Special kinds of morphisms, subobjects, and quotient objects
Last week I was using the word “invertible” as if it was perfectly clear. Well, it should be, more or less, since categories are pretty similar to monoids, but I should be a bit more explicit. First, though, there’s a few other kinds of morphisms we should know.
We want to motivate these definitions from what we know of sets, but the catch is that sets are actually pretty special. Some properties turn out to be the same when applied to sets, though they can be different in other categories.
First of all, let’s look at injective functions. Remember that these are functions where implies . That is, distinct inputs produce distinct outputs. Now we can build a function as follows: if for some we define . This is well-defined because at most one can work, by injectivity. Then for all the other elements of we just assign them to random elements of . Now the composition is the identity function on because for all . We say that the function has a (non-unique) “left inverse”.
Now since has a left inverse there’s something else that happens: if we have two functions and both from to , and if then . That is, is “left cancellable”.
Now in any category we say a morphism is a “monomorphism” (or “a mono”, or “ is monic”) if it is left cancellable, whether or not the cancellation comes from a left inverse as above. If has a left inverse we say is “injective” or that it is “an injection”. By the same argument as above, every injection is monic, but in general not all monos are injective. In the two concepts are the same.
Similarly, a surjective function has a right inverse , and is thus right cancellable. We say in general that a right cancellable morphism is an “epimorphism” (or “an epi”, or “ is epic”). If the right cancellation comes from a right inverse, we say that is “surjective”, or that it is “a surjection”. Again, every surjection is epic, but not all epis are surjective. In the two concepts are again the same.
If a morphism is both monic and epic then we call it a “bimorphism”, and it can be cancelled from either side. If it is both injective and surjective we call it an “isomorphism”. All isomorphisms are bimorphisms, but not all bimorphisms are isomorphisms. If is an isomorphism, then we can show (try it) that the left and right inverses are not only unique, but are the same, and we call the (left and right) inverse . When I said “invertible” last week I meant that such an inverse exists.
Now recall that any subset of a set comes with an injective function “including” into . Similarly, subgroups and subrings come with “inclusion” monomorphisms. We generalize this concept and define a “subobject” of an object in a category to be a monomorphism . In the same way we generalize quotient groups and quotient rings by defining a “quotient objects” of to be epimorphisms .
Notice that we define a subobject to be an arrow, and we allow any monomorphism. Consider the function defined by and . It seems odd at first, but we say that this is a subobject of . The important thing here is that we don’t define these concepts in terms of elements of sets, but in terms of arrows and their relations to each other. We “can’t tell the difference” between and since they are isomorphic as sets. If we just look at the arrow and the usual inclusion arrow of , they pick out the same subset of so we may as well consider them to be the same subset.
Let’s be a little more general here. Let and be two subobjects of . We say that “factors through” if there is an arrow so that . If we take the class of all subobjects of (all monomorphisms into ) we can give it the structure of a preorder by saying if factors through . It should be straightforward to verify that this is a preorder.
Now we can turn this preorder into a partial order as usual by identifying any two subobjects which factor through each other. If and then . Since is monic we can cancel it from the left and find that . similarly we find that . That is, and are inverses of each other, and so and are isomorphic as subobjects of . Conversely, if and are isomorphic subobjects then and factor through each other by an isomorphism . This gives us a partial order on (equivalence classes of) subobjects of . If the class of equivalence classes of subobjects is in fact a proper set for every object we say that our category is “well-powered”.
The preceding two paragraphs can be restated in terms of quotient objects. Just switch the directions of all the arrows and the orders of all the compositions. We get a partial order on (equivalence classes of) quotient objects of . If the class of equivalence classes is a proper set for each object then we say that the category is “co-well-powered”.
It should be noted that even though isomorphic subobjects come with an isomorphism between their objects, just having an isomorphism between the objects is not enough. One toy example is given in the comments below. Another is to consider two distinct one-element subsets of a given set. Clearly the object for each is a singleton, and all singletons are isomorphic, but the two subsets are not isomorphic as subobjects.
As an exercise, consider the category of commutative rings with unit and determine the partial order on the set of quotient objects of .