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.

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

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

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 . β −1 −1 0 The functor ΦP (β): P (b) → P (b ) can therefore be seen as “parallel transport” along the path β from the fiber over b to the fiber over b0. 0 0 Since ΦP (β ) ◦ ΦP (β) = ΦP (β β), a “connection” giving rise to this “parallel transport” would have to be flat. Thus: bundles of categories can be thought of as functors endowed with a flat connection. (The nonflat case corresponds to considering pseudofunctors B → CAT.)

Bundle Theory The “geometric” viewpoint for Categories Kathryn Hess

Bundles of ﬁnite A morphism β : b → b0 in B can be seen as a “path” from sets 0 Bundles of b to b . categories

Bundles of monoidal categories

Bundles of bicategories 0 0 Since ΦP (β ) ◦ ΦP (β) = ΦP (β β), a “connection” giving rise to this “parallel transport” would have to be flat. Thus: bundles of categories can be thought of as functors endowed with a flat connection. (The nonflat case corresponds to considering pseudofunctors B → CAT.)

Bundle Theory The “geometric” viewpoint for Categories Kathryn Hess

Bundles of finite A morphism β : b → b0 in B can be seen as a “path” from sets 0 Bundles of b to b . categories −1 −1 0 Bundles of The functor ΦP (β): P (b) → P (b ) can therefore be monoidal seen as “parallel transport” along the path β from the categories 0 Bundles of fiber over b to the fiber over b . bicategories 0 0 Since ΦP (β ) ◦ ΦP (β) = ΦP (β β), a “connection” giving rise to this “parallel transport” would have to be flat. (The nonflat case corresponds to considering pseudofunctors B → CAT.)

Bundle Theory The “geometric” viewpoint for Categories Kathryn Hess

Bundles of finite A morphism β : b → b0 in B can be seen as a “path” from sets 0 Bundles of b to b . categories −1 −1 0 Bundles of The functor ΦP (β): P (b) → P (b ) can therefore be monoidal seen as “parallel transport” along the path β from the categories 0 Bundles of fiber over b to the fiber over b . bicategories 0 0 Since ΦP (β ) ◦ ΦP (β) = ΦP (β β), a “connection” giving rise to this “parallel transport” would have to be flat. Thus: bundles of categories can be thought of as functors endowed with a flat connection. Bundle Theory The “geometric” viewpoint for Categories Kathryn Hess

Bundles of finite A morphism β : b → b0 in B can be seen as a “path” from sets 0 Bundles of b to b . categories −1 −1 0 Bundles of The functor ΦP (β): P (b) → P (b ) can therefore be monoidal seen as “parallel transport” along the path β from the categories 0 Bundles of fiber over b to the fiber over b . bicategories 0 0 Since ΦP (β ) ◦ ΦP (β) = ΦP (β β), a “connection” giving rise to this “parallel transport” would have to be flat. Thus: bundles of categories can be thought of as functors endowed with a flat connection. (The nonflat case corresponds to considering pseudofunctors B → CAT.) If p : E → B is a covering map of topological spaces, then Π(p) : Π(E) → Π(B) is bundle of categories.

The corresponding local data Φp : Π(B) → Cat is such − that Φp(b) is the fundamental groupoid of p 1(b).

Bundle Theory Example: covering spaces for Categories Kathryn Hess

Bundles of ﬁnite sets

Bundles of For any topological space X, let Π(X) denote its categories 0 Bundles of fundamental groupoid, i.e., Ob Π(X) = X and Π(X)(x, x ) monoidal is the set of based homotopy classes of paths from x to categories 0 Bundles of x . bicategories The corresponding local data Φp : Π(B) → Cat is such − that Φp(b) is the fundamental groupoid of p 1(b).

