Database Reference
In-Depth Information
9
Weak Monoidal DB Topos
9.1
Topological Properties
A traditional axiomatic foundation of mathematics is the set theory, in which all
mathematical objects are ultimately represented by sets (even functions which map
between sets). More recent work in category theory allows this foundation to be
generalized using topoi; each topos completely defines its own mathematical frame-
work. The category of sets forms a familiar topos, and working within this topos
is equivalent to using traditional set-theoretic mathematics. But one could instead
choose to work with many alternative topoi. A standard formulation of the axiom
of choice makes sense in any topos, and there are topoi in which it is invalid. Con-
structivists will be interested to work in a topos without the law of excluded middle.
The definition of
topos
was originally proposed by Lawvere and Tierney, using
terms which were available when they started the topos theory in 1969 [
12
,
24
], it
was required that a category
C
(see Sect.
8.1.3
) satisfy the following properties:
1.
C
is finitely complete (has a terminal object and pullbacks);
2.
C
is finitely co-complete (has an initial object and pushouts);
3.
C
has exponentiation;
4.
C
has a subobject classifier (to be introduced in the next Sect.
9.1.2
).
Subsequently, C. Juul Mikklsen discovered that condition 2 is implied by the com-
bination of conditions 1, 3, and 4 [
19
].
Thus, a topos can be defined as a CCC (Cartesian Closed Category, see
Sect.
8.1.3
) with a subobject classifier.
The word “elementary” or “classic” (which from now on will be understood) has
a special technical meaning having to do with the nature of the definition of topos.
The most known topoi are
Set
and, for any small category
C
, the functor category
Set
C
. The category
Grph
of graphs of the kind permitting multiple directed edges
between two vertices is a topos. A graph
G
(V
G
,E
G
)
consists of two sets, an
edge set
E
G
and a vertex set
V
G
, and two functions
s,t
between those sets, assign-
ing to every edge
e
=
V
G
.
Grph
is thus
equivalent to the functor category
Set
C
, where
C
is the category with two objects
∈
E
G
its source
s(e)
∈
V
G
and target
t(e)
∈
