it follows that every automorphism θ ∈ AutF (A, σ) preserves the decomposition

above, hence also the pair of subalgebras {Q1 , Q2 }. If Q1 Q2 , then θ must

preserve separately Q1 and Q2 ; therefore, it restricts to automorphisms of Q1 and

of Q2 . Let q1 ∈ Q— , q2 ∈ Q— be such that

1 2

θ|Q1 = Int(q1 ), θ|Q2 = Int(q2 ).

Then θ = Int(q1 — q2 ); so

GO(A, σ) = { q1 — q2 | q1 ∈ Q— , q2 ∈ Q— }

1 2

and the map which carries (q1 ·F — , q2 ·F — ) to (q1 —q2 )·F — induces an isomorphism

∼

(Q— /F — ) — (Q— /F — ) ’ PGO(A, σ).

’

1 2

If Q1 Q2 , then we may assume for notational convenience that A = Q—F Q where

Q is a quaternion algebra isomorphic to Q1 and to Q2 . Under the isomorphism

γ— : A ’ EndF (Q) such that γ— (q1 — q2 )(x) = q1 xγ(q2 ) for q1 , q2 , x ∈ Q, the

involution σ = γ — γ corresponds to the adjoint involution with respect to the

reduced norm quadratic form nQ ; therefore GO(A, σ) is the group of similitudes of

the quadratic space (Q, nQ ):

GO(A, σ) GO(Q, nQ ).

(These results are generalized in §??).

Multipliers of similitudes. Let (A, σ) be a central simple algebra with invo-

lution of any kind over an arbitrary ¬eld F . Let G(A, σ) be the group of multipliers

of similitudes of (A, σ):

G(A, σ) = { µ(g) | g ∈ Sim(A, σ) } ‚ F — .

If θ is an involution of the same kind as σ on a division algebra D Brauer-equivalent

to A, we may represent A as the endomorphism algebra of some vector space V

over D and σ as the adjoint involution with respect to some nonsingular hermitian

or skew-hermitian form h on V :

(A, σ) = EndD (V ), σh .

As in the split case (where D = F ), the similitudes of (A, σ) are the similitudes

of the hermitian or skew-hermitian space (V, h). It is clear from the de¬nition

that a similitude of (V, h) with multiplier ± ∈ F — may be regarded as an isometry

∼

(V, ±h) ’ (V, h). Therefore, multipliers of similitudes of (A, σ) can be character-

’

ized in terms of the Witt group W (D, θ) of hermitian spaces over D with respect

to θ (or of the group W ’1 (D, θ) of skew-hermitian spaces over D with respect to θ)

(see Scharlau [?, p. 239]). For the next proposition, note that the group W ±1 (D, θ)

is a module over the Witt ring W F .

§12. GENERAL PROPERTIES 163

(12.20) Proposition. For (A, σ) EndD (V ), σh as above,

G(A, σ) = { ± ∈ F — | (V, h) (V, ±h) }

= { ± ∈ F — | 1, ’± · h = 0 in W ±1 (D, θ) }.

In particular, if A is split and σ is symplectic, then G(A, σ) = F — .

Proof : The ¬rst part follows from the description above of similitudes of (A, σ).

The last statement then follows from the fact that W ’1 (F, IdF ) = 0.

As a sample of application, one can prove the following analogue of Scharlau™s

norm principle for algebras with involution by the same argument as in the classical

case (see Scharlau [?, Theorem 2.8.6]):

(12.21) Proposition. For any ¬nite extension L/F ,

NL/F G(AL , σL ) ‚ G(A, σ).

(12.22) Corollary. If σ is symplectic, then

F —2 · NrdA (A— ) ‚ G(A, σ).

If moreover deg A ≡ 2 mod 4, then this inclusion is an equality, and

G(A, σ) = F —2 · NrdA (A— ) = NrdA (A— ).

Proof : Let D be the division algebra (which is unique up to a F -isomorphism)

Brauer-equivalent to A. Then, NrdD (D— ) = NrdA (A— ) by Draxl [?, Theorem 1,

p. 146], hence it su¬ces to show that NrdD (d) ∈ G(A, σ) for all d ∈ D — to prove

the ¬rst part. Let L be a maximal sub¬eld in D containing d. The algebra AL is

then split, hence (??) shows:

G(AL , σL ) = L— .

