Hall Algebras and Bridgeland Stability Conditions for Stable -categories

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1 Hall Algebras and Bridgeland Stability Conditions for Stable -categories Hiro Lee Tanaka, Harvard University Goodwillie Birthday Conference, June 25, 2014 Dubrovnik, Croatia

3 Today s topic: Factorization homology? Metrics and Goodwillie calculus? Hidden seeds from the s-dot construction: We ll place two ideas (usually treated underived) into derived setting.

5 Defn. An -category C is called stable if 1. C has a zero object, 2. Every morphism f : X Y admits a fiber and a cofiber X f Y, K X 0 X /Y f 0 Y, 3. A square is a pushout if and only if it is a pullback. Examples C = Spectra, Chain complexes. Remark Letting X [1] := 0/X, we have hom(x, Y ) Ω hom(x, Y [1]). A cube is Cartesian if and only if it is cocartesian.

11 Definition The Hall algebra H(A) is the vector space with multiplication H(A) := [M] π 0 A Q [M][N] := [L] # M L such that M = M and M /L = N. [L] Remark (We re counting the number of short exact sequences M L N.) Example If A =Vect f.d. F,coefficients count points in the Grassmanian of m-planes in l dimensions.

12 More generally [M][N] := [L] χ(m, L, N)[L] where χ(m, L, N) is some Euler characteristic of the space of short exact sequences. Examples 1. Just counting (in finite, discrete case). 2. χ(m, L, N) = π 0 {M L N} π 0 Aut(M) i>0 π i hom(m,l) ( 1)i π i hom(m,m) ( 1)i (Toen) 3. Behrend function, which is like an Euler char. for stacks.

13 Why should we care? 1. H(Rep F (Q)) = Un +. Righthand side is positive Borel part of quantum group; Q is associated Dynkin quiver. (Ringel, Schofield, Lusztig.) In fact, one can recover all of the quantum group by taking 2-periodic Q-representations. (Bridgeland) = Another way to create Chern-Simons field theory 2. Relations in the Hall algebra help us count algebraic curves in algebraic geometry. (Bridgeland shows D-T invariants are P-T invariants.) Bridgeland stability conditions! 3. Many results are best understood derived: Wall-crossing formulas. So we should understand more homotopy-theoretic ways to see this algebra. 4. Fix issues: Hall algebra not functorial. Finiteness conditions annoying. (We ll fix both today; somewhat new methods.)

16 Why did associativity work? s 3 A s 2 A s1 A s 2 A s 2 A s1 A s 2 A (Three-term sequences can be glued up from two-term sequences with common terms.) More generally, s n A s 2 A s1 A... s1 A s 2 A, n 3. s A is an example of a 2-Segal space. (Dyckerhoff-Kapranov)

17 Replacing Hall algebra by a Hall category Definition Let C be a stable -category. Let Hall(C) :=Spaces /s1 C Spaces / ob C. Define a monoidal structure by X C Y s 2 C ob C X Y ob C ob C. Theorem (Dyckerhoff-Kapranov) (Hall(C), C ) is an E 1 -monoidal -category.

18 Functoriality Consider a functor F : C D between stable -categories. Assume it preserves colimits, so it is exact. X C Y X D Y s 2 C F s 2 D ob C F X Y = X Y ob C ob C F ob D ob D So instead of an equivalence F (X C Y ) F (X ) D F (Y ), we have a map in D, F(X C Y ) F (X ) D F (Y ). i.e., F defines a colax functor.

19 Functoriality Upshot: Any exact functor from C D gives rise to a colax functor Hall(C) Hall(D). Theorem (Eventually) The construction Hall : St Cat ex E 1 Cat colax is a functor. Remark Why eventually? No easy notion of colax functors yet in -operads. Remark Compare: Hall algebras are functorial only for fully-faithful, extension-closed embeddings. (I.e., very rarely.)

20 Other properties, some pending colax formulation 1. We expect Hall to be lax monoidal; essentially because we always have a map s C s D = s (C D) s (C D) = If C is an stable E n 1 -category, then Hall(C) isan E n -category. (Probably uninteresting example: C = Mod(A) for an E n -algebra A.) (Less uninterestingly, sometimes we can put a comonoidal structure on Hall(C), so using bar construction, we may get an E n+1 algebra out of an E n algebra. Problem: lax functors don t preserve comonoids.) 2. Let be a heart for C. Then one can define Hall( ) using the exact same definition, and there is always a fully faithful, monoidal functor Hall( ) Hall(C). = Since is always discrete, one can see the Abelian case of Hall algebra as sitting inside this stable case.

21 Getting relations in Hall categories: Bridgeland Stability Roughly, a relation says: Creating X by certain extensions is the same thing as creating it out of some other series of extensions. One way to do this is to use Bridgeland stability conditions.

22 Bridgeland stability conditions for an abelian category Let A be an abelian category. A linear function Z : K 0 (A) (C, +) such that 1. Z(ob {0}) H R Z has the Harder-Narasimhan property is called a Stability function for A.

28 Definition A Bridgeland stability condition for C a stable -category is a pair (, Z) where is the heart of a bounded t-structure on C, andz is a stability function on. Theorem (Bridgeland, Kontsevich-Soibelman) Fix a surjective group map K 0 (C) Γ, and consider the set of all stability conditions K 0 (C) C factoring through Γ. Thissethasa topology and an atlas making it a complex manifold. Remark Not every heart admits a stability function. Hard to construct: still unaccomplished in full generality for Calabi-Yau 3-folds and D b Coh. Conjectural for Fukaya categories. Plenty for representations of acyclic quivers. Often, K 0 Γ is Chern character map.

Derivatives of the identity functor and operads

Derivatives of the identity functor and operads University of Oregon Manifolds, K-theory, and Related Topics Dubrovnik, Croatia 23 June 2014 Motivation We are interested in finding examples of categories in which the Goodwillie derivatives of the identity