This time, old chapters on equalizers and co-equalizers have been tidied up, and (I hope!) considerably improved. As with the chapters on products, there is now first a short chapter of pre-categorial reflections on equivalence relations and quotients. This is then used to better motivate the categorial idea of co-equalizers and its dual.

I should perhaps add that on each iteration of these notes there can be scattered tinkering with earlier chapters. This time I have corrected a number of typos I was told about, but also re-re-revised what I say about commuting diagrams (do we allow commuting diagrams to have non-equal parallel arrows?) and re-re-revised what I say about slice categories (reducing the amount of fuss about the conventional-but-not-quite-right story).

So there are now fourteen chapters (pp. 117) available of *Category Theory I: Notes towards a gentle introduction.*

This takes us up to a choice point. I’ve now talked about some familiar ‘ordinary maths’ constructions in categorial terms, in particular looking at products and quotients, which we’ve learnt to see as certain kinds of limiting cases. Should we next dig deeper or cast our net wider? By digging deeper, I mean giving a more abstract treatment of limits and co-limits in general. By casting our net wider, I mean first looking at other sorts of familiar constructions that don’t get regimented as limits, such as forming “power objects”, as when we go from A and B to the all the functions from A to B. There are pluses and minuses with either approach …