From (??), it follows that

NL/F (d) ∈ G(A, σ).

This completes the proof of the ¬rst part, since NL/F (d) = NrdD (d).

Next, assume deg A = n = 2m, where m is odd. Since the index of A divides

its degree and its exponent, we have ind A = 1 or 2, hence D = F or D is a

quaternion algebra. In each case, NrdD (D— ) contains F —2 , hence F —2 ·NrdA (A— ) =

NrdA (A— ). On the other hand, taking the pfa¬an norm of each side of the equation

σ(g)g = µ(g), for g ∈ GSp(A, σ), we obtain NrdA (g) = µ(g)m by (??). Since m is

odd, it follows that

2

µ(g) = µ(g)’(m’1)/2 NrdA (g) ∈ F —2 · NrdA (A— ),

hence G(A, σ) ‚ F —2 · NrdA (A— ).

12.C. Proper similitudes. Suppose σ is an involution of the ¬rst kind on

a central simple F -algebra A of even degree n = 2m. For every similitude g ∈

Sim(A, σ) we have

NrdA (g) = ±µ(g)m ,

as can be seen by taking the reduced norm of both sides of the equation σ(g)g =

µ(g).

164 III. SIMILITUDES

(12.23) Proposition. If σ is a symplectic involution on a central simple F -algebra

A of degree n = 2m, then

NrdA (g) = µ(g)m for all g ∈ GSp(A, σ).

Proof : If A is split, the formula is a restatement of (??). The general case follows

by extending scalars to a splitting ¬eld of A.

By contrast, if σ is orthogonal, we may distinguish two types of similitudes

according to the sign of NrdA (g)µ(g)’m :

(12.24) De¬nition. Let σ be an orthogonal involution on a central simple alge-

bra A of even degree n = 2m over an arbitrary ¬eld F . A similitude g ∈ GO(A, σ)

is called proper (resp. improper ) if NrdA (g) = +µ(g)m (resp. NrdA (g) = ’µ(g)m ).

(Thus, if char F = 2, every similitude of (A, σ) is proper; however, see (??).)

It is clear that proper similitudes form a subgroup of index at most 2 in the

group of all similitudes; we write GO+ (A, σ) for this subgroup. The set of improper

similitudes is a coset of GO+ (A, σ), which may be empty.16 We also set:

PGO+ (A, σ) = GO+ (A, σ)/F — ,

and

O+ (A, σ) = GO+ (A, σ) © O(A, σ) = { g ∈ A | σ(g)g = NrdA (g) = 1 }.

The elements in O+ (A, σ) are the proper isometries.

(12.25) Example. Let Q be a quaternion algebra with canonical involution γ over

a ¬eld F of characteristic di¬erent from 2, and let σ = Int(u) —¦ γ for some invertible

pure quaternion u. Let v ∈ A be an invertible pure quaternion which anticommutes

with u. The group GO(A, σ) has been determined in (??); straightforward norm

computations show that the elements in F (u)— are proper similitudes, whereas

those in F (u)— · v are improper, hence

GO+ (Q, σ) = F (u)— .

However, no element in F (u)— · v has norm 1 unless Q is split, so

O+ (Q, σ) = O(Q, σ) = { z ∈ F (u) | NF (u)/F (z) = 1 } if Q is not split.

(12.26) Example. Let A = Q1 —F Q2 , a tensor product of two quaternion algebras

over a ¬eld F of characteristic di¬erent from 2, and σ = γ1 — γ2 where γ1 , γ2 are

the canonical involutions on Q1 and Q2 .

If Q1 Q2 , then we know from (??) that all the similitudes of (A, σ) are of

the form q1 — q2 for some q1 ∈ Q— , q2 ∈ Q— . We have

1 2

µ(q1 — q2 ) = γ1 (q1 )q1 — γ2 (q2 )q2 = NrdQ1 (q1 ) · NrdQ2 (q2 )

and

NrdA (q1 — q2 ) = NrdQ1 (q1 )deg Q2 · NrdQ2 (q2 )deg Q1 = µ(q1 — q2 )2 ,

so all the similitudes are proper:

GO(A, σ) = GO+ (A, σ) O(A, σ) = O+ (A, σ).

and

16 Fromthe viewpoint of linear algebraic groups, one would say rather that this coset may

have no rational point. It has a rational point over a splitting ¬eld of A however, since hyperplane

re¬‚ections are improper isometries.