Bundle Theory Example: covering spaces for Categories Kathryn Hess

Bundles of ﬁnite sets

Bundles of For any topological space X, let Π(X) denote its categories 0 Bundles of fundamental groupoid, i.e., Ob Π(X) = X and Π(X)(x, x ) monoidal is the set of based homotopy classes of paths from x to categories 0 Bundles of x . bicategories If p : E → B is a covering map of topological spaces, then Π(p) : Π(E) → Π(B) is bundle of categories. Bundle Theory Example: covering spaces for Categories Kathryn Hess

Bundles of ﬁnite sets

Bundles of For any topological space X, let Π(X) denote its categories 0 Bundles of fundamental groupoid, i.e., Ob Π(X) = X and Π(X)(x, x ) monoidal is the set of based homotopy classes of paths from x to categories 0 Bundles of x . bicategories If p : E → B is a covering map of topological spaces, then Π(p) : Π(E) → Π(B) is bundle of categories.

The corresponding local data Φp : Π(B) → Cat is such − that Φp(b) is the fundamental groupoid of p 1(b). Hopf algebroids: a Hopf algebroid (A, Γ) over a commutative ring R gives rise to a functor

AlgR → Gpd ,→ Cat.

(Flores) Classifying spaces for families of subgroups: to a discrete group G and a family F of subgroups of G, there is associated a functor

R : OF → Cat : G/H → G/H.

The nerve of ER is then a model for EFG: it is a G-CW-complex such that every isotropy group belongs to F and the ﬁxed-point space with respect to any element of F is contractible.

Bundle Theory Other examples for Categories Categories ﬁbered over groupoids (and therefore Kathryn Hess stacks) Bundles of ﬁnite sets

Bundles of categories

Bundles of monoidal categories

Bundles of bicategories (Flores) Classifying spaces for families of subgroups: to a discrete group G and a family F of subgroups of G, there is associated a functor

R : OF → Cat : G/H → G/H.

The nerve of ER is then a model for EFG: it is a G-CW-complex such that every isotropy group belongs to F and the ﬁxed-point space with respect to any element of F is contractible.

Bundle Theory Other examples for Categories Categories ﬁbered over groupoids (and therefore Kathryn Hess stacks) Bundles of ﬁnite sets

Hopf algebroids: a Hopf algebroid (A, Γ) over a Bundles of commutative ring R gives rise to a functor categories Bundles of monoidal AlgR → Gpd ,→ Cat. categories Bundles of bicategories Bundle Theory Other examples for Categories Categories ﬁbered over groupoids (and therefore Kathryn Hess stacks) Bundles of ﬁnite sets

Hopf algebroids: a Hopf algebroid (A, Γ) over a Bundles of commutative ring R gives rise to a functor categories Bundles of monoidal AlgR → Gpd ,→ Cat. categories Bundles of bicategories

(Flores) Classifying spaces for families of subgroups: to a discrete group G and a family F of subgroups of G, there is associated a functor

R : OF → Cat : G/H → G/H.

The nerve of ER is then a model for EFG: it is a G-CW-complex such that every isotropy group belongs to F and the ﬁxed-point space with respect to any element of F is contractible. Bundle Theory The local description for Categories Kathryn Hess

Bundles of ﬁnite sets

Bundles of categories Recall that Cat admits a monoidal structure, given by Bundles of monoidal cartesian product. categories Bundles of Let B denote any monoidal category. bicategories Local monoidal bundle data over B is a monoidal functor

Φ: B → Cat. Bundle Theory The global description for Categories Kathryn Hess

Bundles of ﬁnite sets

Bundles of categories

Bundles of Let B and E be monoidal categories. monoidal categories

A strict monoidal functor P : E → B that is a bundle of Bundles of categories is a bundle of monoidal categories if the lifts bicategories satisfy: 0 0 βbe ⊗ βb e0 = β\⊗ β e⊗e0 , for all β : P(e) → b and β0 : P(e0) → b0 in B. Bundle Theory The tautological bundle for Categories Kathryn Hess

Bundles of ﬁnite sets

Bundles of categories

Bundles of monoidal The tautological bundle of categories categories Bundles of bicategories τcat : Cat∗ → Cat

is a bundle of monoidal categories. Bundle Theory Classification for Categories Theorem Kathryn Hess The tautological bundle τ classifies bundles of Bundles of finite cat sets

monoidal categories. Bundles of categories

Bundles of Proof. monoidal Using the constructions of the previous classiﬁcation categories Bundles of theorem, we see that bicategories

