ADDITIVE ∞- CATEGORIES AND CANONICAL MONOIDAL STRUCTURES I

MORITZ GROTH ADDITIVE ∞-CATEGORIESANDCANONICAL

E∞ -MONOIDS MONOIDALSTRUCTURES I ANDGROUPS

ADDITIVE ∞- CATEGORIES Moritz Groth MONOIDSIN PRESENTABLE ∞-CATS Radboud University Nijmegen ∞-VARIANT OFALGEBRAIC K-THEORY 05.04.2013, Progress in Higher Category Theory, Halifax REFERENCE

ADDITIVE ∞- CATEGORIES AND CANONICAL MONOIDAL STRUCTURES I

MORITZ GROTH

E∞ -MONOIDS ANDGROUPS joint with David Gepner and Thomas Nikolaus (arXiv):

ADDITIVE ∞- CATEGORIES ‘Universality of multiplicative infinite loop space machines’ MONOIDSIN PRESENTABLE ∞-CATS

∞-VARIANT OFALGEBRAIC K-THEORY PLAN

ADDITIVE ∞- CATEGORIES AND CANONICAL MONOIDAL STRUCTURES I 1 ∞-MONOIDSAND ∞-GROUPSIN ∞-CATEGORIES MORITZ E E GROTH

E∞ -MONOIDS ANDGROUPS 2 PREADDITIVEANDADDITIVE ∞-CATEGORIES

ADDITIVE ∞- CATEGORIES

MONOIDSIN 3 MONOIDS IN PRESENTABLE ∞-CATEGORIES PRESENTABLE ∞-CATS

∞-VARIANT OFALGEBRAIC 4 ∞-CATEGORICAL VARIANT OF ALGEBRAIC K-THEORY K-THEORY CALIBRATION OF NOTATION AND TERMINOLOGY

ADDITIVE ∞- CATEGORIES AND CANONICAL MONOIDAL Notation and terminology as in Higher Topos Theory and STRUCTURES I Higher Algebra. MORITZ GROTH ∞-category = simplicial set with inner-horn-filling-property E∞ -MONOIDS ANDGROUPS ADDITIVE ∞- Alternative choices: CATEGORIES

MONOIDSIN quasi-category PRESENTABLE ∞-CATS weak Kan complex ∞-VARIANT OFALGEBRAIC inner Kan complex K-THEORY quategory ... E∞-MONOIDSIN ∞-CATEGORIES

ADDITIVE ∞- CATEGORIES AND Segal’s picture on ∞-monoids: CANONICAL E MONOIDAL STRUCTURES I Fin∗ category of finite pointed sets, pointed maps MORITZ GROTH hni ∈ Fin∗ finite pointed set ({0,1,. . . ,n},0) ρ : hni → h1i pointed map with ρ−1(1) = {i} E∞ -MONOIDS i i ANDGROUPS C ∞-category with finite products ADDITIVE ∞- CATEGORIES DEFINITION MONOIDSIN PRESENTABLE C : (F ) → C ∞-CATS An E∞-monoid in is a functor M N in∗ such that

∞-VARIANT the canonical maps OFALGEBRAIC K-THEORY ρ∗ : Mn → M1 × ... × M1, n ≥ 0,

are equivalences. E∞-GROUPSIN ∞-CATEGORIES

ADDITIVE ∞- CATEGORIES AND CANONICAL An E∞-monoid M : N(Fin∗) → C gives rise to MONOIDAL STRUCTURES I multiplication maps µ: M1 × M1 → M1, MORITZ GROTH shear maps σ = (µ, π2): M1 × M1 → M1 × M1.