§12. GENERAL PROPERTIES 165

On the other hand, if Q1 Q2 , then the algebra A is split, hence GO+ (A, σ) is a

subgroup of index 2 in GO(A, σ).

Proper similitudes of algebras with quadratic pair. A notion of proper

similitudes can also be de¬ned for quadratic pairs. We consider only the charac-

teristic 2 case, since if the characteristic is di¬erent from 2 the similitudes of a

quadratic pair (σ, f ) are the similitudes of the orthogonal involution σ.

Thus let (σ, f ) be a quadratic pair on a central simple algebra A of even degree

n = 2m over a ¬eld F of characteristic 2. Let ∈ A be an element such that

f (s) = TrdA ( s) for all s ∈ Sym(A, σ)

(see (??)). For g ∈ GO(A, σ, f ), we have f (gsg ’1 ) = f (s) for all s ∈ Sym(A, σ),

hence, as in (??),

TrdA (g ’1 g ’ )s = 0 for s ∈ Sym(A, σ).

By (??), it follows that

g ’1 g ’ ∈ Alt(A, σ).

Therefore, f (g ’1 g ’ ) ∈ F by property (??) of the de¬nition of a quadratic pair.

(12.27) Proposition. The element f (g ’1 g ’ ) depends only on the similitude g,

and not on the choice of . Moreover, f (g ’1 g ’ ) = 0 or 1.

Proof : As observed in (??), the element is uniquely determined by the quadratic

pair (σ, f ) up to the addition of an element in Alt(A, σ). If = + x + σ(x), then

g ’1 g ’ = (g ’1 g ’ ) + (g ’1 xg ’ x) + σ(g ’1 xg ’ x),

since σ(g) = µ(g)g ’1 . We have

f g ’1 xg ’ x + σ(g ’1 xg ’ x) = TrdA (g ’1 xg ’ x) = 0,

hence the preceding equation yields

f (g ’1 g ’ ) = f (g ’1 g ’ ),

proving that f (g ’1 g ’ ) does not depend on the choice of .

We next show that this element is either 0 or 1. By (??), we have σ( ) = + 1,

hence 2 + = σ( ) . It follows that

g ’1 2 g ’ 2

= µ(g)’1 σ(g)σ( ) g ’ σ( ) + (g ’1 g ’ ),

hence g ’1 2 g ’ 2

∈ Sym(A, σ). We shall show successively:

f (g ’1 g ’ )2 = f (g ’1 g ’ )2 ,

(12.28)

f (g ’1 g ’ )2 = f (g ’1 2 g ’ 2

(12.29) ),

f (g ’1 2 g ’ 2

) = f (g ’1 g ’ ).

(12.30)

By combining these equalities, we obtain

f (g ’1 g ’ )2 = f (g ’1 g ’ ),

hence f (g ’1 g ’ ) = 0 or 1.

We ¬rst show that f (x)2 = f (x2 ) for all x ∈ Alt(A, σ); equation (??) follows,

since g ’1 g ’ ∈ Alt(A, σ). Let x = y + σ(y) for some y ∈ A. Since σ( ) + = 1,

we have

σ(y)y = σ(y) y + σ σ(y) y ,

166 III. SIMILITUDES

hence

f σ(y)y = TrdA σ(y) y .

The right side also equals

TrdA yσ(y) = f yσ(y) ,

hence f σ(y)y + yσ(y) = 0. It follows that

f (x2 ) = f y 2 + σ(y 2 ) = TrdA (y 2 ).

On the other hand, (??) shows that TrdA (y 2 ) = TrdA (y)2 ; since f (x) = TrdA (y),

we thus have f (x)2 = f (x2 ).

To prove (??), it su¬ces to show

f (g ’1 g + g ’1 g) = 0,

since (g ’1 g ’ )2 = (g ’1 2 g ’ 2

) + (g ’1 g + g ’1 g). By the de¬nition of , we

have

f (g ’1 g + g ’1 g) = TrdA (g ’1 g + g ’1 g) ;

the right-hand expression vanishes, since TrdA ( g ’1 g ) = TrdA ( 2 g ’1 g).

To complete the proof, we show (??): since g is a similitude and 2 + =

σ( ) ∈ Sym(A, σ), we have f g ’1 ( 2 + )g = f ( 2 + ), hence

f (g ’1 2 g + g ’1 g + 2

+ )=0

and therefore

f (g ’1 2 g + 2

) = f (g ’1 g + ).

