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I added to initial object the theorem characterizing initial objects in terms of cones over the identity functor.
I drew the obvious corollary, that an object is initial iff it is the limit of the identity functor.
By the way, if I were to decide, I would replace the writeup of the proof “initial object is limit over identity functor” in the entry by what is now in Sandbox 1528 .
The existing proof was admittedly a bit telegraphic in places; I added a bit more explanation that certain triangles commute by the cone condition. But in general I like the more concise version. However, the longer version could be on a breakout subpage?
added pointer to:
Prodded by a comment here I have clarified the wording (here) regarding strict initial objects.
While I was at it I have added a bunch of Definition/Remark environments and re-arranged somewhat, for more systematics.
The remaining examples of strict initial objects I have moved to strict initial object.
Added example to show that the status of being an initial object is dependent on the category considered. Also added the dual version of the example from terminal object about the identity morphism in over categories.
Added example to show that the status of being the initial object is dependent on the category considered. Also added the dual version of the example from terminal object about the identity morphism in over categories.
(I would also like to know: Is there a reason, no extra page Field for the category of fields exists, but only a section on fields? So far, I have already dead-linked it twice, on initial object and on determinant. I certainly would like to create it, and have already written a bit for it, but first want to make sure that there isn’t a reason it doesn’t exist yet.)
(For some reason unknown to me, correcting some errors merely a minute later than the original edit was registered as a new one in the logs. Edit: I just noticed that the original comment is posted here twice, maybe it was automatically copied into the box again.)
Please do create an entry Field on the category of fields, if you have the energy!
The only reason it does not exist yet is that nobody yet got around to doing it (as with so many basic tasks on the nLab).
Regarding the apparent glitch in the logs: I can’t readily tell what happened.
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