What is tautological ring
First defined by Mumford,
What is , moduli space of curves of (arithmetic genus ) and marked (smooth) points, all singularities simple nodes; stability condition: every irreducible component of genus zero must have distinguished points should be stability condition gurantees no automorphisms, it is a DM stack smooth and proper of dim 3g-3+n
Intersection theory means studying Chow rings , there is a cycle map from this ring to the cohomology , tautological ring is a subring of the Chow ring, classes naturally arise in geometry. If we restrict this to tautological ring, is it an isomorphism to its image.
Maps between : 1. forget a marked point (need to restablize if necessary by contracting unstable component) 2. glueing map
Tautological ring is simultaneously defined as the smallest subring of the Chow ring closed under pushforward under the above maps starting with fundamantal class in each space
Essentially tautological divisors are boundary divisors of specific shapes.
But we can also get chern class?
drawing dual graph of stable curves convey same info
There is a permutation representation of on tautological
Something called Grothendieck-Teichmuller group
profinite version
Goal of Grothendieck: understand
uncountable group, what are its elements, action on for certain schemes .
LHS is acted on by , so is ;
E.g. , then
the action is the cyclotomic character
Remove one more point: , now . Belyi prove that this is a faithful representation.
Idea: constrain image by way of various maps behave between moduli space
Drinfeld’s subgroup subgroup of the form : Let be the two generators of , , , such that 1. 2. if . 3. where is generator of . Theorem (Drinfeld-Ihara): Image is contained in conjecture is it is exactly that
Connection to topology: configuration space (ordered configuration space), unordered analog. braid group, . is a fibration with fiber over $(0,1) equal to , is isomorphic to , fibration split so
Braid group appear as automorphisms of objects in braided monoidal category
Drinfeld’s definition: mess with all structures of bracat
Notice: little disks operad
Theorem (Horel, based on Drinfeld, Bar-Natan)
The free algerba is graded (i.e. bracket preserves degree); There should exist an adjunction from category of lie algebras to that of associative algebras making the diagram of free functors commute. Indeed, this is the functor of forming universal envelopping algebra.
A -coalgebra is a vector space equipped with a map and a counit such that it is coassociative and satisifies counitality ( is identity and another one for right counitality) Bialgebra: is ring hom
Monoidal cat - , braiding - , symmetric if Example: product of two sets, tensor product of vector spaces, graded vector space with braiding given by Koszul sign convention
An algebra in a monoidal category is a monoidal object (bilinearity is distributive law)
If is braided and are algebras, then so is .
A bialgebra in a braided category is a algebra and coalgebra s.t. comultiplication is a mp of algebras. It is moreover a Hopf algebra if there is an antipode (like inverse) such that is equal to the composite .
Example: group algebra, universal envelopping algebra of Lie algebra (Milnor-Moore: There is an equivalence of category between the category of Lie algebras to that if primitively generated Hopf algebras if is characterstic zero, one direction is given by taking universal envelopping algebra, the other is given by taking primitive elements)
Let be the set of group-like elements, then there is a map of algebras .
(Cartier-Kostant-Milnor-Moore) If is Hopf algebra over algebraically closed field, and characteristic 0 and co-commutative, then (does use algebraically closed)
impplies if is generated by group like elements, then is isomorphic to
natural isomorphism (same universal property)
Completed tensor product: , but completion of vector spaces are not unique, https://en.wikipedia.org/wiki/Complete_topological_vector_space, though there is a unique Hausdorff completion. We want to make the category of TVS with completed tensor product to be a monoidal category
If is filtered by , we can define the completion to be the inverse limit of and the topology is given by the local base ; can be filtered
If is graded vector space , then
If is graded Lie algebra, then is completed Lie algbra (need to use that is positively graded), but it is not graded vector space.
Filter: be the ideal generated by such that such that . Let . then becomes a completed Hopf algebra, and it is isomorphic to
Let be a complete positive graded Lie algebra, over a field of characteristic 0. Define , we define multiplication . A priori this only makes sense in , but the content of BCH is that this has an alternate expression
Example: If is the completion of the free Lie algebra , there is an isomorphism of groups from to the group like element of the completed Hopf algebra (It is important to complete it, since the only group-like elements in the usual envelopping algebra is 1 from PBW theorem; conversely, the only primitive element in a group Hopf algebra is , as can be checked by direct computation.)
Sketch: , and is primitive iff is group-like (.