(12.31) Example. Suppose (V, q) is a nonsingular quadratic space of even dimen-

sion n = 2m and let (σq , fq ) be the associated quadratic pair on EndF (V ), so

that

GO EndF (V ), σq , fq = GO(V, q),

as observed in (??). If ∈ EndF (V ) is such that fq (s) = tr( s) for all s ∈

Sym EndF (V ), σq , we claim that for all g ∈ GO(V, q) the Dickson invariant ∆(g),

de¬ned in (??), satis¬es

∆(g) = f (g ’1 g ’ ).

Since the right-hand expression does not depend on the choice of , it su¬ces to

prove the claim for a particular . Pick a basis (e1 , . . . , en ) of V which is symplectic

for the alternating form bq , i.e.,

bq (e2i’1 , e2i ) = 1, bq (e2i , e2i+1 ) = 0 and bq (ei , ej ) = 0 if |i ’ j| > 1,

and identify every endomorphism of V with its matrix with respect to that basis.

An element ∈ EndF (V ) such that tr( s) = fq (s) for all s ∈ Sym EndF (V ), σq is

given in (??); the corresponding matrix is (see the proof of (??))

«

0

1

1 q(e2i )

¬ ·

..

= where i = q(e .

.

2i’1 ) 0

0 m

§12. GENERAL PROPERTIES 167

On the other hand, for a matrix M representing the quadratic form q, we may

choose

«

M1 0

q(e2i’1 ) 0

¬ ·

..

M = where Mi = .

. 1 q(e2i )

0 Mm

where W = M + M t . Therefore, for all

It is readily veri¬ed that M = W ·

invertible g ∈ EndF (V ),

g ’1 g + = W ’1 (W g ’1 W ’1 M g + M ).

Since σq (g) = W ’1 g t W , we have W g ’1 W ’1 = µ(g)’1 g t if g ∈ GO(V, q), hence

the preceding equation may be rewritten as

g ’1 g + = µ(g)’1 W ’1 g t M g + µ(g)M .

Let R be a matrix such that g t M g + µ(g)M = R + Rt , so that

∆(g) = tr µ(g)’1 W ’1 R .

We then have

g ’1 g + = µ(g)’1 W ’1 (R + Rt ) = µ(g)’1 W ’1 R + σq µ(g)’1 W ’1 R ,

hence

fq (g ’1 g + ) = tr µ(g)’1 W ’1 R ,

and the claim is proved.

Note that this result yields an alternate proof of the part of (??) saying that

’1

f (g g+ ) = 0 or 1 for all g ∈ GO(A, σ, f ). One invokes (??) after scalar extension

to a splitting ¬eld.

(12.32) De¬nition. Let (σ, f ) be a quadratic pair on a central simple algebra A

of even degree over a ¬eld F of characteristic 2. In view of (??), we may set

∆(g) = f (g ’1 g ’ ) ∈ {0, 1} for g ∈ GO(A, σ, f ),

where ∈ A is such that f (s) = TrdA ( s) for all s ∈ Sym(A, σ). We call ∆ the

Dickson invariant. By (??), this de¬nition is compatible with (??) when (A, σ, f ) =

EndF (V ), σq , fq .

It is easily veri¬ed that the map ∆ is a group homomorphism

∆ : GO(A, σ, f ) ’ Z/2Z.

We set GO+ (A, σ, f ) for its kernel; its elements are called proper similitudes. We

also set PGO+ (A, σ, f ) = GO+ (A, σ, f )/F — .

(12.33) Example. Let Q be a quaternion algebra with canonical involution γ over

a ¬eld F of characteristic 2, and let (γ, f ) be a quadratic pair on Q. Choose ∈ Q

satisfying f (s) = TrdQ ( s) for all s ∈ Sym(Q, γ). As observed in (??), we have

GO(Q, γ, f ) = F ( )— ∪ F ( )— · v

where v ∈ Q— satis¬es v ’1 v = + 1. For g ∈ F ( )— we have g ’1 g + = 0, hence

∆(g) = 0. On the other hand, if g ∈ F ( )— · v, then g ’1 g + = 1, hence ∆(g) = 1,

by (??). Therefore (compare with (??))

GO+ (Q, γ, f ) = F ( )— .

