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Gf (X). This is the usual action of a functor on diagrams in a category.

4.4.2 Since S(G; ) : op Â¡ Cat is a category-valued functor, we can use

!

the Grothendieck construction to form the split Â¯bration of by S(G; ). This

wrG , called the wreath product

Â¯bration consists of a category denoted

with given action G, and a functor Â¦ : wrG Â¡ !.

of by

We now unwind what this implies to give an elementary deÂ¯nition of the

wreath product.

Â¡

!

4.4.3 DeÂ¯nition Given small categories and and a functor G :

G

Cat, the wreath product wr is a category deÂ¯ned as follows:

wrG are pairs (A; P ), where A is an object of

WP{1 The objects of and

P : G(A) Â¡ ! is a functor.

WP{2 An arrow (f; Â¸) : (A; P ) Â¡ (A0 ; P 0 ) of wrG has f : A Â¡ A0 an arrow

! !

0Â±

and Â¸ : P Â¡ P Gf a natural transformation.

!

of

WP{3 If (f; Â¸) : (A; P ) Â¡ (A0 ; P 0 ) and (g; Â¹) : (A0; P 0) Â¡ (A00; P 00 ) are arrows of

! !

G

wr , as in

G(A) - G(A0 ) - G(A00 )

Gf Gg

@ Â¡

P@ Â¡ P 00

0

P

@ Â¡

@ ?Â¡

R Âª

4.4 Wreath products 53

then

(g; Â¹) Â± (f; Â¸) = (g Â± f; Â¹:Gf Â± Â¸) : (A; P ) Â¡ (A00 ; P 00 )

!

To see the meaning of WP{3, observe that Â¸ : P Â¡ P 0 Gf and Â¹ : P 0 Â¡

! !

Â±

00 Â±

P Gg are natural transformations. Then

Â¹:Gf : P 0 Â± Gf Â¡ P 00 Â± Gg Â± Gf = P 00 Â± G(g Â± f )

!

is the natural transformation whose component at an object x of G(A) is the

component of Â¹ at Gf (x) (this was described in Section 4.4). Then

Â¹:Gf Â± Â¸ : P Â¡ P 00 Â± G(g Â± f)

!

is the usual composite of natural transformations (see 4.2.11); it is the natural

transformation whose component at an object x of G(A) is the composite of the

components (Â¹:Gf (x)) Â± Â¸x.

It follows from WP{3 that there is a projection functor

wrG Â¡

!

Â¦:

taking (A; P ) to A and (f; Â¸) to f .

4.4.4 Special cases of the wreath product If the functor G in deÂ¯ni-

tion ES 4.4.3 is set-valued, then one obtains the discrete wreath product of

by with action G. When and are both monoids, the discrete wreath

product is also a monoid. (The general case need not be a monoid.)

4.4.5 DeÂ¯nition For any small category , the right regular representa-

tion of is the functor R : Â¡ Set deÂ¯ned as follows:

!

RR{1 If C is an object of , then R (C) is the set of arrows of with codomain

C.

RR{2 If f : C Â¡ C 0 in and g 2 R (C), then R (f )(g) = f Â± g.

!

For small categories and , the standard wreath product wr is

R

the wreath product wr . This is a generalization of what is called the

standard wreath product for groups and monoids. It is the wreath product used

in [Rhodes and Tilson, 1989]. They also have a two-sided version of the wreath

product.

4.4.6 The action induced by a wreath product Given small categories

Â¡ Cat and H :

! Â¡ Cat, there is an induced

!

and and functors G :

functor G wr H : wrG Â¡ Cat deÂ¯ned as follows:

!

wrG , (G wr H )(A; P ) is the split opÂ¯bration

WF{1 For an object (A; P ) of

induced by H Â± P : G(A) Â¡ Cat.

!

54 Fibrations

WF{2 If (h; Â¸) is an arrow of wrG with domain (A; P ), and (t; x) is an object

of (G wr H)(A; P ), so that x is an object of G(A) and t is an object of

H (P (x)), then

(G wr H )(h; Â¸)(t; x) = (HÂ¸x(t); Gh(x))

WF{3 If (u; f ) : (t; x) Â¡ (t0; x0 ) is an arrow of (G wr H)(P; A), then

!

(G wr H )(h; Â¸)(u; f) = (H (Â¸x0)(u); Gh(f ))

WF{1 can be perceived as saying that G wr H is obtained by composing the shapes

given by G (see the discussion in ES 4.4.1) with H. Indeed, G. M. Kelly, who invented

this concept [1974] called what we call the wreath product the `composite' of the

categories. That is in some ways a better name: the word `product' suggests that the

two factors are involved in the product in symmetric ways, which is not the case, as

the next subsection describes.

wrG

4.4.7 The action G wr H of just deÂ¯ned is said to be triangular

because it is a precise generalization of the action of a triangular matrix. For

example, the action " #" # " #

ax + by

ab x

Â£ =

0c y cy

can be described this way: the eÂ®ect on the Â¯rst coordinate depends on both the

Â¯rst and second coordinates, but the eÂ®ect on the second coordinate depends only

on the second coordinate.

The dependency of the action on the coordinates given in WF{2 and WF{3

is analogous to the dependency for the matrices in the example just given.

The wreath product can be generalized to many factors, using the following

theorem, proved in [Kelly, 1974], Section 7. This theorem allows one to think of

the wreath product as generalizing triangular matrices bigger than 2 Â£ 2.

Â¡ Cat, H :

! Â¡ Cat and K :

! Â¡ Cat

!

4.4.8 Proposition Let G :

be functors. Then there is an isomorphism of categories I making this diagram

commute.

) I- (

wrG ( wrH wrG ) wrG wr H

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