Fix typo: it should be *initial* object of co-slice, not terminal.

Fix typo: it should be *initial* object of co-slice, not terminal.

Fix typo: it should be *initial* object of co-slice, not terminal.

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.

]]>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.)

]]>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 pointer to:

- Francis Borceux, Section 2.3 in Vol. 1:
*Basic Category Theory*of:*Handbook of Categorical Algebra*, Encyclopedia of Mathematics and its Applications**50**Cambridge University Press (1994) (doi:10.1017/CBO9780511525858)

Add “coterminator” to list of possible names.

]]>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?

]]>Add a bit more explanation to “initial = limit of 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 .

]]>I have cross-linked the discussion of initial objects as limits over identity functor here with the discussion at *adjoint functor* where this is put to use: there.

Also, I added pointers to MacLane, lest any reader gets the impression that this weren’t in the literature…

]]>I drew the obvious corollary, that an object is initial iff it is the limit of the identity functor.

]]>I added to initial object the theorem characterizing initial objects in terms of cones over the identity functor.

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