What is a "foo" in category theory? While browsing through several pages  of nlab(mainly on n-Categories), I encountered the notion "foo" several times. However, there seems to be article on nlab about this notion. Is this some kind of category theorist slang? Please explain to me what this term means.
 A: It's slang, which I've mostly seen used in the context of computing rather than category theory; foo is just a placeholder for something else, as is bar. A logician I know likes talking about widgets and wombats $-$ it all serves the same purpose.
For example, you might say "an irreducible foo is a foo with no proper sub-foos".
A: 1.1 More than the sum of their parts.
We motivate this first chapter by noticing that while many real-world structures are compositional, the results of observing them are often not. The reason is that observation is inherently “lossy”: in order to extract information from something, one must drop the details. For example, one stores a real number by rounding it to some precision. But if the details are actually relevant in a given system operation, then the observed result of that operation will not be as expected. This is clear in the case of roundoff error, but it also shows up in non-numerical domains: observing a complex system is rarely enough to predict its behavior because the observation is lossy.
Acentral themein category theory is the study of structures and structure-preserving maps. A map f : X ! Y is a kind of observation of object X via a specified relationship it has with another object, Y. For example, think of X as the subject of an experiment and Y as a meter connected to X, which allows us to extract certain features of X by looking at the reaction of Y.
Asking which aspects of X one wants to preserve under the observation f becomes the question “what category are you working in?.” As an example, there are many functions f from R to R, and we can think of them as observations: rather than view x “directly”, we only observe f(x). Out of all the functions f : R -> R, only some of them preserve the order of numbers, only some of them preserve the distance between numbers, only some of them preserve the sum of numbers, etc. Let’s check in with an exercise; a solution can be found in Chapter 1.
Exercise 1.1. Some terminology: a function f : R -> R is said to be
(a) order-preserving ...
(b) metric-preserving ...
(c) addition-preserving ...
For each of the three properties defined above—call it foo—find an f that is foo-preserving and an example of an f that is not foo-preserving.
From: Seven Sketches in Compositionality: An Invitation to Applied Category Theory by Brendan Fong and David I. Spivak pp. 1-2 (2018)
