Reposting from many many moons ago ….

Ok, so how do you remember which are the epimorphisms, which are the monomorphisms, and which way around the funny arrows get used?

Since the textbooks don’t seem eager to offer helpful mnemonics, I offer a forgetful world the following: go by alphabetical proximity!

L-for-left goes with M-for-mono, and P-almost-for-epi goes almost next to R-for-right. OK?

But what does that mean? Simple.

A mono is of course a left-cancellable morphism, and you signal one using an arrow with an extra decoration (a tail) on the left.

Dually, an epi is a right-cancellable morphism, and you signal one of those using an arrow with an extra decoration (another head) on the right.

Easy, huh? Well, it works for me — and these days, I’m grateful for all the props I can get. You can thank me later.

1. This reflects one of the problems with category theory: opaque terminology (monomorphism) explained by further opaque terminology (left-cancellable morphism).

A problem with your mnemonic is that you have to remember it’s the first letter of ‘mono’ but the second of ‘epi’. I would end up remembering it by remembering that it’s ‘backwards’. The natural relationship would be that alphabetical order based on first letters gives you left – right, but instead it gives you right – left.

Anyway, if someone can remember what left-/right-cancellable means, the arrows go with left/right in a fairly obvious way, although some choices of arrows (↣,↠) reflect that (and the duality between the two concepts) better than others (↪,↠).

Then there’s the suggestion here relating the arrows and the diagrams that express the category-theoretic definitions.

Really, though, the terminology ought to suggest its own meaning. I’ve never been happy with ‘injection’ and ‘surjection’, but they at least do that more clearly (imo) than ‘monomorphism’ and ‘epimorphism’. Still, we have ‘mono’ suggesting one, which goes with 1-1 and so with injection; and ProofWiki suggests reading ‘epi’ as ‘onto’ (which also goes with ‘sur’).

Then the two-headed arrow suggests a degree of heaviness or force on the right / destination end, so you can think “yes, it’s going to hit everything on that side.”

1. I wasn’t being entirely solemn about this! (The repost isn’t a signal that I thought it I important, but I was prompted by a twitter comment.)

I suppose that what generates a certain amount of category theory “opaque terminology” is wanting to avoid terminology that could be misleadingly(?) specific. Thus, for example, the preference for “arrow” rather than “morphism” because while most arrows in interesting cases are indeed morphisms in a familiar sense, not all are.

And we want “mono” or “monomorphism” or something because in the general case monomorphisms don’t have to be injective. And so it goes …

1. Why “mono” in “monomorphism” if it doesn’t indicate 1-1 or injective?

Anyway, I don’t think “arrow” is opaque. It’s terms like “monomorphism” and “left-cancellable morphism” that I think are opaque, though of course it’s possible to learn what they mean and to use them enough that they become familiar.

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