# Functors preserve isomorphisms, sections and retracts. Do they reflect these properties?

Functors preserve isomorphisms, sections and retracts. Which of these properties, if any, do functors reflect?

Are there other fundamental properties preserved and or reflected by functors?

Please specify covariant or contravariant if applicable.

Don't assume I understand any cat theory - all I know is triangles and functional languages.

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What kind of functors? You always have a functor to the category with one object and one morphism and in the vast majority of cases it doesn't reflect anything. – t.b. May 17 '12 at 3:49
@t.b., I meant any functor - and you provided a textbook worthy counterexample. But the situation seems awfully asymmetrical. Could you suggest an improvement to my Q, maybe conditions on domain or codomain cats such that any functor between them reflects one or more of those properties? – alancalvitti May 17 '12 at 15:20
@alancalvitti: A functor that reflects isomorphisms is called a ‘conservative functor’. A fully faithful functor is necessarily conservative, and also reflects sections and retracts. – Zhen Lin May 18 '12 at 8:02
I suggest you forget about contravariant functors. There is only one type of functors: the covariant one (so you do not need to specify what kind of functor you have). Whenever you see a paper with contravariant functor $F:\mathcal{C}\to\mathcal{D}$, think of it as a functor (ie. a covariant one) from $\mathcal{C}^{Op}$ to $\mathcal{D}$ – magma May 18 '12 at 9:36
@alancalvitti: Have you tried to answer your question for constant functors? – Martin Brandenburg May 18 '12 at 10:21

You can find a table of many preserved and reflected properties of functors in wonderful "The Joy of Cats" (JOC) book which you can find online at:

http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf

In general:

• functors need not reflect isomorphisms (JOC 3.22)
• functors need not reflect sections (JOC 7.23)
• functors need not reflect retracts (JOC 7.29)

However: full and faithful functors reflect isomorphisms, sections and retracts (JOC 7.30)

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Thanks magma - would like to see a DB or theorem-proving system that automatically checks all such properties – alancalvitti Aug 15 '12 at 3:06
Although not mentioned in the book, I think fully faithful functors also reflect limits. – Stefan Hamcke Jul 4 '15 at 16:51