Spectral Sequences in Homotopy Type Theory Floris van Doorn Carnegie Mellon University June 4, 2018 Joint work with Jeremy Avigad, Steve Awodey, Ulrik Buchholtz, Egbert Rijke and Mike Shulman. Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 1 / 27 Outline Cohomology in HoTT Spectral sequences Atiyah-Hirzebruch and Serre spectral sequences for cohomology Future work: Spectral sequences for homology Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 2 / 27 Classical theorem: The Eilenberg-MacLane spaces K(A; n) classify cohomology. Recall. The Eilenberg-MacLane space K(A; n) is the unique pointed type X with one nontrivial homotopy group πn(X) ' A. Cohomology How do we define cohomology? The usual constructions of singular cohomology is not homotopy invariant. Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 3 / 27 Cohomology How do we define cohomology? The usual constructions of singular cohomology is not homotopy invariant. Classical theorem: The Eilenberg-MacLane spaces K(A; n) classify cohomology. Recall. The Eilenberg-MacLane space K(A; n) is the unique pointed type X with one nontrivial homotopy group πn(X) ' A. Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 3 / 27 Cohomology We define the reduced cohomology of a pointed type X with coefficients in an abelian group A to be n ∗ He (X; A) :≡ kX ! K(A; n)k0: The unreduced cohomology can be defined similarly for any (not necessarily pointed) type X: n n H (X; A) :≡ kX ! K(A; n)k0 = He (X + 1;A): The group structure comes from the equivalence K(A; n) '∗ ΩK(A; n + 1). Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 4 / 27 Long Exact Sequence of Homotopy Groups Given a pointed map f : X !∗ Y with fiber F . Then we have the following long exact sequence. : : π2(f) π2(F ) π2(X) π2(Y ) π2(p1) π1(δ) π1(f) π1(F ) π1(X) π1(Y ) π1(p1) π0(δ) π0(f) π0(F ) π0(X) π0(Y ) π0(p1) Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 5 / 27 Spectral Sequences p;q DefinitionA spectral sequence consists of a family Er of abelian groups for p; q : Z and r ≥ 2. For a fixed r this gives the r-page of the spectral sequence. ::: q q • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • p;q p p;q p E2 E3 Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 6 / 27 Spectral Sequences p;q p;q p+r;q−r+1 Definition ::: with differentials dr : Er ! Er such that dr ◦ dr = 0 (this is cohomologically indexed) ::: q q • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • p;q p p;q p E2 E3 Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 6 / 27 Spectral Sequences p;q p;q p;q Definition ::: and with isomorphisms αr : H (Er) ' Er+1 p;q p;q p−r;q+r−1 where H (Er) = ker(dr )=im(dr ). q q • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • p;q p p;q p E2 E3 Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 6 / 27 Spectral Sequences p;q p;q p;q Definition ::: and with isomorphisms αr : H (Er) ' Er+1 p;q p;q p−r;q+r−1 where H (Er) = ker(dr )=im(dr ). The differentials of Er+1 are not determined by Er. q q • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • p;q p p;q p E2 E3 Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 6 / 27 And we can get information about the diagonals on the infinity page. Convergence of Spectral Sequences q • • • • • In many spectral sequences the p;q • • • • • pages converge to E1 . • • • • • • • • • • • • • • • D0 D1 D2 D3 D4 p p;q E1 Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 7 / 27 Convergence of Spectral Sequences q • • • • • In many spectral sequences the p;q • • • • • pages converge to E1 . • • • • • And we can get information about the diagonals on the • • • • • infinity page. • • • • • D0 D1 D2 D3 D4 p p;q E1 Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 7 / 27 p;q p;q The second page is E2 = C ; p;q The spectral sequence converges to E1 ; n p;q D is built up from E1 for n = p + q in the following way: We have short exact sequences: 0;n n n;1 0 ! E1 !D ! D ! 