Categories and Functors

136 8. The Morita Theorem

5) multiplication of morphisms is associative, i. e. for every triplet of morphisms f, g, h we have h(g f) = (hg)f, provided that the products are defined; 14

6) for every object X E ObC, there exists a morphism Ix E Hom(X,X) such that fIx = f and lxg = g for all morphisms f : X -+ Y and g : Z -+ X.

It is easy to see that a morphism Ix with the above properties is unique.

It is called the identity morphism of the object X.

Examples of Categories. 1. The category Sets of sets. Objects of this category are sets and morphisms

f :

X -+ Y are maps of the set X into the set Y.

Composition of morphisms is the usual composition of maps. It is evident that all category axioms are satisfied. IS

2. The category Gr of groups. Objects of this category are groups, mor- phisms

f :

X -+ Y are homomorphisms of the group X into the group Y and composition is the usual product of homomorphisms.

3. The category of vector spaces over a field K (denoted by Vect or, spec- ifying the field, by Vect I(), the category of K -algebras Alg (or Algl(), the category mod-A of right modules and the category A-mod of left modules over the algebra A, etc. are defined analogously. In all these examples, morphisms are some maps of the sets with the usual composition. However, the following examples show that there are categories of a different kind.

4. Every semigroup P (with identity) can be regarded as a set of morphisms of a category consisting of a single object. Here, composition of morphisms naturally coincides with their product in the semigroup P.

5. The category Mat of matrices. Objects of this category are natural numbers; the set of morphisms Hom( m, n) is defined to be the set of all n x m matrices with entries from a field K. Composition of the morphisms is the usual product of matrices. Here a verification of all axioms is also trivial.

6. Let M be a partially ordered set. Consider it as the set of objects of a category in which Hom(x, y) consists of a single element when x

S

y and it is empty otherwise. Composition of the morphisms is defined in a natural manner.

7. The path category. Let V be a diagram (see Sect. 3.6). One can associate with V the following category C'D. Put ObC'D

=

^{V and for i,j E V, let}

Hom(i,j) be the set of all paths from i to j. Composition of the paths is defined, as in Chapter 3, by concatenation and 1; is the "empty" path with both starting and terminal object at i (see Chap. 3). Again, we get a category which is called the path category of the diagram V.

14It is easy to see that if one of the sides of this equality is defined, so is the other;

this happens if and only if the terminal object of f coincides with the initial object of 9 and the terminal object of 9 with the initial object of h.

150f course, in this definition Ob(Sets) and Mor(Sets) are not sets. However, for all practical purposes this is not essential: We can always restrict ourselves to the subsets of a fixed set (and their maps). This remark also refers to the other analogous examples.

8.1 Categories and Functors 137 8. The dual category. For any category C, one can construct a new cate- gory Co in the following way: ObCo

=

ObC, MorCo

=

MorC and the initial (terminal) object of a morphism

f

in the category Co is its terminal (initial) object in the category C. The product gf in the category Co is defined to be the product fg in the category C. The category Co is said to be dual (opposite) to the category C. Evidently, Coo = C.

In order to avoid any confusion, objects and morphisms of the category Co are usually marked by a little circle: XO,

r,

etc. Then the above defi- nitions can be written in the form ObCo = (ObC)O, MorCo = (MorC)O, Homc'(Xo,YO) = Homc(Y,X)O and gar = (fg)o.

In every category one can define the concept of an isomorphism. Indeed, a morphism

f :

^{X -+ Y} is said to be an isomorphism if and only if there is a morphism f- 1 : Y ^-+ X such that f- 1 f = Ix and ff- 1 = ly. Evidently, these conditions define the morphism

f-

¹ uniquely. The morphism

f-

¹ is called the inverse of f. Of course, f- 1 is also an isomorphism and (f-l )-1 = f.

Moreover, it is easy to see that a composition of isomorphisms

f

and 9 (if defined) is again an isomorphism and that (g f) ^-1 = f- 1 g-1 .

As much as the concept of a homomorphism plays an important role in the study of groups, algebras and modules, a central concept of category theory is that of a functor.

A functor F from a category C to a category V is a pair of maps Fob: ObC -+ ObV and Fmor : MorC -+ MorV satisfying the following condi- tions:

1) if

f :

X -+ Y, then Fmor(f) : Fob(X) -+ Fob(Y);

2) Fmor(1x) = IFob(x);

3) if gf is defined, then Fmor(gf) = Fmor(g)Fmor(f).

Usually, instead of Fmor(f) and Fob(X) one simply writes F(f) and F(X).

Examples of functors. 1. Let C be a category. Fix an object X E ObC and construct the functor hx : C -+ Sets in the following way. If Y E ObC, define hx(Y)

=

Hom(X, Y). If

f :

^{Y -+ Z,} then hx(f) is the map of the sets Hom(X, Y) -+ Hom(X, Z) assigning to every morphism 9 : X -+ Y the morphism fg : X ^-+ Z. The conditions 1) and 2) are satisfied trivially and 3) follows from the associativity of multiplication of morphisms.¹⁶

If C = mod-A (or A-mod), where A is an algebra over K, then all sets Hom(X, Y) are vector spaces over K and one can see easily that, for any

f,

the map hx(f) is a homomorphism. Therefore hx can be considered in this case as a functor to the category Vect of vector spaces over the field K.

2. Forgetful functors. Let C = Gr, V ⁼ Sets. Define the functor F : C -+ V by F(X) = X and F(f) =

f

for every X E C and

f

E MorC. In other words, we forget the group structure on X and consider X simply as a set and homomorphisms as set maps. This functor is called the forgetful functor from the category of groups to the category of sets.

16The reader not familiar with category techniques is advised to verify the conditions.

