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Thanks for the enhancement, it helps.
In the presentation of categories with a family of collections of morphisms, I notice that composition is referred to as a function. But as the collections of morphisms are collections, would this not make this statement invalid, as functions are specifically between sets? If I’m correct, it might make sense to refer to composition here as an ’operation’ (or perhaps ’assignment’?), rather than a ’function’.
I think if one wants to be hard-nosed about it, ’assignment’ is probably better. One could consult Eilenberg and Mac Lane’s original paper for their wording, to see what they use (they certainly defined categories in an elementary way).
Added: Their definition has (my emphasis):
Certain pairs of mappings … determine uniquely a product mapping …
and they later go on to prove from their definition (which I don’t recall reading in detail before! Or it was a long time ago…) that a mapping has a unique source and target and that pair of mappings have a product precisely when the usual source/target matching condition holds. I guess this is more like the definition of groupoid dating back to Brandt, and that operator algebraists (and other non-category theorists) still use.
Thanks for adding the open-access link to the original paper!
Interestingly, in the original paper, as you say, the phrase ’mapping’ is used. Later though, in ‘Categories for the Working Mathematician’, Mac Lane uses the phrase ’operation’ when defining (meta)categories, instead reserving the phrase ’mapping’ exclusively (as far as I can tell) for when sets are involved.
I realise that this discussion is somewhat based around pedantry, but I don’t think it hurts to have a little exploration of what terminology is used in literature. So I’ve done a little digging (16 books and a couple of papers), and found the following uses (I’m only going to consider cases where objects / morphisms are aggregated via ’collections’ (as the interest here is examining terminology apart from ’function’ and ’set’)):
Apologies for any inaccuracies in the references above!
I did notice that quite a few sources would opt to avoid any debate on terminology here, instead phrasing by “for morphisms $f \in \mathcal{C}(B,C)$ and $g \in \mathcal{C}(A,B)$, there exists the composite morphism $f \circ g \in \mathcal{C}(A,C)$…”, for example.
So the most prevalent uses (from my small sample) appear to be ‘operation’, or to avoid such terminology entirely. I’m going to indulge my pedantry and update the explanation on the page as such. Anyone may (as always, by nature of the medium) revert/edit if they disagree.
In such a formulation, “exists” really means “is given”, because this is not a pure existence statement. Perhaps this should be indicated, or the terminology adjusted.
I should clarify, for E&M, ’mapping’ is what we would now call a morphism or arrow, and ’product mapping’ just means the composite. The key word is ’determines’, so you can translate what they wrote as: “certain pairs of arrows $f$ and $g$ determine a composite arrow $fg$”.
Ah, thank you for the clarification. I misinterpreted ’product mapping’ as meaning an operation from pairs (the composition operation), as opposed to the composite itself.
Yes, it may be been possible they assumed that every category was concretisable (the counterexample due to Isbell was a few years away!), so “mapping” for an arbitrary abstract arrow makes sense, in that light.
added pointer to
for the definition “with a single set of morphisms”, i.e. as internal categories in Sets.
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