ñòð. 8 |

FLT-2 and FLT-3.

The comments concerning Theorem 7.5.1 apply here too. See [Barr and Wells,

1985], Section 4.4, Theorem 2 (p. 156). The constructions in [Ehresmann, 1968]

and [Bastiani and Ehresmann, 1972] are more general since they allow cones over

inÂ¯nite diagrams. Barr and Wells [1994] provide a construction of that shows

that every object of is a limit of a Â¯nite diagram in the graph of .

2.3.2 Example The sketch for semigroups (see Section 7.2) is an FP sketch

and therefore an FL sketch whose cones all go to discrete diagrams. The FL

theory of this sketch contains a node v that is the limit of this diagram:

- sÂ£s

Â¢

s

@ Â¡c

id @ Â¡

@

R Â¡

Âª

(2.2)

s

Since M0 is injective, we write s for M0(s), s Â£ s for M0(s Â£ s), and so on.

A model M of the sketch induces a limit preserving functor F which must

take v to a set. Since v is the equalizer of ids and c Â± Â¢, F (v) must be the set

of elements e of M (s) that satisfy the requirement ee = e, that is, the set of

idempotents in the semigroup M (s). (This set is not in general a subsemigroup.)

2.4 General deÂ¯nition of sketch 21

2.3.3 Example The FL theory of the sketch for categories (see ES 2.1.5) con-

tains a node v that is the limit of the diagram

c1

Â¡ @

sÂ¡ @t

Â¡

Âª @

R

c0 c0 (2.3)

@

I Â¡

Âµ

s@ Â¡t

@ Â¡

c1

If is a small category, hence a model of the sketch, the induced limit-preserving

functor F takes v to the set of all parallel pairs of arrows in .

2.3.4 Exercises

1. The following diagram in the sketch for categories has a limit v in the FL

theory of that sketch. What is its value in a model?

c1 c1 c1

s@t Â¡s t @ sÂ¡ t

@Â¡ @Â¡

Â¡@ Â¡@

Â¡ @ Â¡ @

?Â¡Âª R

@ Â¡

Âª @?

R

c0 c0 c0

2. Same question as Exercise ES 1 for the following diagram:

c2

Â¡ @

Â¡ @

p1 p2

c

Â¡ @

Â¡ @

Â¡

Âª ? @

R

c1 c1 c1

s@t Â¡s t @ sÂ¡ t

@Â¡ @Â¡

Â¡@ Â¡@

Â¡ @ Â¡ @

?Â¡Âª @

R Âª

Â¡ @?

R

c0 c0 c0

is isomorphic to a model M 0 for which

3. Show that any model M of a sketch

M 0(c) and M 0(d) have no elements in common if c 6d.

=

2.4 General deÂ¯nition of sketch

Although we will not be using the most general notion of sketch, we give the

deÂ¯nition here.

22 More about sketches

2.4.1 DeÂ¯nition A sketch = ( ; ; ; ) consists of a graph , a set

of diagrams in , a set of cones in and a set of cocones in .

2.4.2 DeÂ¯nition A model M of a sketch = ( ; ; ; ) in a category

is a homomorphism from to the underlying graph of that takes every

diagram in to a commutative diagram, every cone in to a limit cone and

every cocone in to a colimit cocone.

2.4.3 DeÂ¯nition Let M , N be models of a sketch in a category .A

homomorphism of models Â® : M Â¡ N is a natural transformation from M

!

to N .

Sketches in general do not have initial algebras, or even families of initial

algebras. What they do have is `locally free diagrams' as described by Guitart

and Lair [1981, 1982].

2.4.4 Regular sketches Chapter 8 of [Barr and Wells, 1985] describes special

classes of sketches with cones and cocones that have in common the fact that

their theories are embedded in a topos, and so inherit the nice properties of a

topos. (Toposes are discussed in Chapter ES 5.) These are the regular sketches,

coherent sketches and geometric sketches. We describe regular sketches here to

illustrate the general pattern. (Note: The French school uses the phrase \regular

sketch" for a sketch in which no node is the vertex of more than one cone.)

2.4.5 An arrow f : A Â¡ B is a regular epimorphism if and only if there is a

!

cocone diagram

g

Â¡! A Â¡f B

Â¡ Â¡!

C Â¡!Â¡

h

It follows that the predicate of being a regular epimorphism can be stated within

the semantics of a Â¯nite limits sketch. Note that an arrow can be required to

become an epimorphism (not necessarily regular) in a model by using the dual

of Theorem 8.3.3.

Let us say that a regular cocone is one of the form

cÂ¡ aÂ¡ b

!!

Â¡

!

2.4.6 DeÂ¯nition A regular sketch = ( ; ; ; ) consists of a graph,

a set of diagrams, a set of Â¯nite cones and a set of regular cocones.

There is one undesirable feature to the deÂ¯nition above. The introduction

of a regular sketch requires the introduction of a sort c and two arrows c Â¡!

Â¡

!

a. A model will have to provide a value for c as well as the two arrows. This

is generally irrelevant information that one would not normally want to have

to provide. Worse, the deÂ¯nition of natural transformation is such that arrows

2.4 General deÂ¯nition of sketch 23

between models will have to preserve this additional information and that is

deÂ¯nitely undesirable. The way in which this is usually dealt with is by adding,

with each cocone

d0 d

c Â¡ Â¡ a Â¡! b

Â¡! Â¡

ñòð. 8 |