The Unapologetic Mathematician

Mathematics for the interested outsider

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 f:X\rightarrow Y where f(x_1)=f(x_2) implies x_1=x_2. That is, distinct inputs produce distinct outputs. Now we can build a function g:B\rightarrow A as follows: if y=f(x) for some x\in X we define g(y)=x. This is well-defined because at most one x can work, by injectivity. Then for all the other elements of Y we just assign them to random elements of X. Now the composition g\circ f is the identity function on X because g(f(x))=x for all x\in X. We say that the function f has a (non-unique) “left inverse”.

Now since f has a left inverse g there’s something else that happens: if we have two functions h_1 and h_2 both from Y to X, and if f\circ h_1=f\circ h_2 then h_1=g\circ f\circ h_1=g\circ f\circ h_2=h_2. That is, f is “left cancellable”.

Now in any category \mathcal{C} we say a morphism f is a “monomorphism” (or “a mono”, or “f is monic”) if it is left cancellable, whether or not the cancellation comes from a left inverse as above. If f has a left inverse we say f 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 \mathbf{Set} the two concepts are the same.

Similarly, a surjective function f has a right inverse g, and is thus right cancellable. We say in general that a right cancellable morphism is an “epimorphism” (or “an epi”, or “f is epic”). If the right cancellation comes from a right inverse, we say that f is “surjective”, or that it is “a surjection”. Again, every surjection is epic, but not all epis are surjective. In \mathbf{Set} 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 f 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 f^{-1}. When I said “invertible” last week I meant that such an inverse exists.

We’ve already seen these terms in other categories. In groups and rings we have monomorphisms and epimorphisms, which are monos and epis in the categories \mathbf{Grp} and \mathbf{Ring}.

Now recall that any subset T of a set S comes with an injective function T\rightarrow S “including” T into S. Similarly, subgroups and subrings come with “inclusion” monomorphisms. We generalize this concept and define a “subobject” of an object C in a category \mathcal{C} to be a monomorphism S\rightarrow C. In the same way we generalize quotient groups and quotient rings by defining a “quotient objects” of C to be epimorphisms C\rightarrow Q.

Notice that we define a subobject to be an arrow, and we allow any monomorphism. Consider the function f:\{a,b\}\rightarrow\{1,2,3\} defined by f(a)=1 and f(b)=3. It seems odd at first, but we say that this is a subobject of \{1,2,3\}. 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 \{a,b\} and \{1,3\} since they are isomorphic as sets. If we just look at the arrow f and the usual inclusion arrow of \{1,3\}\subseteq\{1,2,3\}, they pick out the same subset of \{1,2,3\} so we may as well consider them to be the same subset.

Let’s be a little more general here. Let f_1:S_1\rightarrow C and f_2:S_2\rightarrow C be two subobjects of C. We say that f_1 “factors through” f_2 if there is an arrow g:S_1\rightarrow S_2 so that f_1=f_2\circ g. If we take the class of all subobjects of C (all monomorphisms into C) we can give it the structure of a preorder by saying f_1\leq f_2 if f_1 factors through f_2. 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 f_1=f_2\circ g_2 and f_2=f_1\circ g_1 then f_1=f_1\circ g_1\circ g_2. Since f_1 is monic we can cancel it from the left and find that 1_{S_1}=g_1\circ g_2. similarly we find that 1_{S_2}=g_2\circ g_1. That is, g_1 and g_2 are inverses of each other, and so S_1 and S_2 are isomorphic as subobjects of C. Conversely, if S_1 and S_2 are isomorphic subobjects then f_1 and f_2 factor through each other by an isomorphism g:S_1\rightarrow S_2. This gives us a partial order on (equivalence classes of) subobjects of C. If the class of equivalence classes of subobjects is in fact a proper set for every object C 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 C. If the class of equivalence classes is a proper set for each object C 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 \mathbf{CRing} of commutative rings with unit and determine the partial order on the set of quotient objects of \mathbb{Z}.

May 29, 2007 Posted by | Category theory | 20 Comments

New This Week’s Finds

Now, back to the math. John Baez has posted a new This Week’s Finds. He continues his “Tale of Groupoidification”. It features a great comparison of group actions on sets and those on vector spaces (which we’ll get to soon enough). Even better, he gives a translation “for students trying to learn a little basic category theory” into the language of categories.

May 29, 2007 Posted by | Uncategorized | Leave a comment