E∞ -MONOIDS ANDGROUPS

ADDITIVE ∞- DEFINITION CATEGORIES

MONOIDSIN An E∞-monoid in C is an E∞-group if equivalently: PRESENTABLE ∞-CATS 1 There is an inversion map M1 → M1. ∞-VARIANT 2 OFALGEBRAIC The shear maps are equivalences. K-THEORY 3 The underlying commutative monoid in Ho(C) is a group object. PREADDITIVE ∞-CATEGORIES

ADDITIVE ∞- CATEGORIES AND CANONICAL MONOIDAL STRUCTURES I DEFINITION

MORITZ GROTH Let C be a pointed ∞-category with finite coproducts and finite products. Then C is preadditive if the canonical maps E∞ -MONOIDS ANDGROUPS X t Y → X × Y are equivalences.

ADDITIVE ∞- CATEGORIES For such pointed ∞-category C with finite coproducts and MONOIDSIN PRESENTABLE finite products the following are equivalent: ∞-CATS 1 The ∞-category C is preadditive. ∞-VARIANT OFALGEBRAIC K-THEORY 2 The homotopy category Ho(C) is preadditive.

3 The forgetful functor MonE∞(C) → C is an equivalence. PREADDITIVE ∞-CATEGORIES II

ADDITIVE ∞- CATEGORIES AND CANONICAL MONOIDAL STRUCTURES I EXAMPLE MORITZ If C has finite products, then Mon (C) is preadditive. GROTH E∞

E∞ -MONOIDS ANDGROUPS COROLLARY

ADDITIVE ∞- CATEGORIES If C has finite products, then the forgetful functor

MONOIDSIN MonE∞(MonE∞(C)) → MonE∞(C) is an equivalence. PRESENTABLE ∞-CATS

∞-VARIANT OFALGEBRAIC In preadditive C, canonical ∞-monoid structures arise from K-THEORY E fold maps ∇: X ⊕ X → X. Associated to these we have shear maps σ = (∇, π2): X ⊕ X → X ⊕ X. ADDITIVE ∞-CATEGORIES

ADDITIVE ∞- CATEGORIES AND CANONICAL MONOIDAL DEFINITION STRUCTURES I

MORITZ A preadditive ∞-category is additive if the shear maps GROTH σ : X ⊕ X → X ⊕ X are equivalences.

E∞ -MONOIDS ANDGROUPS PROPOSITION ADDITIVE ∞- CATEGORIES

MONOIDSIN 1 A preadditive ∞-category C is additive iff Ho(C) is PRESENTABLE CATS additive iff Grp (C) → C is an equivalence. ∞- E∞ ∞-VARIANT 2 If C has finite products, then Grp (C) is additive and OFALGEBRAIC E∞ K-THEORY the functor Grp (Grp (C)) → Grp (C) is an E∞ E∞ E∞ equivalence. PRESENTABILITYOFMONOIDSANDGROUPS

ADDITIVE ∞- L CATEGORIES Let Pr be the ∞-category of presentable ∞-categories AND CANONICAL with morphisms the left adjoint functors. MONOIDAL STRUCTURES I PROPOSITION MORITZ GROTH Let C be a presentable ∞-category. Then the ∞-categories Mon (C) and Grp (C) are presentable. E∞ -MONOIDS E∞ E∞ ANDGROUPS

ADDITIVE ∞- CATEGORIES N(Fin∗) MonE∞(C) is an accessible localization of C . MONOIDSIN PRESENTABLE Grp (C) is an accessible localization of Mon (C). E∞ E∞ ∞-CATS

∞-VARIANT OFALGEBRAIC COROLLARY (GROUPCOMPLETION) K-THEORY Given C ∈ PrL then there are adjunctions:

C Mon (C) Grp (C)  E∞  E∞ Mon (−), Grp (−) AS LOCALIZATIONS E∞ E∞

ADDITIVE ∞- CATEGORIES AND The assignments C 7→ Mon ∞(C) and C 7→ Grp (C) are CANONICAL E E∞ MONOIDAL obviously functorial in product-preserving functors. STRUCTURES I MORITZ ROPOSITION GROTH P The assignments C 7→ Mon (C) and C 7→ Grp (C) define E∞ -MONOIDS E∞ E∞ ANDGROUPS functors Mon (−): PrL → PrL and Grp (−): PrL → PrL. E∞ E∞ ADDITIVE ∞- CATEGORIES

