Bundle Theory for Categories Kathryn Hess Bundles of finite sets Bundles of Bundle Theory for Categories categories Bundles of monoidal categories Kathryn Hess Bundles of bicategories Institute of Geometry, Algebra and Topology Ecole Polytechnique Fédérale de Lausanne IV Seminar on Categories and Applications Bellaterra, 9 June 2007 Bundle Theory for Categories Kathryn Hess Bundles of finite sets Bundles of categories Bundles of monoidal categories Joint work with Steve Lack. Bundles of bicategories Bundle Theory Outline for Categories Kathryn Hess Bundles of finite sets Bundles of 1 Bundles of finite sets categories Bundles of monoidal categories 2 Bundles of Bundles of categories bicategories 3 Bundles of monoidal categories 4 Bundles of bicategories The bundle p : E → B gives rise to local data: −1 Φp : B → FinSet : b → p (b), where FinSet is the set of finite sets. Any set map Φ: B → FinSet can be globalized: let EΦ = {(b, x) | x ∈ Φ(b), b ∈ B} and pΦ : EΦ → B :(b, x) 7→ b. Bundle Theory Global and local descriptions for Categories A bundle of finite sets is a set map Kathryn Hess Bundles of finite p : E → B, sets Bundles of categories −1 such that p (b) is a finite set for all b ∈ B. Bundles of monoidal categories Bundles of bicategories Any set map Φ: B → FinSet can be globalized: let EΦ = {(b, x) | x ∈ Φ(b), b ∈ B} and pΦ : EΦ → B :(b, x) 7→ b. Bundle Theory Global and local descriptions for Categories A bundle of finite sets is a set map Kathryn Hess Bundles of finite p : E → B, sets Bundles of categories −1 such that p (b) is a finite set for all b ∈ B. Bundles of monoidal The bundle p : E → B gives rise to local data: categories Bundles of −1 bicategories Φp : B → FinSet : b → p (b), where FinSet is the set of finite sets. Bundle Theory Global and local descriptions for Categories A bundle of finite sets is a set map Kathryn Hess Bundles of finite p : E → B, sets Bundles of categories −1 such that p (b) is a finite set for all b ∈ B. Bundles of monoidal The bundle p : E → B gives rise to local data: categories Bundles of −1 bicategories Φp : B → FinSet : b → p (b), where FinSet is the set of finite sets. Any set map Φ: B → FinSet can be globalized: let EΦ = {(b, x) | x ∈ Φ(b), b ∈ B} and pΦ : EΦ → B :(b, x) 7→ b. −1 Observe that τset (X) = X for all X ∈ FinSet. Bundle Theory The tautological bundle for Categories Kathryn Hess Bundles of finite sets Bundles of categories Let Bundles of FinSet∗ = {(X, x) | X ∈ FinSet, x ∈ X}. monoidal categories The tautological bundle of finite sets is the map Bundles of bicategories τset : FinSet∗ → FinSet :(X, x) 7→ X. Bundle Theory The tautological bundle for Categories Kathryn Hess Bundles of finite sets Bundles of categories Let Bundles of FinSet∗ = {(X, x) | X ∈ FinSet, x ∈ X}. monoidal categories The tautological bundle of finite sets is the map Bundles of bicategories τset : FinSet∗ → FinSet :(X, x) 7→ X. −1 Observe that τset (X) = X for all X ∈ FinSet. Proof. The globalization pΦ : EΦ → B of Φ: B → FinSet fits into a pullback square EΦ / FinSet∗ . pΦ τset Φ B / FinSet Moreover, it is obvious that pΦp = p and ΦpΦ = Φ for all p : E → B and for all Φ: B → FinSet. Bundle Theory Classification for Categories Kathryn Hess Proposition Bundles of finite The bundle τset classifies bundles of finite sets. sets Bundles of categories Bundles of monoidal categories Bundles of bicategories Bundle Theory Classification for Categories Kathryn Hess Proposition Bundles of finite The bundle τset classifies bundles of finite sets. sets Bundles of categories Proof. Bundles of monoidal The globalization pΦ : EΦ → B of Φ: B → FinSet fits into categories a pullback square Bundles of bicategories EΦ / FinSet∗ . pΦ τset Φ B / FinSet Moreover, it is obvious that pΦp = p and ΦpΦ = Φ for all p : E → B and for all Φ: B → FinSet. Bundle Theory The local description for Categories Kathryn Hess Bundles of finite sets Bundles of categories Bundles of Let Cat denote the category of small categories monoidal categories Let B denote any category. Local category bundle data Bundles of bicategories over B is a functor Φ: B → Cat. , i.e.: (Existence of a lift) e / β∗e ∈ E ∃ βbe ↓ P P P(e) / ∈ B. ∀ β b Bundle Theory The global description for Categories Kathryn Hess Bundles of finite A functor P : E → B is a bundle of categories if it is a split sets Bundles of opfibration with small fibers categories Bundles of monoidal categories Bundles of bicategories Bundle Theory The global description for Categories Kathryn Hess Bundles of finite A functor P : E → B is a bundle of categories if it is a split sets Bundles of opfibration with small fibers , i.e.: categories Bundles of (Existence of a lift) monoidal