MONOIDAL INFINITY-CATEGORIES

PATRICK SCHULTZ

1. Monoidal category Let C be a category with finite products. Definition. A monoid in C is • An object A ∈ C • Maps m : A × A → A and u : 1 → A, satisfying associativity and unit axioms. Definition. A monoidal category is • A category C • Functors m : C × C → C, u : 1 → C, • Natural isomorphisms ∼ ax,y,z :(X ⊗ Y ) ⊗ Z → X ⊗ (Y ⊗ Z) and ∼ ∼ lx : 1 ⊗ X → X, rx : X ⊗ 1 → X. These have to satisfy the pentagon axiom for associativity and a trian- gle for the unit.

Example: for vector spaces HomVect(V ⊗ W, X) consists of bilinear maps from V × W → X. Definition. A monoidal functor F : C → D is an ordinary functor plus ∼ ∼ isomorphisms Φx,y : F (X) ⊗ F (Y ) → F (X ⊗ Y ) and φ : 1D → F (1C). Example: the free functor F : Set → Vect is monoidal. The forgetful functor U : Vect → Set is not monoidal, but lax monoidal: there is a natural map U(V )×U(W ) → U(V ⊗W ), but it is not an isomorphism. Exercise: right adjoint of a monoidal functor is lax monoidal. Definition. A monoid object in a monoidal category C is a lax functor 1 → C. Definition. A multicategory C is • A set of objects Ob(C) • Morphisms C(x1, ..., xn; y) 1 2 PATRICK SCHULTZ

• Compositions for all possible combinations.

Definition. A multicategory C is representable if for all tuples x1, ..., xn ∼ there are x1 ⊗ ... ⊗ xn, such that C(x1, ..., xn, y) = C(x1 ⊗ ... ⊗ xn, y) natural in y. Remark: a functor F : C → D between representable multicategories is a lax monoidal functor.

2. Presentable categories Definition. A category C is accessible if for some regular cardinal κ if •C has κ-filtered colimits • There is a set of κ-compact objects which generate C under κ-filtered colimits. Usually κ = ω. Then κ-filtered colimits are finite colimits. Definition. C is presentable if in addition it has all (small) colimits. Let Catpr be the category of presentable categories with morphisms colimit-preserving functors. pr R op Cat is symmetric monoidal with C1 ⊗ C2 = Fun (C1 , C2) with unit Set. Here FunR are limit-preserving and FunL are colimit-preserving func- tors. L,L L Proposition 1. Fun (C1 × C2, D) = Fun (C1 ⊗ C2, D). pr,st pr Let Cat∞ ⊂ Cat∞ be the subcategory of stable ∞-categories. This subcategory inherits the symmetric monoidal structure with unit the category of spectra.

Definition. An E∞-ring is a commutative monoidal object in Sp.