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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeJul 17th 2014
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   Author: Mike Shulman
   Format: MarkdownItexIt's well-known that a set-valued functor is representable iff its category of elements has an initial object. Today I noticed (I think) that a set-valued functor is a *retract* of a representable iff its category of elements admits a cone over the identity functor, i.e. there is an object $e$ and a natural transformation $c:const_e \to Id$. (Additionally asking that $c_e = 1_e$ would make $e$ initial.) Is that true? If so, is it written down anywhere?

   It’s well-known that a set-valued functor is representable iff its category of elements has an initial object. Today I noticed (I think) that a set-valued functor is a retract of a representable iff its category of elements admits a cone over the identity functor, i.e. there is an object $e$ and a natural transformation $c:const_e \to Id$. (Additionally asking that $c_e = 1_e$ would make $e$ initial.) Is that true? If so, is it written down anywhere?

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeJul 17th 2014
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   Author: Zhen Lin
   Format: MarkdownItexI've not seen this before. It seems to be a variation on Freyd's initial object lemma / the general adjoint functor theorem.

   I’ve not seen this before. It seems to be a variation on Freyd’s initial object lemma / the general adjoint functor theorem.