Φ: B → Cat monoidal ⇒

PΦ : EΦ → B bundle of monoidal categories and

P : E → B bundle of monoidal categories ⇒

ΦP : B → Cat monoidal . Bundle Theory The “geometric” viewpoint for Categories Kathryn Hess

Bundles of ﬁnite sets

Bundles of If P : E → B is a bundle of monoidal categories, then the categories Bundles of associated “parallel transport” is such that monoidal categories −1 −1 0 µ −1 0 Bundles of P (b0) × P (b0) / P (b0 ⊗ b0) bicategories

0 0 ΦP (β)×ΦP (β ) ΦP (β⊗β ) −1 −1 0 µ −1 0 P (b1) × P (b1) / P (b1 ⊗ b1)

0 0 0 commutes for all “paths” β : b0 → b1 and β : b0 → b1. Let X be the category with one object ∗ and with morphism set equal to X.

Bundle Theory Example: modules over a ﬁxed ring for Categories Kathryn Hess

Let R be a ring, and let X be a left R-module. Bundles of ﬁnite sets

Let (R, ⊗, I) be the monoidal category where Bundles of categories Ob R = ; N Bundles of monoidal R(m, n) = Mnm(R), the set of (n × m)-matrices with categories

coefﬁcients in R; Bundles of bicategories composition is given by matrix multiplication; 0 0 m ⊗ m := m + m , I := 0 and for all M ∈ Mnm(R), 0 M ∈ Mn0m0 (R)

M 0 M ⊗ M0 := . 0 M0 Bundle Theory Example: modules over a ﬁxed ring for Categories Kathryn Hess

Let R be a ring, and let X be a left R-module. Bundles of ﬁnite sets

Let (R, ⊗, I) be the monoidal category where Bundles of categories Ob R = ; N Bundles of monoidal R(m, n) = Mnm(R), the set of (n × m)-matrices with categories

coefﬁcients in R; Bundles of bicategories composition is given by matrix multiplication; 0 0 m ⊗ m := m + m , I := 0 and for all M ∈ Mnm(R), 0 M ∈ Mn0m0 (R)

M 0 M ⊗ M0 := . 0 M0

Let X be the category with one object ∗ and with morphism set equal to X. Bundle Theory Example: modules over a ﬁxed ring for Categories Kathryn Hess Let Φ : R → Cat denote the functor given by X Bundles of ﬁnite sets ×m ΦX (m) = X Bundles of categories

Bundles of and monoidal ( categories ×m ×n ∗ 7→ ∗ ΦX (M): X → X : Bundles of ~x 7→ M~x bicategories   x1 ~  .  ×n for all x =  .  ∈ X . xn

It is easy to see that ΦX is monoidal and therefore gives rise to a bundle of monoidal categories

PX : EX → R. Bundle Theory Example: modules over a ﬁxed ring for Categories Kathryn Hess

Bundles of ﬁnite sets

Bundles of categories

Bundles of monoidal Proposition categories Bundles of The categories of left and of right modules over a ﬁxed bicategories ring R embed into the category of bundles of monoidal categories over R. Horizontal composition

MAT(U, V )×MAT(V , W ) −→ MAT(U, W ):(A, B) 7→ A∗B

is given by matrix multiplication, i.e., a (A ∗ B)(u, w) = A(u, v) × B(v, w) v∈V

for all u ∈ U and w ∈ W .