138 8. The Morita Theorem

In a similar way, we may construct a variety of examples of forgetful functors taking for C a category of sets with "more" structure and for V a category of sets with "less" structure.

Take, for example: a) C

=

^{AlgK , V}

=

^{VectK; b) C}

=

^{mod-A, V}

=

^Vect;

c) C = AlgL , V = AlgK , where L is an extension of the field g, etc.

3. Let A be an algebra, B

=

^Mn(A). Construct a functor G : mod-A -+

mod-B in the following way. For every A-module M, put G(M)

=

nM. We endow G(M) with a B-module structure in a natural way: Considering an element x E G(M) as an n-dimensional vector with coordinates from M, define xb for b E B using the ordinary matrix multiplication rule. If f : M -+ N is an A-module homomorphism, define G(f) : G(M) -+ G(N) coordinatewise:

For x = (XI,X2, ... ,Xn) we put G(f)x = (fxl,jX2, ... ,fxn ). It is easy to verify that G(f) is a homomorphism of B-modules and that this construction indeed defines a functor.

4. If L is an extension of a field g, then it is possible to construct a functor F: AlgK -+ AlgL defining F(A) to be the L-algebra AL

=

^{A 0 L and F(f),}

where f : A -+ B, to be the L-algebra homomorphism f 01: AL -+ BL.

5. Let C be a semigroup with identity regarded as a category with a single object (Example 4 of a category). Let us clarify the meaning of a functor from the category C into the category VectK . Since ObC consists of a single element,

Fob is determined by a single vector space V. Then, for every element a of the semigroup, F(a) E E(V); moreover, F(l) = Iv and F(ab) = F(a)F(b).

Hence, Frnor is a representation of the semigroup C on a vector space V.

6. If Co is the dual (opposite) category of a category C, the functors F : Co -+ V are called contravariant functors from the category C to the category V (and in order to emphasize that it preserves the direction of ar- rows, the ordinary functors from C to V are called covariant functors). Since there is a one-to-one correspondence between ObCo and ObC, and also be- tween MorCo and MorC, the maps Fob and Frnor for a contravariant functor can be interpreted also as maps ObC -+ ObV and MorC -+ MorV. However, then the axioms in the definition of a functor take on the following form:

1°) if f : X -+ Y, then F(f) : F(Y) -+ F( X) (i. e. F "reverses the arrows");

2°) F(lx)

=

^{IF(X) ;}

3°) F(gf) = F(f)F(g) (i. e. F "reverses the order of the arrows").

An important example of a contravariant functor is obtained in analogy to Example 1. For a fixed object X E ObC, one can construct the functor h'X : Co -+ Sets by setting h'XCYO) = Hom(Y, X) and defining h'X(r) with f : Y -+ Z to be the map Hom(Z,X) -+ Hom(Y,X) assigning to a morphism 9 : Z -+ X the morphism gf : Y -+ X. If C

=

^{mod-A (or A-mod), then h'X}

can be interpreted as a functor Co -+ Vect.

Categories of modules over algebras (and many other categories) have an additional structure: they are linear in the following sense.

A category C is called a linear category over a field g (or g -linear or simply linear if there is no danger of misunderstanding) if, for every pair

8.1 Categories and Functors 139 (X, Y) of its objects, the set of morphisms Hom(X, Y) is endowed with the structure of a vector space over K and the composition of morphism is K- linear, i. e.

(J

+

g)h = fh

+

gh, f(g

+

h) = fg

+

fh and

(>..f)g = f(>.g) = >.(Jg)

for any morphisms

f,

g, h such that the corresponding formulae make sense, and for any >. E K.

Of course, a K-linear category with one object is just a K-algebra (cf.

Example 4 above). If a category Cis K-linear, then so is its dual Co (with the same linear structure).

A functor F : C -+ 'D between two linear categories is said to be linear (K -linear if we need to specify the field K) if

F(J

+

g) = F(J)

+

F(g) and F( >..f) = >.F(J)

for any morphisms f,g such that f

+

9 is defined, and for any>. E K. One can easily check that for every object X of a linear category C the functors hx and h'X (considered as functors to Vect) are linear.

An important property of linear functors is the fact that they preserve direct sums. Namely, we have the following "categorical" characterization of direct sums of modules.

Proposition 8.1.1. M c:::: Ml EEl M2 EEl ... EEl Mn if and only if there exist morphisms ik : Mk ^-+ M and Pk : M ^-+ Mk for all k ⁼ 1,2, ... , n such that Pkik

=

^{1Mk ,} Pkif

=

⁰ if k

i

£ and i1Pl

+

i2P2

+ ... +

inPn

=

^{1M .}

Proof. If M c:::: Ml EEl M2 EEl ... EEl JvIn , we can take for ik and Pk the natural embedding Mk ^-+ M and projection AI[ -+ Mk, respectively. On the other hand, given ik and Pk, the homomorphisms

C)M ^{~ ,~,M.}

are mutually inverse isomorphisms.

o

Now we are able to define a direct sum of objects M1 , M2, ... ,Mk ^{of any} linear category C as an object M such that morphisms ik : Mk ^-+ M and Pk : M ^-+ Mk satisfying the relations of Proposition 8.1.1 exist. One can easily verify (and we recommend to do it) that such M is defined up to an isomorphism (in C).

Corollary 8.1.2 Let F : C ^-+ 'D be a linear functor between two linear eate-

n n

gories and M c:::: EEl Mk in C. Then F(M) c:::: EEl F(Mk) in 'D.

k=l k=l

140 8. The Morita Theorem

In what follows, all categories and functors will be assumed to be linear and we shall use Corollary 8.1.2 frequently without any reference.

No documento Finite Dimensional Algebras (páginas 145-150)