Exercise 5.17c (Sets for Mathematics, p. 109)

The assignments (-)^T(A) = A^T for any set A and (-)^T(f) = f ^T for any function f: A –> B define a functor (-)^T: S –> S.

 

First, S is the category of sets, whose objects and morphisms are sets and functions, respectively.  Next, a functor

P: V –> W

is an assignment of objects and morphisms of the codomain category W to objects and morphisms, respectively, of the domain category V in a way respectful of the domain, codomain, identity, and composition structure of the category.

Let’s look at the case of the functor of our present concern, which has the category S of sets as both domain and codomain category i.e. a functor

P: S –> S

The functor P assigns sets to sets, which we denote as

P_Ob: S_Ob –> S_Ob

and functions to functions, which we denote as

P_Mp: S_Mp –> S_Mp

These two functions

P_Ob: S_Ob –> S_Ob

P_Mp: S_Mp –> S_Mp

together are required to satisfy, in order to constitute a functor

P: S –> S

the following four conditions corresponding to preserving (i) domain, (ii) codomain, (iii) identity, and (iv) composition.

(i) Preserving Domain

S_Ob       –P_Ob–>            S_Ob

^                                  ^

|                                   |

domain                        domain

|                                   |

S_Mp      –P_Mp–>           S_Mp

The commutativity of the above diagram stated as

domain o P_Mp = P_Ob o domain

guarantees the preserving of domain (where ‘o’ denotes composition).  This commutativity equation is read as: the domain set (in top-right S_Ob) of the function (in bottom-right S_Mp) to which a function (in bottom-left S_Mp) is assigned to by P_Mp is same as the set (in top-right S_Ob) to which the domain set (in top-left S_Ob), of the function (in bottom-left S_Mp), is assigned to by P_Ob.

(ii) Preserving Codomain

S_Ob       –P_Ob–>            S_Ob

^                                  ^

|                                   |

codomain                    codomain

|                                   |

S_Mp      –P_Mp–>           S_Mp

The commutativity of the above diagram stated as

codomain o P_Mp = P_Ob o codomain

guarantees the preserving of codomain.  This commutativity equation is read as: the codomain set (in top-right S_Ob) of the function (in bottom-right S_Mp) to which a function (in bottom-left S_Mp) is assigned to by P_Mp is same as the set (in top-right S_Ob) to which the codomain set (in top-left S_Ob), of the function (in bottom-left S_Mp), is assigned to by P_Ob.

(iii) Preserving Identity

S_Mp      –P_Mp–>           S_Mp

^                                  ^

|                                   |

identity                         identity

|                                   |

S_Ob       –P_Ob–>            S_Ob

The commutativity of the above diagram stated as

identity o P_Ob = P_Mp o identity

guarantees the preserving of identity.  This commutativity equation is read as: the identity function (in top-right S_Mp) of the set (in bottom-right S_Ob) to which a set (in bottom-left S_Ob) is assigned to by P_Ob is same as the function (in top-right S_Mp) to which the identity function (in top-left S_Mp), of the set (in bottom-left S_Ob), is assigned to by P_Mp.

(iv) Preserving Composition

P_Mp (g o f) = P_Mp (g) o P_Mp (f)

The function (P_Mp (g o f)) to which the composite function (g o f) is assigned to by P_Mp is same as the composite of the functions (P_Mp(g) and P_Mp(f)) to which the functions (g and f, respectively) are assigned to by P_Mp.

Now that we know what it takes to be a functor, let’s see if what we are given i.e. the functions

(-)^T(A) = A^T

(-)^T(f) = f ^T

together constitute a functor

(-)^T: S –> S

The function

(-)^T_Ob: S_Ob –> S_Ob

assigns to each set

A

(in the domain set S_Ob) the map set

A^T

of T-shaped figures in A i.e. the functions

a: T –> A

from [a fixed set] T to A.

The function

(-)^T_Mp: S_Mp –> S_Mp

assigns to each function

f: A –> B

(in the domain set S_Mp) the induced function

f ^T: A^T –> B^T

where A^T and B^T are map sets whose elements are T-shaped figures in A and B, respectively.

The function

f ^T: A^T –> B^T

assigns to each element (a T-shaped figure in A)

a: T –> A

in the domain map set

A^T

the element

T —a–> A —f–> B

i.e. a T-shaped figure in B

fa: T –> B

in the codomain set

B^T

Thus

f ^T(a) = fa

for all

a: T –> A

in the domain map set

A^T

of the function

f ^T: A^T –> B^T

Now we have to check to see if the object function

(-)^T_Ob (A) = A^T

and the morphism function

(-)^T_Mp (f: A –> B) = f ^T: A^T –> B^T

together constitute a functor

(-)^T: S –> S

i.e. preserve (i) domain, (ii) codomain, (iii) identity, and (iv) composition.

(i) Preserving Domain

S_Ob       –(-)^T_Ob–>         S_Ob

^                                  ^

|                                   |

domain                        domain

|                                   |

S_Mp      –(-)^T_Mp–>        S_Mp

domain o (-)^T_Mp = (-)^T_Ob o domain

LHS

domain o (-)^T_Mp (f: A –> B) = domain (f ^T: A^T –> B^T) = A^T

RHS

(-)^T_Ob o domain (f: A –> B) = (-)^T_Ob (A) = A^T

(ii) Preserving Codomain

S_Ob       –(-)^T_Ob–>         S_Ob

^                                  ^

|                                   |

codomain                    codomain

|                                   |

S_Mp      –(-)^T_Mp–>        S_Mp

codomain o (-)^T_Mp = (-)^T_Ob o codomain

LHS

codomain o (-)^T_Mp (f: A –> B) = codomain (f ^T: A^T –> B^T) = B^T

RHS

(-)^T_Ob o codomain (f: A –> B) = (-)^T_Ob (B) = B^T

(iii) Preserving Identity

S_Mp      –(-)^T_Mp–>        S_Mp

^                                  ^

|                                   |

identity                         identity

|                                   |

S_Ob       –(-)^T_Ob–>         S_Ob

identity o (-)^T_Ob = (-)^T_Mp o identity

LHS

identity o (-)^T_Ob (A) = identity (A^T) = 1_AT: A^T –> A^T

RHS

(-)^T_Mp o identity (A) = (-)^T_Mp (1_A: A –> A) = 1_A^T: A^T –> A^T

where

1_A^T (a: T –> A) = T —a–> A –1_A–> A = 1_A o a = a: T –> A

(iv) Preserving Composition

(-)^T_Mp (g o f) = (-)^T_Mp (g) o (-)^T_Mp (f)

where

f: A –> B, g: B –> C, and g o f: A –> C

LHS

(-)^T_Mp (g o f: A –> C) = (gf)^T: A^T –> C^T

where

(gf)^T(a: T –> A) = T —a–> A —gf–> C = (gf)a: T –> C

RHS

(-)^T_Mp (g) o (-)^T_Mp (f)

= g^T: B^T –> C^T o f ^T: A^T –> B^T

= A^T —f ^T–> B^T —g^T–> C^T

= g^T o f ^T: A^T –> C^T

where

g^T o f ^T (a: T –> A) = g^T (fa: T –> B) = g(fa): T –> C

Thanks to the associative law

(gf)a = g(fa)

of composition

T —a–> A —f–> B —g–> C

the morphism component

(-)^T_Mp: S_Mp –> S_Mp

of the functor

(-)^T: S –> S

preserves composition.  Thus with (i) domain, (ii) codomain, (iii) identity, and (iv) composition preserved by the assignments

(-)^T(A) = A^T

(-)^T(f) = f ^T

we do have a functor

(-)^T: S –> S