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• CommentRowNumber1.
• CommentAuthorzskoda
• CommentTimeMay 15th 2011
• (edited May 15th 2011)

New entry representable morphism, in the sense of Grothendieck school. The notion is used at closed immersion of schemes where I just made some changes.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMay 16th 2011

I have added a link to the existing representable morphism of stacks

• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeMay 16th 2011

Oh thanks, I forgot of its existence. It will need more work though… :=)

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeOct 5th 2012

I clarified representable morphism as suggested by a query, and removed the query.

• CommentRowNumber5.
• CommentAuthorAli Caglayan
• CommentTimeDec 21st 2018

Are there any category theoretic references on this notion? Everything I’ve found has been geometry related.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeDec 22nd 2018

Well, there’s Natural models of type theory

• CommentRowNumber7.
• CommentAuthorAli Caglayan
• CommentTimeDec 22nd 2018

Yes thats what I’ve been reading, I haven’t made this clear. It seems strange to me no category theorists have studied this notion since Grothendieck. In fact Awodey is the only writing I could find that uses it for reasons other than geometry.

• CommentRowNumber8.
• CommentAuthorGuest
• CommentTimeJun 11th 2019
The sentence "The main role of properties of spaces have to be done in a relative setup" doesn't really mean anything.
• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeJun 11th 2019

I believe I know what the intended meaning was, but I certainly agree that it was not very clear. I tried to clarify it.

• CommentRowNumber10.
• CommentAuthorMike Shulman
• CommentTimeJun 11th 2019

By the way, regarding #7, I believe representable morphisms (of $Cat$-valued (pre)sheaves, rather than sheaves of sets) are implicitly present in the discussion of “comprehension schemes” in section B1.3 of Sketches of an Elephant. But it’s some work to extract them.