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deÂ¯ning two functors

F : SO( ) Â¡ Func( ; Cat)

!

and

G : Func( ; Cat) Â¡ SO( )

!

so that F is an equivalence with pseudo-inverse G as deÂ¯ned in Section 3.4.

50 Fibrations

4.3.4 DeÂ¯nition For a category , deÂ¯ne the functor

F : SO( ) Â¡ Func( ; Cat)

!

as follows:

FI{1 If P : Â¡ ! is a split opÂ¯bration with splitting Â·, then F(P; Â·) : Â¡ Cat

!

is the functor F satisfying FF{1 through FF{3 deÂ¯ned in ES 4.1.7.

FI{2 If Â³ : (P; Â·) Â¡ (P 0; Â·0 ) is a homomorphism of opÂ¯brations, FÂ³ : F(P; Â·)

!

Â¡ F(P 0 ; Â·0 ) is the natural transformation whose component at an object C

!

of is the functor Â³ restricted to P Â¡1 (C).

To show that F Â³ is a natural transformation, it is necessary to show that for

every f : C Â¡ D in

! the following diagram commutes:

FÂ³C-

F(P 0 ; Â·0)(C)

F(P; Â·)(C)

F(P 0; Â·0)(f )

F(P; Â·)(f) (4.8)

? ?

- F(P 0 ; Â·0 )(D)

F(P; Â·)(D)

FÂ³D

Let u : X Â¡ X 0 be in F(P; Â·)(C). Note that u is an arrow in the inverse image

!

P Â¡1 C, so P u = idC . Moreover, Â³u is an arrow for which P 0 (Â³u) = idC .

By deÂ¯nition of cleavage, there are unique arrows u such that P (u) = idD and

e e

u such that P 0 (u) = idD , for which

b b

u Â± Â·(f; X) = Â·(f; X 0 ) Â± u

e

and

u Â± Â·0 (f; Â³(X)) = Â·(f; Â³(X 0 )) Â± Â³u

b

The top route in Diagram (ES 4.8) takes u to u and the bottom route takes it to

b

Â³u. The following calculation shows that the diagram commutes:

e

u Â± Â·0(f; Â³(X)) = Â·0 (f; Â³(X 0 )) Â± Â³(u)

b

Â³(Â·(f; X 0 )) Â± Â³(u)

=

Â³(Â·(f; X 0 ) Â± u)

=

(4.9)

Â³(u Â± Â·(f; X))

= e

Â³(u) Â± Â³(Â·(f; X))

= e

Â³(u) Â± Â·0 (f; Â³(X))

= e

so that u = Â³(u) by the uniqueness requirement in the deÂ¯nition of u.

b e b

4.3.5 The Grothendieck functor To deÂ¯ne the functor going the other way

we extend the Grothendieck construction.

4.4 Wreath products 51

4.3.6 DeÂ¯nition For a category , deÂ¯ne the functor

G : Func( ; Cat) Â¡ SO( )

!

as follows:

GR{1 For F : Â¡ Cat, G(F ) = G( ; F ).

!

GR{2 For a natural transformation Â® : F Â¡ G :

! Â¡ Cat,

!

GÂ®(x; C) = (Â®Cx; C)

for (x; C) an object of G( ; F ) (so that C is an object of and x is an

object of F C), and

GÂ®(u; f) = (Â®C 0u; f )

for (u; f) an arrow of G( ; F ) (so that f : C Â¡ C 0 in and u : F f x Â¡ x0

! !

in F C 0 ).

Note that in GR{2, Â®C 0 u has domain Â®C 0 (F f x), which is Gf (Â®Cx) because

Â® is a natural transformation. The veriÂ¯cation that GÂ® is a functor is omitted.

4.3.7 Theorem The functor F : SO( ) Â¡ Func( ; Cat) deÂ¯ned in ES 4.3.4

!

is an equivalence of categories with pseudo-inverse G.

There is a similar equivalence of categories between split Â¯brations and con-

travariant functors. The details are in [Nico, 1983]. Moreover, the nonsplit case for

both Â¯brations and opÂ¯brations corresponds in a precise way to `pseudo-functors',

which are like functors except that identities and composites are preserved only

up to natural isomorphisms. See [Gray, 1966] (the terminology has evolved since

that article).

4.3.8 Exercises

1. Verify that GÂ® as deÂ¯ned by GR{2 is a functor.

2. Verify that G as deÂ¯ned by GR{1 and GR{2 is a functor.

4.4 Wreath products

In this section, we introduce the idea of the wreath product of categories (and

of functors), based on an old construction originating in group theory. In the

monoid case, this construction allows a type of series-parallel decomposition of

Â¯nite state machines (the Krohn{Rhodes Theorem). This section is not needed

later.

52 Fibrations

Â¡ Cat a functor. With

!

4.4.1 Let and be small categories and G :

these data we deÂ¯ne the shape functor S(G; ) : op Â¡ Cat as follows. If A

!

is an object of , then S(G; )(A) is the category of functors from the category

G(A) to with natural transformations as arrows.

Thus an object of S(G; )(A) is a functor P : G(A) Â¡ ! and an arrow from

to P 0 : G(A) Â¡ is a natural transformation from P to P 0 . It

P : G(A) Â¡ ! !

is useful to think of S(G; )(A) as the category of diagrams of shape G(A) (or

models of G(A)) in ; the arrows between them are homomorphisms of diagrams,

in other words natural transformations.

Embedding or modeling a certain shape (diagram, space, structure, etc.) into a cer-

tain workspace (category, topological space, etc.) in all possible ways is a tool used

all over mathematics. In particular, what we have called the shape functor is very

reminiscent of the singular simplex functors in algebraic topology.

We must say what S(G; ) does to arrows of op . If f : A Â¡ A0 is an arrow

!

of , then S(G; )(f ) : Func(G(A0); ) Â¡ Func(G(A); ) takes a functor H :

!

G(A0) Â¡

! to H Â± Gf : G(A) Â¡ ! and it takes a natural transformation Â® : H

Â¡ H 0 : G(A0 ) Â¡ to the natural transformation Â®Gf : H Â± Gf Â¡ H 0 Â± Gf :

! ! !

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