The arithmetic mean is perhaps one of the oldest mathematical ideas, intimately tied to fairness in mutual endeavor. The arithmetic mean answers the question “what payout would we have gotten if we all got the same payout”, or in other words, which makes p_1 + \cdots + p_n = A + \cdots + A$ true? Most people are familiar with the arithmetic mean, and a few math nerds might know about the geometric or harmonic means. But what is a mean, in general?
When explaining category theory to my friends (which happens more than you might think), I often find myself summing the theory up as “the abstract study of variable quantities”. We analyse the notion of a variable quantity into two components: what the quantity is, and with what the quantity varies. To say that a quantity is a number that varies with the time (say, if is the position of a beetle as it walks along the edge of a ruler, or the angle in degrees that the sun makes with the horizon), we write where is our notion of time and our notion of number.
All Functions Are Smooth!? What about…