# What is an 'elementary, pure language'?

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?

• Compare with Mathematical Logic: Language Jan 22, 2020 at 15:56
• 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. Jan 23, 2020 at 7:11
• 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. Jan 23, 2020 at 8:32