Let be a symmetric monoidal cateogory. Let be an object and let . There is an action of on (transposition acts by the braiding, which is ok since it is symmetric; we also need to check if and which follows from the braiding axiom)
There is an insertion operation: for every and ,
Sometimes has more structure than just a set, can ask that is enriched over another monoidal category , i.e. is an object in , and composition is a morphism in , and we also want to ask is cotensored over , i.e. we can take tensor product of an object in and and also as objects in satisfying various axiom. For example, if is a subcategory of , e.g. the category of vector space is cotensored over that of sets by defining .
Let be the groupoid of finite sets. The skeletons are and morphisms are from an object to itself and equal to . A symmetric sequence is a -module in a category is a functor , determined up to natural equivalence by restriction to , so it is nothing but a colletion of objects with an action of on . An operad in is a symmetric sequence with maps, for any finite sets, , we have an morphism , and it should be natural in (equivariant condition), which are associative: 1. Plugging in two different slots of commute 2. Plugging in then in is the same as first plugging into then plugging into . There is also a unital axiom.
Example: if is enriched, cotensored over , then is an operad. If is a finite set, then where is the tensor product of times, each copy labelled by elements of .
Let nature do its work and forget
An algebra structure on an object over an operad is a morphism between operads. If there is a hom-tensor adjunction then this is saying .
Examples: , with right action of on itself; , is given by block insertion
Why is it called ? Because an object in a monoidal category is an associative algebra iff it has the structure of -algebra, i.e. monoidal structure on ( gives the multiplication. The other direction is permuting the factor); Also note that the associativity axioms in operads means that insertion of operations are associative, it doesn’t mean that the operation is associative in general (but it doesn in this case).
If is an operad in , and is a (lax) monoidal functor, then is an operad in . The associative operad in -modules is , explicitly, the -th term is
Operad of parenthesized mutations , e.g. , ,((4(23))1)$ are elements of , equivalently is the set of binary rooted trees labelled by .
Little disk operads: A -map is a function from the open unit disk to itself of the form where and . , this can be topologized as a subspace of
Commutative operad: . Structure maps are forced by definition. There is a map of operad , so if is a Com-algebra, it is also a Assoc-algebra. THere exists map of operads by forgetting the parentheses. What is a -algebra, it is just binary operations on .
In , recall , then the Lie operad is the suboperad generated by . Jacobi idenitty is forced by the suboperad. We will see that Lie is freely generated by with the sign representation of subject to the Jacobi identity.
In this language universal envelopping algebras are adjoint to forgetful functor
is free operad generated by .
Free operads: Left adjoint to the forgetful functor from operads to symmetric sequences
Graph:= collection of half-edges (connected to one vertex), a vertex is an equivalence class of half edges, an involution on the half edges, fixed points are legs, internal edges are pairs of half edges swapped under involution
We can produce a 1D CW complex homeomorphic to a graph. A tree is a graph whose top space is simply connected, and rooted tree is a tree with a leg picked as a root, the other legs are leaves. Since the graph is a 1-dimensional simply connected there exists unique path from any vertex from leaf to the root, and we direct the edge according to it. for any vertex , we have the incoming half edges and the outgoing half edges.
Digression: how to start from each one with an action of to get a functor : , and this natural in (by postcomposing with )
Explicit description of free operad: Let be a symmetric sequence: Define for some finite set . The idea is direct summing over all trees and all labels on leaves by of a certain object in . The object is and we can define
Ideals of operads: collection of subobject closed under insertion from left. A presentation of an operad is an isomorphism
Examples: In Set, is the free operad on where is a symmetric sequence with at degree 2 and otherwise. In vec, is where is the nonidentity element of (see Ginzburg-Kapranov Koszul duality for operads for reference). Similarly for the Lie operad (proof?)
Recall is and similarly for , it is independent of if is a connected manifold with
The intuition is that we can think of a loop in the (paranthesized) configuration space of as braids in three dimensions by thinking of time as the vertical axis.
Yetter-Drinfeld modules: group, be the category with object right -modules which decompose such that where and morphisms are linear maps preserving the action and the grading
Monoidal structure: with action via diagonal action.
Braiding: is going to send to (similar to semidirect product) where
Monoidal functor (lax): is monoidal if there exists natural transformation (colax if it is the other direction)
If is an algebra in and is lax monidal then is an algebra in (coalgebra if it is colax monoidal)
operad - trees + graph complexes (Ginzburg - Kapranov) Koszul duality for operad
May’s Geometry of iterated loop space (older definition)
smirnov - operad as monoid in certain category (symmetric sequence)