is a ﬁnite set of discrete cones in E giving the rela
tionships and S a ﬁnite set of discrete cocones in E
specifying the attributes. For example the omission
of a diagram from L means that it is not required to
commute, so this diagram is effectively punctured.
Sketches lack ﬂexibility as all structures and con
straints have to be prespeciﬁed. In difﬁcult ar
eas such as interoperability, sketches are inadequate
as they do not offer natural closure. (Johnson and
Rosebrugh, 2000) attempt to adapt their sketches to
achieve interoperability but the aim is to achieve only
logical independence, as in the threelevel architec
ture of Figure 1, not semantic interoperability, as in
the fourlevel architecture of Figure 2. The difference
between a natural structure and a sketch is like that be
tween typing and labelling. A graph is richer than an
entityrelationship model as its arrows are typed with
identity functors. Labelling in the entityrelationship
model is an informal typing whereas the identity ar
row is a formal typing.
6 NATURAL COMPOSITION
Some problems with partial functions can be avoided
by altering the data design so that the partial functions
only operate in the assignment to the end of the chain
(the terminal object). For instance an alternative de
sign can be considered for Figure 11. Here the natural
order would be to consider ﬁrst accessions, which are
then put into the stack and can be issued later. For this
schema the composition diagram would be as in Fig
ure 12. These are full categories without composition
failure and the puncture sign can be removed. There
are no punctured diagrams if ISS is the codomain of
each of x
′
, t
′
and z
′
. This is because these are all par
tial functions, mapping onto a category which is last
in the sequence, the terminal object. There is a type
change but it occurs just once, in the ﬁnal step. It
is when partial functions map onto intermediate cat
egories in a chain that typing problems are likely to
occur, because of the ﬂuctuations of the types.
7 CONCLUSIONS
The use of a formal fourlevel architecture, based on
category theory, provides an encouraging framework
for tackling both semantic and organisational inter
operability. The use of the Godement calculus, in
particular, enables many different paths at a number
of level to be compared and analysed. A number of
problems remain. Failure of composition, particularly
due to the existence of partial functions, needs to be
ACC ST K
ISSCAT
x
′
u
′
y
′
t
′
z
′


? ?
@
@
@
@
@
@
@
@
@
@
@R
Figure 12: Nonpunctured Commuting Diagram for Library
Example
ACC = accessions, ST K = stock, ISS = issues, CAT =
catalogue
identiﬁed. Punctured categorical diagrams are used
for this purpose in preference to lifted categories or
sketches. Semantic annotation remains a challenging
area where the open Heyting logic may be of assis
tance.
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