MONOIDSIN THEOREM PRESENTABLE ∞-CATS 1 The functor Mon (−): PrL → PrL is a localization with ∞-VARIANT E∞ OFALGEBRAIC local objects the preadditive, presentable ∞-categories. K-THEORY 2 The functor Grp (−): PrL → PrL is a localization with E∞ local objects the additive, presentable ∞-categories. SOMEIMMEDIATECONSEQUENCES

ADDITIVE ∞- CATEGORIES AND CANONICAL Let C, D be presentable ∞-categories, and let S be the MONOIDAL 0 STRUCTURES I ∞-category of spaces (‘free homotopy theory on ∆ ’).

MORITZ GROTH COROLLARY (‘PREADDITIVIZATION’)

E∞ -MONOIDS If D is preadditive, then C → Mon (C) induces a canonical ANDGROUPS E∞ equivalence FunL(Mon (C), D) → FunL(C, D). In particular, ADDITIVE ∞- E∞ CATEGORIES L we obtain Fun (MonE∞(S), D) 'D. MONOIDSIN PRESENTABLE ∞-CATS COROLLARY (‘ADDITIVIZATION’) ∞-VARIANT OFALGEBRAIC If D is additive, then C → Grp (C) induces a canonical K-THEORY E∞ equivalence FunL(Grp (C), D) → FunL(C, D). In particular, E∞ we obtain FunL(Grp (S), D) 'D. E∞ A REFINED PICTURE OF THE STABILIZATION

ADDITIVE ∞- CATEGORIES L AND PrPt pointed presentable ∞-categories CANONICAL L MONOIDAL PrPre preadditive presentable ∞-categories STRUCTURES I L PrAdd additive presentable ∞-categories MORITZ L GROTH PrSt stable presentable ∞-categories

E∞ -MONOIDS ANDGROUPS THEOREM (STABILIZATION)

ADDITIVE ∞- CATEGORIES 1 The stabilization of presentable ∞-categories factors MONOIDSIN L L L L L PRESENTABLE as Pr  PrPt  PrPre  PrAdd  PrSt. ∞-CATS 2 In particular, for C ∈ PrL, the suspension spectrum ∞-VARIANT OFALGEBRAIC functor factors as K-THEORY Σ∞ : C → C → Mon (C) → Grp (C) → Sp(C). + ∗ E∞ E∞ ∞-CATEGORICAL ‘DIRECTSUM’K-THEORY

ADDITIVE ∞- CATEGORIES AND CANONICAL Classical picture: MONOIDAL STRUCTURES I

MORITZ C symmetric, monoidal category GROTH Ce largest subgroupoid of C E∞ -MONOIDS |C | associated -space ANDGROUPS e E∞

ADDITIVE ∞- K (C) K-theory spectrum via group-completion CATEGORIES

MONOIDSIN PRESENTABLE Some steps towards an ∞-categorical variant are: ∞-CATS ∞-VARIANT 1 The inclusion S = Grpd∞ → Cat∞ admits a right adjoint OFALGEBRAIC K-THEORY given by C 7→ Ce.

2 The ∞-categories SymMonCat∞ and MonE∞(Cat∞) are equivalent, compatibly with ∞-groupoids. ∞-CATEGORICAL ‘DIRECTSUM’K-THEORY II

ADDITIVE ∞- CATEGORIES AND CANONICAL MONOIDAL STRUCTURES I DEFINITION MORITZ GROTH The algebraic K-theory SymMonCat∞ → Sp is defined as the composition E∞ -MONOIDS ANDGROUPS

ADDITIVE ∞- Mon ∞(Cat∞) → Mon ∞(S) → Grp (S) → Sp(S). CATEGORIES E E E∞

MONOIDSIN PRESENTABLE ∞-CATS Will be discussed further by Thomas Nikolaus in the second ∞-VARIANT OFALGEBRAIC part of this talk! K-THEORY Thanks for your attention!