What does "data" mean?

Question

I see the word "data" from time to time in the context of category theory.
(e.g. Glossary of category theory "A category consists of the following data ...")
But I've never seen the definition of "data".

Question : What does "data" mean?

Thanks in advance!

Answer 1

Data means all the ingredients which constitute a category, as opposite to axioms, which describe the behavior of the given data. It is similar to when you find "Given a topological space $X$..." or "Given a function $f:A\to B$..." in a definition or a theorem. Data is what you have to consider to talk about a category, namely objects, arrows, a rule to compose two arrows together and an identity arrow for each object. This is what you need to consider to talk about categories, it is in a sense your input information. Axioms describe how this input is processed to have an actual category. For example, in an abstract group, data are the element of the group, I mean, the group itself regarded as just a set, and axioms are the axioms for a group. When the data satisfy the axioms, the set becomes a group. Similarly, when a class of objects and arrows, with a composition and identity morphisms, satisfy the axioms for a category, your data become a category.