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The word "local" in category theory does not seem to have a precise definition in itself but it often appears as part of other terminology. To my understanding, it is then used in the following ways, which AFAIU are quite unrelated:

  • A category is locally $P$ if all of its slice categories are $P$.

    Example: locally cartesian closed categories.

    This is applied to functors in two ways:

    • A functor $F : \mathcal C \to \mathcal D$ is locally $P$ if its restriction to slice categories $F/x : \mathcal C / x \to \mathcal D / Fx$ is $P$.

      Example: locally cartesian closed functor.

    • A functor $F : \mathcal C \to \mathcal D$ is locally $P$ if its restriction $F/\top : \mathcal C \to \mathcal D / F \top$ (where $\top$ is the terminal object) is $P$.

      Example: local right adjoint functor.

  • For (2-)categories and (2-)functors, locally $P$ means that $P$ holds on Hom-sets/categories.

    Examples: locally small, locally fully faithful, local adjunction, ...

  • Locally $P$ means that $P$ holds after localization/sheafification (or similar).

    Examples: localization, local epimorphism, local object, local equivalence, ...

Am I correct that these uses are fairly unrelated? If so, are there synonyms that I can use to disambiguate? The current situation makes it difficult to come up with good terminology for things. For example, I am looking for names for the following concepts:

  1. A functor $F : \mathcal C \to \mathcal D$ whose restriction $F/\top : \mathcal C \to \mathcal D / F \top$ is (a) full / (b) faithful / (c) fully faithful,
  2. A functor $F : \mathcal C \to \mathcal D$ whose restrictions to slice categories $F/x : \mathcal C / x \to \mathcal D / Fx$ are all (a) full / (b) faithful / (c) fully faithful.

I don't care that 1b is equivalent to 2b and 1c to 2c. What I'm looking for is a general way to come up with good names for concepts. Currently, the best names I can come up with are:

  1. (a) locally full / (b) locally faithful / (c) locally fully faithful,
  2. (a) locally full / (b) locally faithful / (c) locally fully faithful, (yes, that's the same).

Unfortunately, it turns out that locally fully faithful is already the term used for 2-functors that are fully faithful on Hom-categories. So I have 6 concepts, for which I can come up with only 3 terms which moreover are already taken.

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    $\begingroup$ I think there is no real question here. It is as it is, and yes the notions are not equivalent. "What I'm looking for is a general way to come up with good names for concepts." Every mathematician does and arrives at different naming conventions. It's a sad truth that even in category theory a full unified and consistently named theory is nothing more than a dream. It already fails with the definition of a category! $\endgroup$
    – Martin Brandenburg
    Commented Jan 24, 2023 at 17:46
  • $\begingroup$ What do you mean when you say "It already fails with the definition of a category"? $\endgroup$
    – Sridhar Ramesh
    Commented Jan 25, 2023 at 23:22
  • $\begingroup$ @Sridhar: perhaps that "category" could mean any of small category, locally small category, or not-necessarily-locally-small category depending on an author's conventions. Or maybe there are even other options, I don't know. $\endgroup$
    – Qiaochu Yuan
    Commented Jan 26, 2023 at 3:49
  • $\begingroup$ Yes. Numerous non-equivalent definitions of "category" appear in standard textbooks. For one this is because some authors restrict to locally small categories (including myself), others not, and also it's because of the choice of set-theoretic foundations (of which there are many). Also disjointness of hom-sets is an issue which not every author does consistently. When you think about it, this is really a mess. $\endgroup$
    – Martin Brandenburg
    Commented Jan 27, 2023 at 22:03

2 Answers 2

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The terminology situation is certainly unfortunate. There is an nLab page for locally, which lists the different usages of the term (it doesn't currently include those for "local" rather than "locally"). The page suggests "slice-wise" to disambiguate the first case, and "hom-wise" to disambiguate the second case.

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    $\begingroup$ The section titles are proposed solutions, as I wrote in my answer. Consensus on synonyms is driven by usage. I'm not aware of alternative proposals for alternative terminology, and the terminology proposed on the nLab page seems reasonable. $\endgroup$
    – varkor
    Commented Jan 24, 2023 at 16:30
  • $\begingroup$ (Deleted my comment with redundant question.) $\endgroup$
    – anuyts
    Commented Jan 24, 2023 at 16:42
  • $\begingroup$ I am currently inclined to use 1a $\top$-full, 1b $\top$-faithful, 1c $\top$-ff, and 2 slicewise full/faithful/ff. $\endgroup$
    – anuyts
    Commented Jan 26, 2023 at 22:08
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The English language has a vast vocabulary, most of which is irrelevant to mathematics. Unfortunately, we drive the few words that are directly relevant to exhaustion: local is one of them, another is cartesian.

My view is that we should try to employ more of the English vocabulary in mathematics, being careful to match words and meanings appropriately.

So in this case I propose laminated for notions that work according to slices.

(When I proposed prone and supine as replacements for the uses of cartesian in the theory of fibred categories, I got the wrath of Jean Bénabou.)

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    $\begingroup$ In the old days of soft-question-friendly MO, it would have been fun to make up a list of most-abused mathematical terminology. I propose ‘generic’. $\endgroup$
    – LSpice
    Commented Jan 25, 2023 at 23:25
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    $\begingroup$ We already have that! I asked it... dear god, 14 years ago: mathoverflow.net/questions/7389/… $\endgroup$
    – Qiaochu Yuan
    Commented Jan 26, 2023 at 3:50
  • $\begingroup$ Laminated makes an awkward adverb, but laminar sounds nice. Then 2 becomes laminarly full/faithful/ff. Any suggestions for when it only applies to the slice over $F\top$, like a local right adjoint? $\endgroup$
    – anuyts
    Commented Jan 26, 2023 at 22:02

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