categories β Bundles of e / ∗e ∈ E bicategories ∃ βbe ↓ P P P(e) / ∈ B. ∀ β b Bundle Theory The global description for Categories Kathryn Hess Bundles of finite (Universal property of the lift) sets Bundles of δ categories Bundles of monoidal categories " 0 e / β∗e / e β ∃! γ Bundles of be b bicategories ↓ P β ∀ γ P(e) / b / P(e0) ; P(δ) Bundle Theory The global description for Categories Kathryn Hess Bundles of finite sets (γβ)∗(e) Bundles of γβ kk5 categories ce kkkk kkkk Bundles of kkkk monoidal kkk categories e k / β∗e / γ∗β∗e γ Bundles of βbe bβ∗e bicategories ↓ P ∀ β ∀ γ P(e) / b / c and Id\ = Id for all e. P(e)e e Each P−1(b) is a small category. The proof is highly analogous to the usual proof that any continuous map can be factored as a homotopy equivalence followed by a fibration. Bundle Theory Associated bundles for Categories Kathryn Hess Bundles of finite Proposition sets Bundles of For any F : A → B, there is a natural factorization categories Bundles of F monoidal A / B categories N ?? ? Bundles of % ?? bicategories S ?? P ? ? Q Y _ E such that P is bundle of categories, QS = IdA and there is a natural transformation SQ ⇒ IdE. Bundle Theory Associated bundles for Categories Kathryn Hess Bundles of finite Proposition sets Bundles of For any F : A → B, there is a natural factorization categories Bundles of F monoidal A / B categories N ?? ? Bundles of % ?? bicategories S ?? P ? ? Q Y _ E such that P is bundle of categories, QS = IdA and there is a natural transformation SQ ⇒ IdE. The proof is highly analogous to the usual proof that any continuous map can be factored as a homotopy equivalence followed by a fibration. Bundle Theory Invariance under pullback for Categories Kathryn Hess Bundles of finite sets Bundles of Proposition categories Bundles of monoidal If P : E → B is a bundle of categories and F : A → B is categories Bundles of any functor, then the pullback bicategories F ∗P : E × A → A B of P along F is also a bundle of categories. The proof is very straightforward. The tautological bundle of categories is the functor ( (A, a) 7→ A τcat : Cat∗ → Cat : (F, f ) 7→ F. −1 ∼ Observe that τcat (A) = A for all small categories A. Bundle Theory The tautological bundle for Categories Kathryn Hess Bundles of finite Let Cat∗ be the category of pointed, small categories: sets Bundles of categories Ob Cat∗ = {(A, a) | A ∈ Ob Cat, a ∈ Ob A}; Bundles of monoidal categories Cat∗ (A, a), (B, b) = {(F, f ) | F : A → B, f : F(a) → b}. Bundles of bicategories Bundle Theory The tautological bundle for Categories Kathryn Hess Bundles of finite Let Cat∗ be the category of pointed, small categories: sets Bundles of categories Ob Cat∗ = {(A, a) | A ∈ Ob Cat, a ∈ Ob A}; Bundles of monoidal categories Cat∗ (A, a), (B, b) = {(F, f ) | F : A → B, f : F(a) → b}. Bundles of bicategories The tautological bundle of categories is the functor ( (A, a) 7→ A τcat : Cat∗ → Cat : (F, f ) 7→ F. −1 ∼ Observe that τcat (A) = A for all small categories A. Proof. Local to global: Given local category bundle data Φ: B → Cat, consider the pullback EΦ / Cat∗ PΦ τcat Φ B / Cat. Since τcat is a bundle of categories, PΦ is also a bundle of categories. (PΦ is exactly the Grothendieck construction on Φ.) Bundle Theory Classification for Categories Kathryn Hess Theorem Bundles of finite The bundle τcat classifies bundles of categories. sets Bundles of categories Bundles of monoidal categories Bundles of bicategories (PΦ is exactly the Grothendieck construction on Φ.) Bundle Theory Classification for Categories Kathryn Hess Theorem Bundles of finite The bundle τcat classifies bundles of categories. sets Bundles of categories Proof. Bundles of monoidal Local to global: Given local category bundle data categories Φ: B → Cat, consider the pullback Bundles of bicategories EΦ / Cat∗ PΦ τcat Φ B / Cat. Since τcat is a bundle of categories, PΦ is also a bundle of categories. Bundle Theory Classification for Categories Kathryn Hess Theorem Bundles of finite The bundle τcat classifies bundles of categories. sets Bundles of categories Proof. Bundles of monoidal Local to global: Given local category bundle data categories Φ: B → Cat, consider the pullback Bundles of bicategories EΦ / Cat∗ PΦ τcat Φ B / Cat. Since τcat is a bundle of categories, PΦ is also a bundle of categories. (PΦ is exactly the Grothendieck construction on Φ.) Bundle Theory Classification for Categories Proof. Kathryn Hess Bundles of finite Global to local: Given a bundle of categories sets −1 P : E → B, define ΦP : B → Cat by ΦP (b) = P (b) Bundles of and categories Bundles of βbe1 monoidal e1 / β∗e1 categories O O Bundles of ξ Iddb0 =Φ(β)(ξ) bicategories βbe0 e0 / β∗e0 ↓ P P(βbe1 ◦ξ) ) 0 b 5 b .