don’t count on me being very available…

This, in turn, leads to several different themes for me to discuss.

Cryptographic hashes

In the weeks finds, John Baez comes up to speed with the cryptographic community on the broken state of SHA-1 and MD-5. Now, this is a drama that has been developing with quite some speed during the last 1-1½ years. It all began heating up seriously early 2005 when Wang, Yin and Yu presented a paper detailing a serious attack against SHA-1. Since then, more and more tangible evidence for the inadvisability of MD-5 and upcoming problems with SHA-1 have arrived – such as several example objects with different contents and identical MD-5 hashes: postscript documents (Letter of Recommendation and Access right granting), X.509 certificates et.c.

The attacks that occur are what is called collision attacks by those in the trade. They find data and such that . The other serious attack is the preimage attack, where to a specific hash , a message is found such that . Once preimages can be found, it’s time to bail ship quickly. Up until then, it may possibly be enough to pack the life boats and look toward the horizon for possible rescues or deserted islands.

Very interesting reads and link sources for this can be found, as with most things cryptographic, with Bruce Schneier.

Topology of computation

John Baez talks about some lectures he gave at the Geometry of Computation conference in the section of Universal Algebra and Diagrammatic Reasoning. Reading through what he did and looking over the conference as such, I realize I would very much have wanted to be there. They’re doing linguistics and algebraic topology together, for one!

John Baez’ talk (well worth to review the slides! Go do it!) goes through a lot of the themes I’m busying myself with from the vantage point of computer science and computation theory. He talks about PROPs, about PROs, about Monads and categorical monoids, about diagrams and algebraic theories et.c. Most of it very interesting, and rather close to my own work. If I now can work out some good method to display that the Koszul resolutions of PROPs are the same kind of resolutions that Baez uses to get simplicial objects representing computations, I might just close in to something that may give a good explanation as to why the things I like are relevant.

I will bring my post on Operads and PROPs at some point in the future. As will I bring my expose over Koszulness. But not today.

Geometry of roots

The third eye-catcher I found was an absolutely gorgeous picture of all roots of polynomials of degree at most 5 and with integer coefficients in . It makes me want to build more pictures. Now, if I may. Dunno if this is what I -should- be spending the processor cycles of my workplace on, but damn, they are gorgeous!

I wonder how hard it would be to hack a Pari/GP script that spews out coordinates and colour tags for this with coefficients in in a format amenable to producing pretty graphs. Some chosen thoughts awake with the comments from John Baez – among other things, he states that

Odlyzko and Poonen proved some interesting things about the set of all roots of all polynomials with coefficients 0 or 1. If we define a fancier Christensen set to be the set of roots of all polynomials of degree with coefficients ranging from to , Odlyzko and Poonen are studying in the limit . They mention some known results and prove some new ones: this set is contained in the half-plane and contained in the annulus where is the golden ratio . In fact they trap it, not just between these circles, but between two subtler curves. They also show that the closure of this set is path connected but not simply connected.

Now, if the set isn’t simply connected, then I’m immediately growing interested in the homology and homotopy of the set.