Loday construction in functor categories

Size: px

Start display at page:

Download "Loday construction in functor categories"

Patience Harper
5 years ago
Views:

1 Wayne State University AMS Bloomington April, 1st 2017

2 Motivation Goal: Describe a computational tool (using some categorical machinery) for computing higher order THH, which is an approximation to iterated algebraic K-theory.

3 Motivation Goal: Describe a computational tool (using some categorical machinery) for computing higher order THH, which is an approximation to iterated algebraic K-theory. Algebraic K-theory K(R) iterated algebraic K-theory K (n) (R) ) vector bundles n-vector bundles K(2-vector bundles /k) K(K(k)) (Baas-Dundas-Richter-Rognes, Osorno) E 1 -ring spectra E n -ring spectra Deligne conjecture for K-theory (Blumberg-Gepner-Tabuada, Barwick) Quillen-Lichtenbaum Ausoni-Rognes red-shift conjecture K (n) : ht. m spectra ht. m + n spectra

4 Approximating iterated algebraic K-theory For a commutative ring spectrum R, we can approximate iterated K-theory using the iterated trace map K(K(... K(R)... )) THH(THH(... THH(R)... )) where THH(R) = S 1 R.

5 Approximating iterated algebraic K-theory For a commutative ring spectrum R, we can approximate iterated K-theory using the iterated trace map K(K(... K(R)... )) THH(THH(... THH(R)... )) where THH(R) = S 1 R. Now we observe that S 1 (S 1... (S 1 R)) = T n R since CommS has all colimits weighted in ssets (McClure-Staffeldt). We d therefore like to have tools for computing T n R and related invariants.

6 The Loday construction Let X be a simplicial (finite) set and I commutative monoid in a symmetric monoidal model category C.

7 The Loday construction Let X be a simplicial (finite) set and I commutative monoid in a symmetric monoidal model category C. Definition Define X I to be the simplicial object in C with n-simplices (X I ) n = x X n I {x} and with face maps defined by d i : x X n I {x} y X n 1 I {y} x X n I {x} y X n 1 x d 1 i (y) I {x} y X n 1 I {y} and similarly for degeneracy maps.

8 Functor categories: symmetric monoidal product Let (D, ) and (C, ) be symmetric monoidal categories enriched in C.

9 Functor categories: symmetric monoidal product Let (D, ) and (C, ) be symmetric monoidal categories enriched in C. Let I, J Fun(D, C), then the Day convolution is defined on objects using the coend (I Day J)(c) = (a,b) D D D(a b, c) I (a) I (b) and on morphisms by induced maps of coends.

10 Functor categories: symmetric monoidal product Let (D, ) and (C, ) be symmetric monoidal categories enriched in C. Let I, J Fun(D, C), then the Day convolution is defined on objects using the coend (I Day J)(c) = (a,b) D D D(a b, c) I (a) I (b) and on morphisms by induced maps of coends. Theorem (Day) The category (Fun(D, C), Day ) is a closed symmetric monoidal category and the category of commutative monoids in Fun(D, C) is equivalent to the subcategory category of lax symmetric monoidal functors Fun LS (D, C).

11 Functor categories: model structure Let C be a combinatorial cofibrantly generated symmetric monoidal model category satisfying the SS monoid axiom and let D have virtually cofibrant function spaces. Theorem (Isaacson) The category Fun(D, C) with the projective model structure and Day convolution is a symmetric monoidal model category that satisfies the SS monoid axiom. Following White, we say M satisfies the strong commutative monoid axiom (SCMA) if for any (acyclic) cofibration h the map h n /Σ n is also an (acyclic) cofibration.

12 Functor categories: model structure Lemma (A-K) In addition, let C satisfy the SCMA, and let D be a POSet enriched in C (D(a, b) = 1 C or 0). Then the functor category Fun(D, C) with the projective model structure satisfies the strong commutative monoid axiom. White proved that if a model category M satisfies (SCMA) then CommM has the model structure inherited from M and cofibrations in CommM with cofibrant source forget to cofibrations in M. In particular, cofibrant objects in Fun LS (D, C) forget to cofibrant objects in Fun(D, C).

13 Filtered commutative ring spectra Definition A filtered commutative ring spectrum is a cofibrant object in Fun LS (N op, S) with the model structure inherited from the projective model structure on Fun(N op, S)

14 Filtered commutative ring spectra Definition A filtered commutative ring spectrum is a cofibrant object in Fun LS (N op, S) with the model structure inherited from the projective model structure on Fun(N op, S) This is the same data as a sequence of cofibrations... I (2) I (1) I (0) between cofibrant objects with structure maps ρ i,j : I (i) I (j) I (i + j) satisfying commutativity, associativity, unitality, and compatibility.

