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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeAug 2nd 2018

Add some discussion of the equivalence between the two definitions, and how in practice we usually use the family-of-collections-of-morphisms one.

• CommentRowNumber2.
• CommentAuthorjesuslop
• CommentTimeAug 2nd 2018

Thanks for the enhancement, it helps.

• CommentRowNumber3.
• CommentAuthorbgm
• CommentTimeJul 25th 2020

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’.

• CommentRowNumber4.
• CommentAuthorDavidRoberts
• CommentTimeJul 26th 2020
• (edited Jul 26th 2020)

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.

• CommentRowNumber5.
• CommentAuthorDavidRoberts
• CommentTimeJul 26th 2020

Added doi link to Eilenberg and Mac Lane’s article, which goes to the open access AMS-hosted version. The JSTOR version linked is not free to read!

• CommentRowNumber6.
• CommentAuthorbgm
• CommentTimeJul 26th 2020

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’)):

• ‘General theory of natural equivalences’ (Eilenberg and Mac Lane, 1945)
As you say, uses ‘mapping’ when defining categories
• ‘Categories for the Working Mathematician’ (Mac Lane, 1978)
Mac Lane makes a clear distinction between the general presentation of (meta)category (which uses operation), and a presentation of category in terms of sets
• ‘Basic Category Theory For Computer Scientists’ (Pierce, 1991)
Uses ‘operation’ when defining categories, though I believe Pierce specialises to sets in Remark 1.1.2
• ‘Category Theory’ (Awodey, 2010)
Uses ‘operation’ when referring to composition
• ‘An Introduction to Category Theory’ (Simmons, 2011)
Uses the phrase ’assignment’ for source/target/composition
• ‘Category Theory in Context’ (Riehl, 2016)
Uses ‘operation’ to refer to composition

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.

• CommentRowNumber7.
• CommentAuthorbgm
• CommentTimeJul 26th 2020

Rephrase definitions of a category in an attempt to avoid set-specific terminology.

• CommentRowNumber8.
• CommentAuthorDmitri Pavlov
• CommentTimeJul 26th 2020

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.

• CommentRowNumber9.
• CommentAuthorbgm
• CommentTimeJul 26th 2020

Rephrase to avoid ’existence’. E.g., “[a category consists of…] for every object x, a morphism id_x” rather than “[…] for every object x, there exists a morphism id_x”.

• CommentRowNumber10.
• CommentAuthorDavidRoberts
• CommentTimeJul 26th 2020

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 $fg$”.

• CommentRowNumber11.
• CommentAuthorbgm
• CommentTimeJul 27th 2020

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.

• CommentRowNumber12.
• CommentAuthorDavidRoberts
• CommentTimeJul 27th 2020

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.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeJun 18th 2021

added pointer to

• Alexander Grothendieck, Section 4 of: Techniques de construction et théorèmes d’existence en géométrie algébrique III: préschémas quotients, Séminaire Bourbaki: années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212, (numdam:SB_1960-1961__6__99_0, pdf)

for the definition “with a single set of morphisms”, i.e. as internal categories in Sets.

1. typo

julian rohrhuber

2. adding section on foundational issues, which is orthogonal to size issues.

Anonymous

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeAug 31st 2022

I have added to this paragraph (here) some more hyperlinks to technical terms, notably to mathematical foundations.

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