168 III. SIMILITUDES

12.D. Functorial properties. Elaborating on the observation that simili-

tudes of a given bilinear or hermitian space induce automorphisms of its endo-

morphism algebra with adjoint involution (see (??)), we now show how similitudes

between two hermitian spaces induce isomorphisms between their endomorphism

algebras. In the case of quadratic spaces of odd dimension in characteristic dif-

ferent from 2, the relationship with endomorphism algebras takes the form of an

equivalence of categories.

Let D be a division algebra with involution θ over an arbitrary ¬eld F . Let

K be the center of D, and assume F is the sub¬eld of θ-invariant elements in K

(so F = K if θ is of the ¬rst kind). Hermitian or skew-hermitian spaces (V, h),

(V , h ) over D with respect to θ are called similar if there exists a D-linear map

g : V ’ V and a nonzero element ± ∈ F — such that

h g(x), g(y) = ±h(x, y) for x, y ∈ V .

The map g is then called a similitude with multiplier ±.

Assuming (V, h), (V , h ) nonsingular, let σh , σh be their adjoint involutions

on EndD (V ), EndD (V ) respectively.

(12.34) Proposition. Every similitude g : (V, h) ’ (V , h ) induces a K-isomor-

phism of algebras with involution

g— : EndD (V ), σh ’ EndD (V ), σh

de¬ned by g— (f ) = g —¦ f —¦ g ’1 for f ∈ EndD (V ). Further, every K-isomorphism

of algebras with involution EndD (V ), σh ’ EndD (V ), σh has the form g— for

some similitude g : (V, h) ’ (V , h ), which is uniquely determined up to a factor

in K — .

Proof : It is straightforward to check that for every similitude g, the map g— is an

isomorphism of algebras with involution. On the other hand, suppose that

¦ : EndD (V ), σh ’ EndD (V ), σh

is a K-isomorphism of algebras with involution. We then have dimD V = dimD V ,

and we use ¦ to de¬ne on V the structure of a left EndD (V ) —K Dop -module, by

(f — dop ) — v = ¦(f )(v )d for f ∈ EndD (V ), d ∈ D, v ∈ V .

The space V also is a left EndD (V ) —K Dop -module, with the action de¬ned by

(f — dop ) — v = f (v)d for f ∈ EndD (V ), d ∈ D, v ∈ V .

Since dimK V = dimK V , it follows from (??) that V and V are isomorphic as

EndD (V ) —K Dop -modules. Hence, there exists a D-linear bijective map

g: V ’ V

such that f — g(v ) = g —¦ f (v ) for all f ∈ EndD (V ), v ∈ V ; this means

¦(f ) —¦ g = g —¦ f for f ∈ EndD (V ).

It remains to show that g is a similitude, and that it is uniquely determined up to

a factor in K — . To prove the ¬rst part, de¬ne a hermitian form h0 on V by

h0 (v, w) = h g(v), g(w) for v, w ∈ V .

For all f ∈ EndD (V ), we then have

h0 v, f (w) = h g(v), ¦(f ) —¦ g(w) .

§12. GENERAL PROPERTIES 169

Since ¦ is an isomorphism of algebras with involution, σh ¦(f ) = ¦ σh (f ) ,

hence the right-hand expression may be rewritten as

h ¦ σh (f ) —¦ g(v), g(w) = h0 σh (f )(v), w .

Therefore, σh is the adjoint involution with respect to h0 . By (??), it follows that

h0 = ±h for some ± ∈ F — , hence g is a similitude with multiplier ±, and ¦ = g— .

If g, g : (V, h) ’ (V , h ) are similitudes such that g— = g— , then g ’1 g ∈

EndD (V ) commutes with every f ∈ EndD (V ), hence g ≡ g mod K — .

(12.35) Corollary. All hyperbolic involutions of the same type on a central simple

algebra are conjugate. Similarly, all hyperbolic quadratic pairs on a central simple

algebra are conjugate.

Proof : Let A be a central simple algebra, which we represent as EndD (V ) for some

vector space V over a division algebra D, and let σ, σ be hyperbolic involutions

of the same type on A. These involutions are adjoint to hyperbolic hermitian or

skew-hermitian forms h, h on V , by (??). Since all the hyperbolic forms on V are

isometric, the preceding proposition shows that (A, σ) (A, σ ), hence σ and σ

are conjugate.

