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Consider a functor $F:C\to D$ between two categories $C$ and $D$. Suppose $F$ satisfies the following property: for any $a, b\in C$, $F(a)\cong F(b)\iff a\cong b$.

Of course, $a\cong b\Rightarrow F(a)\cong F(b)$, so it is the other direction tricky.

The question is then: is there a name for such functors? Have they been studied?

I know that there are conservative functors, which reflects isomorphisms. However, the property of $F$ mentioned above only reflects "isomorphism relations." This question was asked here before, but I don't see a satisfactory answer.

Thanks for the coming help!

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    $\begingroup$ Just to highlight the difference between conservative functor and the notion you're talking about, if we let $\text{Set}_{n}$ be the full subcategory of $\text{Set}$ whose objects have exactly $n$ elements, then every functor $\text{Set}_{n}\to D$ for any category $D$ has the property you're describing, but will usually not be conservative. (And of course there's nothing special about sets here--the same holds for any $C$ and full subcategory on a given isomorphism class.) $\endgroup$
    – Matt Feller
    Commented Feb 20, 2020 at 16:27

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Some authors name this "isomorphism reflecting", for example in Noncommutative rings and their applications (p. 153) and Models, Modules and Abelian Groups: In Memory of A. L. S. Corner (p.480). But it is kind of dangerous to use this since some (maybe even more!) authors also mean "conservative" by "isomorphism reflecting", for example in Basic concepts of enriched category theory (p.8) and Homology and homotopy in semi-abelian categories (p.9), also the nlab.

I don't know if there is an English term for the isomorphism relation, in German it's just "Isomorphie" which could be translated to "isomorphy". Then the property should be called "isomorphy reflecting".

Of course, you could always just say "injective on isomorphism classes".

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At several places in the book "Category Theory in context", E. Riehl calls this "$F$ creates isomorphisms" (in analogy to the more common term "$F$ creates (co)limits").

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  • $\begingroup$ I am not sure if this analogy fits well. $F$ creates isomorphisms should mean that if $F(f)$ is inverse to $g$, then there is a unique morphism $f'$ with $g=F(f')$ such that $f$ is inverse to $f'$, right? In other words, $F$ is conservative. $\endgroup$
    – Martin Brandenburg
    Commented Feb 21, 2020 at 8:08
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    $\begingroup$ @MartinBrandenburg I guess it depends on the definition of creating limits, there seems to be some debate around that as well: mathoverflow.net/questions/103065/… If you use Def. 1 in the above link, then I agree with what you are saying. If you use Def. 2, then it fits better this version (then the statement in both cases is "if it exists in the target, then it exists in the domain, and $F$ preserves it"). $\endgroup$
    – Pavel Čoupek
    Commented Feb 21, 2020 at 14:50
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    $\begingroup$ However, it seems to be the case that either way, "creating isomorphisms" should also include reflecting them, so there is (should be) some difference between this notion and "injectivity on isom. classes." $\endgroup$
    – Pavel Čoupek
    Commented Feb 21, 2020 at 14:55
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One paper about this is Elliott's Towards a theory of classification, where such functors are called classification functors, and there is a nice collection of examples.

(Though note that a category theorist would perhaps approach some of the material in that paper a bit differently, e.g. by considering inner automorphisms as 2-morphisms. And Elliott's claim that every category is concrete is wrong for size reasons.)

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