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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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    @alancalvitti: Have you tried to answer your question for constant functors?2012-05-18

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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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    Although not mentioned in the book, I think fully faithful functors also reflect limits.2015-07-04