Consider next two hyperbolic quadratic pairs (σ, f ) and (σ , f ) on A. The

involutions σ and σ are hyperbolic, hence conjugate, by the ¬rst part of the proof.

After a suitable conjugation, we may thus assume σ = σ. By (??), there are

idempotents e, e ∈ A such that f (s) = TrdA (es) and f (s) = TrdA (e s) for all

s ∈ Sym(A, σ).

Claim. There exists x ∈ A— such that σ(x)x = 1 and e = xe x’1 .

The claim yields

f (s) = TrdA (x’1 exs) = TrdA (exsx’1 ) = f (xsx’1 ) for all s ∈ Sym(A, σ),

hence x conjugates (σ, f ) into (σ , f ).

To prove the claim, choose a representation of A:

(A, σ) = EndD (V ), σh

for some hyperbolic hermitian space (V, h) over a division algebra D. As in the

proof of (??), we may ¬nd a pair of complementary totally isotropic subspaces U ,

W (resp. U , W ) in V such that e is the projection on U parallel to W and e is the

projection on U parallel to W . It is easy to ¬nd an isometry of V which maps U

to U and W to W ; every such isometry x satis¬es σ(x)x = 1 and e = xe x’1 .

There is an analogue to (??) for quadratic pairs:

(12.36) Proposition. Let (V, q) and (V , q ) be even-dimensional and nonsingular

quadratic spaces over a ¬eld F . Every similitude g : (V, q) ’ (V , q ) induces an

F -isomorphism of algebras with quadratic pair

g— : EndF (V ), σq , fq ’ EndF (V ), σq , fq

de¬ned by

g— (h) = g —¦ h —¦ g ’1 for h ∈ EndF (V ).

Moreover, every F -isomorphism

EndF (V ), σq , fq ’ EndF (V ), σq , fq

170 III. SIMILITUDES

of algebras with quadratic pair is of the form g— for some similitude g : (V, q) ’

(V , q ), which is uniquely determined up to a factor in F — .

Proof : The same arguments as in the proof of (??) apply here. Details are left to

the reader.

Proposition (??) shows that mapping every hermitian or skew-hermitian space

(V, h) to the algebra EndD (V ), σh de¬nes a full functor from the category of

nonsingular hermitian or skew-hermitian spaces over D, where the morphisms are

the similitudes, to the category of central simple algebras with involution where

the morphisms are the isomorphisms. In the particular case where the degree is

odd and the characteristic is di¬erent from 2, this functor can be used to set up an

equivalence of categories, as we show in (??) below.

A particular feature of the categories we consider here (and in the next chapter)

is that all the morphisms are invertible (i.e., isomorphisms). A category which has

this property is called a groupoid . Equivalences of groupoids may be described in a

very elementary way, as the next proposition shows. For an arbitrary category X ,

let Isom(X ) be the class17 of isomorphism classes of objects in X . Every functor

S : X ’ Y induces a map Isom(X ) ’ Isom(Y ) which we also denote by S.

(12.37) Proposition. Let X , Y be groupoids. A covariant functor S : X ’ Y

de¬nes an equivalence of categories if and only if the following conditions hold :

(1) the induced map S : Isom(X ) ’ Isom(Y ) is a bijection;

(2) for each X ∈ X , the induced map AutX (X) ’ AutY S(X) is a bijection.

Proof : The conditions are clearly necessary. Suppose that the covariant functor S

satis¬es conditions (??) and (??) above. If X, X ∈ X and g : S(X) ’ S(X ) is a

morphism in Y , then S(X) and S(X ) are in the same isomorphism class of Y , hence

(??) implies that X and X are isomorphic. Let f : X ’ X be an isomorphism.

Then g —¦ S(f )’1 ∈ AutY S(X ) , hence g —¦ S(f )’1 = S(h) for some h ∈ AutX (X ).

It follows that g = S(h —¦ f ), showing that the functor S is full. It is also faithful: if

f , g : X ’ X are morphisms in X , then S(f ) = S(g) implies S(f ’1 —¦ g) = IdS(X) ,

hence f = g by (??). Since every object in Y is isomorphic to an object of the form

S(X) with X ∈ X , it follows that S is an equivalence of categories (see Mac Lane [?,

p. 91]).

(12.38) Remarks. (1) The proof above also applies mutatis mutandis to con-

travariant functors, showing that the same conditions as in (??) characterize the