Is it known who coined this term and what he meant? By comparison, the association between "full" and "surjective on $\mathrm{Hom}$" doesn't sound so cryptic. (I understand, of course, that the term can be random, in which case this will be the answer).
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asked Jul 1, 2023 at 8:06
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2$\begingroup$ I'm sorry, I just found the answer here: encyclopediaofmath.org/wiki/Faithful_functor.I looked in several places before asking the question (general search, wikipedia, nlab) $\endgroup$– Arshak AivazianCommented Jul 1, 2023 at 8:10
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$\begingroup$ It was the confusion in the translation into Russian (which is mentioned in the attached link) that caused my question. There are two different words in English "exact" and "faithfull". The first term is always translated into Russian as "точный". The second was translated earlier (for example, in MacLane) as "унивалентный" ("univalent" - which now causes completely wrong associations!), later they began to translate as "строгий" ("strict"). $\endgroup$– Arshak AivazianCommented Jul 1, 2023 at 8:31
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7$\begingroup$ How is faithful cryptic? It’s also used for representations and actions. $\endgroup$– Fernando MuroCommented Jul 1, 2023 at 8:57
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1$\begingroup$ In Russian, "faitfhull" for representation and action is translated by the same word as "exact" for an exact functor (in homological algebra). And in the English-language literature, I simply either did not encounter faitfhull actions, or did not pay attention. So I thought that "faitfhull" for "faitfhull functor" should refer to a different shade of the meaning of the word "faitfhull" (like loyal, devoted, or believer?) and it sounded weird to me. Regardless, I apologize again. $\endgroup$– Arshak AivazianCommented Jul 1, 2023 at 13:16
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1$\begingroup$ "Faithful" also has a related meaning like "accurate" or "without distortion," as in "faithful translation," which I think is similar to точный. $\endgroup$– Elizabeth HenningCommented Jul 1, 2023 at 20:47
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1 Answer
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The name derives from the representation theory of groups: a permutation (respectively, $R$-linear) representation of a group $G$ is faithful if and only if it is faithful when considered as a functor $G \to \mathrm{Set}$ (respectively $G\to \mathrm{Mod}_R$)