In "Category Theory, A Gentle Introduction", by Peter Smith, one can find the following sentence: "Take $\mathcal{L}$ to be the elementary pure language of categories" - what is a 'language', not to mention a 'elementary pure' one?
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2$\begingroup$ Compare with Mathematical Logic: Language $\endgroup$– Mauro ALLEGRANZAJan 22, 2020 at 15:56
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$\begingroup$ I agree this is hard to parse, but I think I know what is meant. "Elementary language" means language in the sense of first-order logic (elementary is often a synonym for first-order in the context of logic). And "pure" means that the only symbols in the language are those needed to axiomatized categories - no extra structure. $\endgroup$– Alex KruckmanJan 23, 2020 at 7:11
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$\begingroup$ I think my problem is that I have no context - or maybe the wrong contexts. I speak several languages, and I program in a handful, but although I can imagine some sort of mathematical language theory, I have never actually been exposed to any such thing. $\endgroup$– j4nd3r53nJan 23, 2020 at 8:32
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An elementary language is another term for a first-order theory. The word "elementary" used in this way is somewhat archaic, but it lives on in terms like elementary embedding (an embedding of one first-order theory into another), elementary topos (a first-order theory whose models are a certain kind of category) the Elementary Theory of the Category of Sets (a first-order theory roughly equivalent to ZFC), and elementary class (a term for the class of all models of a fixed first-order theory).
The author uses the phrase "pure sets" to mean that the basic objects of the theory are sets. By analogy, then, a "pure language of categories" means a theory where the basic objects are categories. The author goes on to describe exactly what he's talking about with two basic sorts of objects and morphisms. There are a few unary and binary operations added in, and some equalities enforced. This signature is precisely what's meant by an elementary language (first-order theory).