Bundle Theory The matrix bicategory for Categories Kathryn Hess MAT is speciﬁed by Bundles of ﬁnite MAT0 = Ob Set and sets Bundles of for all U, V ∈ MAT0, categories Bundles of U×V monoidal MAT(U, V ) = Cat , categories

Bundles of where U and V are seen as discrete categories. bicategories Bundle Theory The matrix bicategory for Categories Kathryn Hess MAT is speciﬁed by Bundles of ﬁnite MAT0 = Ob Set and sets Bundles of for all U, V ∈ MAT0, categories Bundles of U×V monoidal MAT(U, V ) = Cat , categories

Bundles of where U and V are seen as discrete categories. bicategories Horizontal composition

MAT(U, V )×MAT(V , W ) −→ MAT(U, W ):(A, B) 7→ A∗B

is given by matrix multiplication, i.e., a (A ∗ B)(u, w) = A(u, v) × B(v, w) v∈V

for all u ∈ U and w ∈ W . This is a sort of “parametrized” version of the local data for a bundle of monoidal categories. In particular, local bicategory bundle data is obtained when local data for a bundle of monoidal categories is “suspended.”

Bundle Theory The local description for Categories Kathryn Hess

Bundles of ﬁnite sets

Bundles of categories

Let B be any small bicategory. Bundles of monoidal Local bicategory bundle data over B consist of a lax categories Bundles of functor Φ: B → MAT. bicategories Bundle Theory The local description for Categories Kathryn Hess

Bundles of ﬁnite sets

Bundles of categories

Let B be any small bicategory. Bundles of monoidal Local bicategory bundle data over B consist of a lax categories Bundles of functor Φ: B → MAT. bicategories This is a sort of “parametrized” version of the local data for a bundle of monoidal categories. In particular, local bicategory bundle data is obtained when local data for a bundle of monoidal categories is “suspended.” The composition functors

c E(e, e0) × E(e0, e00) / E(e, e00)

Π×Π Π 0 0 00 c 00 B Π(e), Π(e ) × B Π(e ), Π(e ) / B Π(e), Π(e )

are morphisms of bundles of categories. The associator and the unitors in B lift to the associator and the unitors in E.

Bundle Theory The global description for Categories Kathryn Hess

A strict homomorphism of bicategories Π: E → B is a Bundles of ﬁnite bundle of bicategories if sets Bundles of The induced functor on hom-categories categories 0 0 Bundles of Π: E(e, e ) → B Π(e), Π(e ) is a bundle of monoidal categories for all 0-cells e, e0 in E. categories Bundles of bicategories The associator and the unitors in B lift to the associator and the unitors in E.

Bundle Theory The global description for Categories Kathryn Hess

c E(e, e0) × E(e0, e00) / E(e, e00)

Π×Π Π 0 0 00 c 00 B Π(e), Π(e ) × B Π(e ), Π(e ) / B Π(e), Π(e )

are morphisms of bundles of categories. Bundle Theory The global description for Categories Kathryn Hess

c E(e, e0) × E(e0, e00) / E(e, e00)

Π×Π Π 0 0 00 c 00 B Π(e), Π(e ) × B Π(e ), Π(e ) / B Π(e), Π(e )

are morphisms of bundles of categories. The associator and the unitors in B lift to the associator and the unitors in E. Bundle Theory The pointed matrix bicategory for Categories Kathryn Hess

Bundles of ﬁnite sets MAT∗ is speciﬁed by Bundles of categories

(MAT∗)0 = Set∗ and Bundles of monoidal for all (U, u), (V , v) ∈ (MAT∗)0, categories Bundles of bicategories U×V ,(u,v) MAT∗ (U, u), (V , v) = Cat∗ ,

where U × V , (u, v) is seen as a discrete, pointed category.

Horizontal composition is again given by matrix multiplication. Bundle Theory The tautological bundle for Categories Kathryn Hess

Bundles of ﬁnite sets

Bundles of categories

Bundles of monoidal The tautological bundle of bicategories is the strict categories

homomorphism Bundles of bicategories

τbicat : MAT∗ → MAT

given by forgetting basepoints. Proof. Local to global: Given local bicategory bundle data Φ: B → MAT, consider the pullback

EΦ / MAT∗

ΠΦ τbicat Φ B / MAT.

Then ΠΦ is a bundle of bicategories, a sort of parametrized Grothendieck construction on Φ.

