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Stream: learning: questions

Topic: confusion between two notations for arrows

Ignat Insarov (Sep 13 2026 at 04:20):

It is often written that $f: x → y$. Is this the same as $f ∈ \mathcal C [x, y]$? If so, can I choose one and use it exclusively? For example, can I say either $y: \mathcal C → (\mathcal C^{\text {op}} → \mathcal S)$ or $y: \text{Cat}[\mathcal C, \text{Cat}[\mathcal C^{\text {op}}, \mathcal S]]$ for the embedding of Yoneda?

Ignat Insarov (Sep 13 2026 at 04:23):

Perhaps it would even be enough to write $y: [\mathcal C, [\mathcal C^{\text {op}}, \mathcal S]]$, without naming the category the arrows are to be taken from? After all, we always know what category our objects belong to, so there is no ambiguity.

fosco (Sep 13 2026 at 08:37):

I have not seen a situation where confusion arises from mixing the two notations, but I would say there are cases where one may feel a tiny bit weird using the second. The feeling I have is that $f : x \to y$ is a meta-linguistic expression that doesn't require you to commit to a flavour or foundation, and can mean, for example, that "$f$", whatever it is, is a proof that "$x \le y$". At the same time, one could say that $f \in \mathcal C[x,y]$ because in a poset $(\mathcal C,\le)$ there is at most one element in each hom- set. The difference is almost imperceptible but I feel something has changed.

After all, we always know what category our objects belong to, so there is no ambiguity.

Sometimes you don't: finite sets, but only isomorphisms/injections/surjections; finite sets, with partial functions; finite sets, and all functions; finite sets, with all relations, not just the functional ones; these are quite different categories. The difference is so noticeable that in the early days Ehresmann pushed to call a category with the name of its arrows rather than with the name of it objects. So, one should say ${\bf pFun}$, ${\bf Inj}$, ${\bf Rel}$, ${\bf Cont}$ (topological spaces and continuous maps)...

Ignat Insarov (Sep 13 2026 at 13:30):

in the early days Ehresmann pushed to call a category with the name of its arrows

Good to know, do you have a reference for this?

fosco (Sep 13 2026 at 13:45):

Mac Lane acknowledges it in the historical section of chapter 1 of CWM:

fosco (Sep 13 2026 at 13:46):

fosco (Sep 13 2026 at 13:47):

I would say "Catégories et Structures" is not easy to find; I saw only one copy, not in good condition, in the university library back in 2010.

Ignat Insarov (Sep 13 2026 at 13:48):

After all, we always know what category our objects belong to, so there is no ambiguity.

Sometimes you don't: finite sets, but only isomorphisms/injections/surjections; finite sets, with partial functions; finite sets, and all functions; finite sets, with all relations, not just the functional ones; these are quite different categories.

I was thinking that we should think of objects as formal symbols marked with the category they belong to. Otherwise, we should get into all kinds of trouble. For example, we want to have a functor $\mathcal C [–, –]: \mathcal C^{\text {op}} × \mathcal C → \mathcal S$, and for that to happen we must have a function on objects that sends them to appropriate arrow sets. If objects are not labelled with the category they belong to, we cannot define that function.

Ignat Insarov (Sep 13 2026 at 14:11):

I would say "Catégories et Structures" is not easy to find; I saw only one copy, not in good condition, in the university library back in 2010.

How about Catégories structurées from back in 1963? https://www.numdam.org/item/10.24033/asens.1125.pdf It says, for instance:

une catégorie d'homomorphismes à produits finis, au-dessus d'une catégorie $\mathfrak M$ d'applications

This is «a category of homomorphisms with finite products on top of a category $\mathfrak M$ of functions».

John Baez (Sep 13 2026 at 16:03):

Anna's Archive, which of course is illegal, has Catégories et structures: extraits.

David Michael Roberts (Sep 14 2026 at 02:12):

@John Baez that article is available legitimately via https://eudml.org/doc/112200

John Baez (Sep 14 2026 at 09:25):

Good! I was looking to see if they had Catégories et structures, and saw only these "extracts".