4. CATEGORIES OF FILTERS

22

We need to check that

p

is a

Filt

-morphism that is

p

(

A ×

RLD

B

)

v A

what is

obvious.

Similarly for the right projection

q

.

It remains to check the universal property: Let

C

be a filter and

f

:

C → A

,

g

:

C → B

. We need to prove that there are a unique

u

:

C → A ×

RLD

B

such that

f

=

p

u

and

g

=

q

u

. Denote

h

(

z

) = (

f

(

z

)

, g

(

z

)).

h

is the unique function Base(

C

)

Base(

A

)

×

Base(

B

) such that

f

=

p

h

and

g

=

q

h

, so it remains to check that

h

is a morphism of

Filt

that is

h

h

iC v A×

RLD

B

,

what obviously follows from

h

f

iC v A

and

h

g

iC v B

.

Theorem

2035

.

Q

RLD

together with projections Pr

k

is a categorical product

in

Filt

.

Proof.

Consider an indexed family

A

of objects.

Denote

p

k

the

k

-th projection from

Q

i

dom

A

Base(

A

i

).

We need to check that

p

k

s a

Filt

-morphism that is

p

k

Q

RLD

A

v A

k

what

is obvious.

It remains to check the universal property: Let

C

be a filter and

f

k

:

C → A

k

.

We need to prove that there are a unique

u

:

C →

Q

RLD

A

such that

f

k

=

p

k

u

.

Denote

h

(

z

) =

λi

dom

A

:

f

i

z

.

h

is the unique function Base(

C

)

Q

i

dom

A

Base(

A

i

) such that

f

k

=

p

k

h

,

so it remains to check that

h

is a morphism of

Filt

that is

h

h

iC v

Q

RLD

A

. It

follows from

RLD

Pr

i

h

h

iC

=

l

h

P r

i

i

h

h

i

up

C

=

l

h

Pr

i

h

i

up

C

=

l

h

f

i

i

up

C

=

h

f

i

iC v A

i

.