One day while doing a computation on the board in front of my students, I accidentally wrote 1 + 1 = 1. (No idea why.)
Student: Um, don't you mean 1 + 1 = 2?
Me (embarrassed): Oh right, thanks.
[Erases mistake. Pauses.]
Wait. Is there a universe in which 1 + 1 = 1?
A Diagram is a Functor
Last week was the start of a mini-series on limits and colimits in category theory. We began by answering a few basic questions, including, "What ARE (co)limits?" In short, they are a way to construct new mathematical objects from old ones. For more on this non-technical answer, be sure to check out Limits and Colimits, Part 1. Towards the end of that post, I mentioned that (co)limits aren't really related to limits of sequences in topology and analysis (but see here). There is however one similarity. In analysis, we ask for the limit of a sequence. In category theory, we also ask for the (co)limit OF something. But if that "something" is not a sequence, then what is it?
Answer: a diagram.