Looking for student's guide to diagram chasing

Part of the problem is that a lot of mathematicans who are heavy into categorical constructions and thinking think that most of it is just obvious if you look at the diagrams-just follow the arrows,they say. I think most beginners would disagree, but for some reason it never penetrates.

For relatively simple constructions-like commutative triangles and squares-it's easy to say it's obvious since you can just reexpress them in terms of good old fashioned functional arguments( write out the compositions explicitly). But in the real world, you can't really get far with this machinery if you can't read much more complicated diagrams. Even the proof of the Snake Lemma in homological algebra-which confuses the crap out of many graduate students-is baby stuff compared to some of the diagrams you'll see other areas of this subject. Including a number of 3-dimensional arrow chases.

So I agree-there's a need for a good practice source on this material for beginners.Until someone writes one,you'll need to do some digging around and putting together a patchwork source. Paulo Aluffi's Algebra:Chapter 0 has probably the best introduction to diagram chasing and its relation to categories that currently exists. It's really designed as a presentation for complete beginners-it does a very good job and has many good exercises for practice. A more general and equally helpful introduction can be found in Harold Simmons' An Introduction To Category Theory. 2 other,less comprehensive,but equally helpful sources are Chapter 4 of Ash's Basic Abstract Algebra ,which is available online at Ash's website and pages 43-53 of P.M. Cohn's An Introduction To Ring Theory. I think you'll find all these sources to be helpful.

Well, the first problem is that there aren't strict conventions on the use and meaning of diagrams. Although a formal definition exists—a diagram of objects and arrows in a category $\mathbf{C}$ is a functor $\mathscr{F} : \mathbf{J} \to \mathbf{C}$, for some (small) indexing category $\mathbf{J}$—in practice, a printed diagram, without clarifying remarks, may not be sufficient to determine what $\mathbf{J}$ is. For example, consider the traditional equaliser diagram $$\bullet \rightarrow \bullet \rightrightarrows \bullet$$ It is ‘obvious’ that the two parallel arrows are not equal. So this is an exception to the usual rule that every path through a diagram between a pair of vertices should be equal. Yet it is precisely this convention which is in force when it is ‘obvious’ that the composite arrows are equal. But I digress.

It is true that pasting two commutative diagrams along a matching pair of paths yields a commutative diagram. This is essentially by a discrete deformation of paths. For example, consider the diagrams

$$\begin{matrix} \bullet & \xrightarrow{f} & \bullet \newline {\scriptstyle g} \downarrow & & \downarrow {\scriptstyle h} \newline \bullet & \xrightarrow{k} & \bullet \end{matrix} \qquad \begin{matrix} \bullet & \xrightarrow{f'} & \bullet \newline {\scriptstyle g'} \downarrow & & \downarrow {\scriptstyle h'} \newline \bullet & \xrightarrow{k'} & \bullet \end{matrix} $$ Suppose $h = g'$. In order to verify that the diagram $$\begin{matrix} \bullet & \xrightarrow{f} & \bullet & \xrightarrow{f'} & \bullet \newline {\scriptstyle g} \downarrow & & \scriptstyle{g'} \downarrow {\scriptstyle h} & & \downarrow {\scriptstyle h'} \newline \bullet & \xrightarrow{k} & \bullet & \xrightarrow{k'} & \bullet \end{matrix} $$ also commutes, we need to show that the three paths between the top-left and bottom-right corners are equal. But this is obvious, since $k' \circ k \circ g = k' \circ h \circ f = k' \circ g' \circ f = h' \circ f' \circ f $.

The troublesome operation is adding in new arrows, since commutativity is not quite a local property. But it is possible to make life a little bit easier, by observing that we can subdivide the diagram into fragments, add in the arrow in each fragment and checking commutativity, then pasting the fragments back together.

As for existence and uniqueness: the only arrows which are guaranteed to exist are those obtained by composing arrows. Anything more than that requires specific knowledge of the objects and arrows in question.

I think diagram chasing is nothing but using injective and surjective homomorphism and exact sequences. If you learn those well, you can find those in diagrams. Try to reduce diagrams into simpler diagrams, morphisms or previously proved lemmas.