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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 9th 2009
   • (edited Nov 9th 2009)
   • PermaLink
   Author: Urs
   Format: MarkdownI created an entry called [[(infinity,1)-Yoneda extension]]. Currently the point of this entry is to present one specific _presentation_ by model category means of what should more abstractly be an (oo,1)-version of the standard [[Yoneda extension]]. The statement is (supposed to be) a simple consequence of the proposition recalled at [[Quillen bifunctor]]. I started preparing this entry on my personal web, but then thought that this kind of material should be on the main nLab. Let me know if you disagree. I put a standout-box cautioning the reader that this is stuff I dreamed up. But even in as far as the statement so far given is right, I would like the entry to be understood as something in search of a bigger and more abstract picture.

   I created an entry called (infinity,1)-Yoneda extension.

   Currently the point of this entry is to present one specific presentation by model category means of what should more abstractly be an (oo,1)-version of the standard Yoneda extension. The statement is (supposed to be) a simple consequence of the proposition recalled at Quillen bifunctor.

   I started preparing this entry on my personal web, but then thought that this kind of material should be on the main nLab. Let me know if you disagree.

   I put a standout-box cautioning the reader that this is stuff I dreamed up. But even in as far as the statement so far given is right, I would like the entry to be understood as something in search of a bigger and more abstract picture.

2. • CommentRowNumber2.
   • CommentAuthorspitters
   • CommentTimeDec 15th 2017
   • PermaLink
   Author: spitters
   Format: MarkdownItexThis is about the ordinary [[Yoneda extension]]. It looks like there is a typo on that page, under Remarks. The precomposition functor described there as the _direct image_, is in fact the _inverse image_ of a geometric morphism. See Elephant A4.1.4. However, the page [[restriction and extension of sheaves]] refers to this as _direct image_ too. So, I may be overlooking something. In any case, this seems to be worth clarifying.

   This is about the ordinary Yoneda extension. It looks like there is a typo on that page, under Remarks. The precomposition functor described there as the direct image, is in fact the inverse image of a geometric morphism. See Elephant A4.1.4. However, the page restriction and extension of sheaves refers to this as direct image too. So, I may be overlooking something. In any case, this seems to be worth clarifying.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeDec 16th 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexPrecomposition with an arbitrary functor is indeed always an inverse image functor between presheaf toposes. But the page [[restriction and extension of sheaves]] is talking about functors that are [[morphisms of sites]], in which case precomposition is also a *direct* image functor between *sheaf* toposes. When the topologies are trivial, being a morphism of sites just means being flat, so that left Kan extension is left exact and hence an inverse image functor. However, I don't think it's appropriate to carry this terminology over to [[Yoneda extension]], so I've removed it.

   Precomposition with an arbitrary functor is indeed always an inverse image functor between presheaf toposes. But the page restriction and extension of sheaves is talking about functors that are morphisms of sites, in which case precomposition is also a direct image functor between sheaf toposes. When the topologies are trivial, being a morphism of sites just means being flat, so that left Kan extension is left exact and hence an inverse image functor.

   However, I don’t think it’s appropriate to carry this terminology over to Yoneda extension, so I’ve removed it.