The case when the monoid is a group
Let be a group (written multiplicatively). We are interested in , the category of (set) representations of . Objects in this category are pairs where is a set and is a right action of on — let’s immediately agree to simply write instead of . Recall that an action must verify the axioms:
- ; and
- .
Arrows in this category are set functions between the underlying sets that respect the group action: for all , we must have . Composition of arrows is just the usual composition of the underlying functions.
Let denote the singleton set, and the set with two elements, both with the trivial action . For any set , the unique function respects the group action. Hence, the representation is the terminal object in . We want to show that the representation morphism induced by the inclusion is the subobject classifier in the category .
Notice also that the pullback of any diagram always exists in , and is given by the usual pullback in with the obvious coordinate-wise action. More precisely, with the action defined to be . The existence of a terminal object and all pullbacks is sufficient for (in fact, equivalent to) having all finite limits in , and these may be constructed as limits of the carrier sets. In other words, the forgetful functor preserves all finite limits.
Let be a (representative for a) subobject of . The group acts on itself by multiplication on the right, and for any fixed element , we have a morphism of representations defined as . We can leverage these morphisms to prove the function underlying must be injective. Let be two elements of such that . This relation, together with the fact preserves the group action, means that we have . Hence . In particular, equality holds when these morphisms are evaluated at the identity element of , so . Since, as we’ve just shown, the function underlying any subobject must be injective, each subobject has a representative where is a subset of , with the group action on being the group action on restricted to . Therefore, any subobject of in is (represented by) a subset of which is stable by the group action, i.e. for all and for all , we have .
Given a subrepresentation , the characteristic set function , defined as if and only if , respects the group actions. Indeed, suppose ; then, for all , we have because is closed under the action of , while because the action on is trivial. On the other hand, the complement of in is also a subrepresentation because is a group, so when , we also have for all , whence in this case.
Because , it follows that this is a pullback square:
The only thing left to show for to be a subobject classifier is that is the only representation morphism such that the previous diagram is a pullback diagram. Suppose is another one. Then , so that if and only if . Since there are only two possible values that these functions can take, we must have .
The previous discussion could have been made perhaps clearer by the use of the forgetful functor , which, as we’ve seen, preserves limits. For instance, pick some monic arrow . An arrow is monic if and only if is a pullback for the pair . Therefore any functor which preserves limits also preserves monic arrows. In our particular case, we see that is a monic arrow in . It is known that monic arrows in are precisely the injective functions. Since is the set function underlying the morphism of representations , we must have that is injective, as we’ve shown in a more roundabout way a couple of paragraphs ago. The uniqueness of the characteristic function may also be proven quickly using the functor : the image of any pullback diagram in through is also a pullback diagram in , and we already know that the characteristic function is the unique set function such that is the pullback of along it.
The general case
Most of our previous discussion carries through when we consider to be simply a monoid (not necessarily a group). We are interested in , the category of (set) representations of the monoid . As before, subobjects of are identified with subsets that are stable under the monoid action, and are called “subrepresentations”. The same arguments also give that with the trivial action is the terminal object in , and the pullback of any pair of arrows always exists (the forgetful functor still preserves finite limits).
In fact, from our previous discussion of group representations, there is only one thing that changes: the characteristic function is not necessarily a morphism of representations anymore. This breaks a lot of stuff, and we don’t have such an easy time finding a subobject classifier. The fundamental reason of why the characteristic function doesn’t respect the monoid action, is that the complement of a subrepresentation may not be a subrepresentation: in the group case, the complement was always a stable under the action, but for monoids in general this is not true. A somewhat artificial example is given by the action of the monoid on the set of relative integers in the obvious way: for all and all , we define to be . Then, for any , the set of all integers in greater or equal to is a subrepresentation, but the complement of such a set is never a subrepresentation.
Let’s introduce some terminology. We say that a subset of is an ideal if, for every and every , we have . Let be a representation of , and let be a subrepresentation of . We say that an element kills an element (relative to ) when we have . For any element , let be the set of elements in which kill relative to . This set is an ideal in . Notice that for all , we have ; moreover, when is a group, we have for every . In fact, when is a group, there are only two ideals in : the empty set and the whole group . However, when is a monoid, it is not true that for all . For instance, in our previous artificial example,
We define to be the set of all ideals of . In our notation, for any subrepresentation of , we have a function . We wish to define a monoid action on so that is a morphism of representations. In other words, we want the ideal of elements that kill to be precisely . This forces us to define, for all ideals and all , the action This action makes into an object of and each function becomes a morphism of representations. When is a group, reduces to the two-elements set and reduces to the characteristic function . To continue this analogy further, we define by sending the unique element of to the top element .
Because if and only if , we see that . Therefore, the following diagram is a pullback square:
Now we need to show is the only arrow such that the previous diagram is a pullback square. Suppose is another one, and pick any . We want to show that .
Let be an element of , so that . By the commutativity of the previous diagram with in place of , this means . Since respects the monoid action, we have . Hence, by the definition of the monoid action on , we have . In particular, since , we have . Because was an arbitrary element of , we find that .
Consider the subset of defined as the set of elements of the form with . Since is an ideal of , this subset is stable under the monoid action and thus is a subrepresentation of . Moreover, for any , we have . Indeed, any may be written in the form for some ; then, () is by definition the set of all such that , and this condition is always verified since is an ideal. Thus the universal property of the pullback gives us an inclusion . In particular, all elements have the property that , so that .
The two previous paragraphs prove, by double inclusion, that . Because was an arbitrary element of , this shows is the unique function such that is the pullback of along . Therefore, , defined by sending the unique element of to the top element , is the subobject classifier in the category of representations of a fixed monoid .