15 Filtered commutative ring spectra Definition A filtered commutative ring spectrum is a cofibrant object in Fun LS (N op, S) with the model structure inherited from the projective model structure on Fun(N op, S) This is the same data as a sequence of cofibrations... I (2) I (1) I (0) between cofibrant objects with structure maps ρ i,j : I (i) I (j) I (i + j) satisfying commutativity, associativity, unitality, and compatibility. Now, for a simplicial (finite) set X we can form X I where I Fun LS (N op, S) where S is the category of symmetric spectra of simplicial sets (with the positive flat stable model structure).

16 May filtration of the generalized bar construction Let I Fun LS (N op, S), then the May filtration of X I (0) is the Loday construction in Fun LS (N op, S);

17 May filtration of the generalized bar construction Let I Fun LS (N op, S), then the May filtration of X I (0) is the Loday construction in Fun LS (N op, S); i.e X I. It is the filtered simplicial spectrum ( I )(0) Day x X 0 ( Day x X 1 I )(0) ( Day x X 2 I )(0)... ( I )(1) Day x X 0 ( I )(1) Day x X 1 ( Day x X 2 I )(1)... ( I )(2) Day x X 0 ( I )(2) Day x X 1 ( Day x X 2 I )(2)......

18 Example: The object (I Day I )(n) is the colimit of the n-th truncation of the diagram I (2) I (2) I (1) I (2) I (0) I (2)... I (2) I (1) I (1) I (1) I (0) I (1)... I (2) I (0) I (1) I (0) I (0) I (0).

19 THH-May spectral sequence Theorem (A-K, Salch) If I is a cofibrant object in Fun LS (N op, S) then X I is again a cofibrant object in Fun LS (N op, S).

20 THH-May spectral sequence Theorem (A-K, Salch) If I is a cofibrant object in Fun LS (N op, S) then X I is again a cofibrant object in Fun LS (N op, S). Given a cofibrant object I in Fun LS (N op, S) we can form a commutative ring spectrum E 0 I, which is additively E 0 I = I (i)/i (i + 1). i 0

21 THH-May spectral sequence Theorem (A-K, Salch) If I is a cofibrant object in Fun LS (N op, S) then X I is again a cofibrant object in Fun LS (N op, S). Given a cofibrant object I in Fun LS (N op, S) we can form a commutative ring spectrum E 0 I, which is additively E 0 I = I (i)/i (i + 1). i 0 Theorem (A-K, Salch) There is a spectral sequence G, (X E 0 I ) G (X I (0)) for any generalized homology theory G.

22 THH-May spectral sequence The main step in constructing this spectral sequence is the proof that the construction of the associated graded E 0 commutes with the functor X. Theorem (A-K, Salch) There is an equivalence in Comm S E 0 (X I ) X E 0 I.

23 THH-May spectral sequence The main step in constructing this spectral sequence is the proof that the construction of the associated graded E 0 commutes with the functor X. Theorem (A-K, Salch) There is an equivalence in Comm S E 0 (X I ) X E 0 I. (The object on the left is the associated graded object in Comm S of the filtered commutative ring spectrum X I. A priori, this is the E 1 -page of the THH-May spectral sequence.)

24 Computation Let j = K(F q ) p where p 5 and q is a prime power that topologically generates Z p.

25 Computation Let j = K(F q ) p where p 5 and q is a prime power that topologically generates Z p. Theorem (A-K) There is an isomorphism of graded rings V (1) THH(j) = P(µ 2 ) Γ(σb) F p {1, α 1, λ 1, λ 2 α 1, λ 2 λ 1α 1 λ 1λ 2 }.

26 Computation Let j = K(F q ) p where p 5 and q is a prime power that topologically generates Z p. Theorem (A-K) There is an isomorphism of graded rings V (1) THH(j) = P(µ 2 ) Γ(σb) F p {1, α 1, λ 1, λ 2 α 1, λ 2 λ 1α 1 λ 1λ 2 }. Proof sketch. Construct the Whitehead tower j of j as an object in Fun LS (N op, S), then compute V (1), THH(E 0 j ) V (1) THH(j).

27 Thank You Wayne State University

Towards THH of the algebraic K-theory spectrum of a

Towards THH of the algebraic K-theory spectrum of a finite field Gabe Angelini-Knoll Wayne State University Graduate Student Topology and Geometry Conference March 29, 2015 Gabe Angelini-Knoll (WSU) Towards