0 : : p;q n;p n;p+1 0 ! E1 !D ! D ! 0 p+1;q−1 n;p+1 n;p+2 0 ! E1 !D ! D ! 0 : : n;0 n;n 0 ! E1 !D ! 0 p;q if there exists a spectral sequence Er such that Convergence of Spectral Sequences For a bigraded abelian group Cp;q and graded abelian group Dn we write p;q p;q p+q E2 = C ) D Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 8 / 27 p;q p;q The second page is E2 = C ; p;q The spectral sequence converges to E1 ; n p;q D is built up from E1 for n = p + q in the following way: We have short exact sequences: 0;n n n;1 0 ! E1 !D ! D ! 0 : : p;q n;p n;p+1 0 ! E1 !D ! D ! 0 p+1;q−1 n;p+1 n;p+2 0 ! E1 !D ! D ! 0 : : n;0 n;n 0 ! E1 !D ! 0 Convergence of Spectral Sequences For a bigraded abelian group Cp;q and graded abelian group Dn we write p;q p;q p+q E2 = C ) D p;q if there exists a spectral sequence Er such that Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 8 / 27 p;q The spectral sequence converges to E1 ; n p;q D is built up from E1 for n = p + q in the following way: We have short exact sequences: 0;n n n;1 0 ! E1 !D ! D ! 0 : : p;q n;p n;p+1 0 ! E1 !D ! D ! 0 p+1;q−1 n;p+1 n;p+2 0 ! E1 !D ! D ! 0 : : n;0 n;n 0 ! E1 !D ! 0 Convergence of Spectral Sequences For a bigraded abelian group Cp;q and graded abelian group Dn we write p;q p;q p+q E2 = C ) D p;q if there exists a spectral sequence Er such that p;q p;q The second page is E2 = C ; Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 8 / 27 n p;q D is built up from E1 for n = p + q in the following way: We have short exact sequences: 0;n n n;1 0 ! E1 !D ! D ! 0 : : p;q n;p n;p+1 0 ! E1 !D ! D ! 0 p+1;q−1 n;p+1 n;p+2 0 ! E1 !D ! D ! 0 : : n;0 n;n 0 ! E1 !D ! 0 Convergence of Spectral Sequences For a bigraded abelian group Cp;q and graded abelian group Dn we write p;q p;q p+q E2 = C ) D p;q if there exists a spectral sequence Er such that p;q p;q The second page is E2 = C ; p;q The spectral sequence converges to E1 ; Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 8 / 27 We have short exact sequences: 0;n n n;1 0 ! E1 !D ! D ! 0 : : p;q n;p n;p+1 0 ! E1 !D ! D ! 0 p+1;q−1 n;p+1 n;p+2 0 ! E1 !D ! D ! 0 : : n;0 n;n 0 ! E1 !D ! 0 Convergence of Spectral Sequences For a bigraded abelian group Cp;q and graded abelian group Dn we write p;q p;q p+q E2 = C ) D p;q if there exists a spectral sequence Er such that p;q p;q The second page is E2 = C ; p;q The spectral sequence converges to E1 ; n p;q D is built up from E1 for n = p + q in the following way: Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 8 / 27 Convergence of Spectral Sequences For a bigraded abelian group Cp;q and graded abelian group Dn we write p;q p;q p+q E2 = C ) D p;q if there exists a spectral sequence Er such that p;q p;q The second page is E2 = C ; p;q The spectral sequence converges to E1 ; n p;q D is built up from E1 for n = p + q in the following way: We have short exact sequences: 0;n n n;1 0 ! E1 !D ! D ! 0 : : p;q n;p n;p+1 0 ! E1 !D ! D ! 0 p+1;q−1 n;p+1 n;p+2 0 ! E1 !D ! D ! 0 : : n;0 n;n 0 ! E1 !D ! 0 Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 8 / 27 Serre Spectral Sequence (Special Case) Theorem. Suppose f : X ! B and b0 : B and let F :≡ fibf (b0). Suppose that B is simply connected and A is an abelian group. Then p;q p q p+q E2 = H (B; H (F; A)) ) H (X; A): This is only true for unreduced cohomology. Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 9 / 27 Example: Cohomology of K(Z; 2) 1 We will compute the cohomology groups of B = K(Z; 2) (which is CP ). f We define the map 1 −! K(Z; 2) determined by the basepoint ? : K(Z; 2).