Bundle Theory Classiﬁcation for Categories Kathryn Hess

Theorem Bundles of ﬁnite sets The bundle τ classiﬁes bundles of bicategories. bicat Bundles of categories

Bundles of monoidal categories

Bundles of bicategories Bundle Theory Classiﬁcation for Categories Kathryn Hess

Theorem Bundles of ﬁnite sets The bundle τ classiﬁes bundles of bicategories. bicat Bundles of categories

Bundles of Proof. monoidal categories Local to global: Given local bicategory bundle data Bundles of Φ: B → MAT, consider the pullback bicategories

EΦ / MAT∗

ΠΦ τbicat Φ B / MAT.

Then ΠΦ is a bundle of bicategories, a sort of parametrized Grothendieck construction on Φ. Bundle Theory The “geometric” viewpoint: ﬁbers over 1-cells for Categories Kathryn Hess

Bundles of ﬁnite sets

Let Π: E → B be a bundle of bicategories. Bundles of categories 0 0 Let f : b → b be a 1-cell in B. Let e, e be 0-cells of E Bundles of 0 0 monoidal such that Π(e) = b, Π(e ) = b . categories f 0 Bundles of The ﬁber category Fibe,e0 over f with respect to (e, e ): bicategories

ˆ f ˆ 0 ˆ f ∈ Ob Fibe,e0 ⇒ f : e → e and Π(f ) = f

and

f ˆ ˆ0 ˆ ˆ0 α ∈ Fibe,e0 (f , f ) ⇒ α : f → f and Π(α) = Idf . Bundle Theory The “geometric” viewpoint: parallel transport for Categories along 2-cells Kathryn Hess

Bundles of ﬁnite sets

0 0 Bundles of Since Π: E(e, e ) → B Π(e), Π(e ) is a bundle of categories categories, for each 2-cell Bundles of monoidal categories

f Bundles of bicategories $ b ⇓ τ 0 : b g

there is a functor

τ f g ∇e,e0 : Fibe,e0 −→ Fibe,e0 . Bundle Theory The “geometric” viewpoint: parallel transport for Categories and composition Kathryn Hess

Bundles of ﬁnite Furthermore, for all sets Bundles of categories f f 0 Bundles of monoidal $ % categories b ⇓ τ b0 ⇓ τ 0 b00 : 9 Bundles of bicategories g g0 in B,

f f 0 C f 0f Fibe,e0 × Fibe0,e00 / Fibe,e00

0 0 ∇τ ×∇τ ∇τ τ e,e0 e0,e00 e,e00 g g0 C g0g Fibe,e0 × Fibe0,e00 / Fibe,e00 commutes. Bundle Theory Examples for Categories Kathryn Hess

Bundles of ﬁnite sets

Bundles of Charted bundles with coefﬁcients in a topological categories Bundles of bicategory (cf., Baas-Dundas-Rognes or monoidal Baas-Bökstedt-Kro) naturally give rise to bundles of categories Bundles of bicategories. bicategories

Parametrized Kleisli constructions.

The domain projection from the Bénabou bicategory of cylinders in a ﬁxed bicategory B down to B is a bundle of bicategories. Homotopy theory: How do these bundle notions interact with the homotopy theory of Cat and of Bicat?

Bundle Theory Work in progress for Categories Kathryn Hess

Bundles of ﬁnite sets

Bundles of categories

K-theory: All these categories of bundles admit Bundles of monoidal “Whitney sum” and “tensor product”-type operations. categories

What information does the associated “bundle Bundles of K-theory” carry? Should englobe both topological bicategories and algebraic K-theory. Bundle Theory Work in progress for Categories Kathryn Hess

Bundles of ﬁnite sets

Bundles of categories

K-theory: All these categories of bundles admit Bundles of monoidal “Whitney sum” and “tensor product”-type operations. categories

What information does the associated “bundle Bundles of K-theory” carry? Should englobe both topological bicategories and algebraic K-theory.

Homotopy theory: How do these bundle notions interact with the homotopy theory of Cat and of Bicat?