In the first post of this series, we saw how subobjects of representable presheaves correspond to sieves on
. Last week, we saw how sieves give a convenient language for defining Grothendieck topologies on a small category. In next week’s (hopefully) final instalment of this series, we will generalise this even further to Lawvere–Tierney topologies on an arbitrary topos. Today’s post defines the last object we need to do this, which we will show generalises the presheaf
from last week:
Definition. Let be a (possibly large) category with a terminal object
. Then a subobject classifier on
is a monomorphism
in
such that for every monomorphism
in
, there exists a unique arrow
such that there is a pullback diagram
Example. If , then the two-point set
with its natural inclusion
given by
is a subobject classifier: the monomorphism
corresponds to the indicator function
that is
on
and
on its complement. (In other situations I would denote this by
, but that notation was already used in this series to denote the representable presheaf
.)
It’s even more natural to take to be the power set
of
. As in the first post of this series, we think of
representing “true” and
representing “false”. The generalisation of the power set of
to presheaf categories is the presheaf
of subpresheaves of
defined last week:
Lemma. Let be a small category. Then the presheaf
together with the map
taking the unique section
to the maximal sieve
for any
is a subobject classifier in
.
Proof. Note that the prescribed map is a morphism of presheaves, since the inverse image of the maximal sieve
under any morphism
in
is the maximal sieve
. Again using the notation from last week, if
is any monomorphism of presheaves, we get a morphism of presheaves
defined on
by
We will discuss some other properties of subobject classifiers in future posts.