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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeJul 1st 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI added to [[initial object]] the theorem characterizing initial objects in terms of cones over the identity functor.

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

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 1st 2017
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI drew the obvious corollary, that an object is initial iff it is the limit of the identity functor.

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

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 22nd 2018
   • (edited Jun 22nd 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have cross-linked the discussion of initial objects as limits over identity functor [here](https://ncatlab.org/nlab/show/initial+object#RelevanceForAdjointFunctorTheorem) with the discussion at _[[adjoint functor]]_ where this is put to use: [there](https://ncatlab.org/nlab/show/adjoint+functor#PointwiseExpression). Also, I added pointers to MacLane, lest any reader gets the impression that this weren't in the literature... <a href="https://ncatlab.org/nlab/revision/diff/initial+object/23">diff</a>, <a href="https://ncatlab.org/nlab/revision/initial+object/23">v23</a>, <a href="https://ncatlab.org/nlab/show/initial+object">current</a>

   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…

   diff, v23, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 22nd 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItexBy 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](https://ncatlab.org/nlab/revision/Sandbox/1528) .

   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 .

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeJun 23rd 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexAdd a bit more explanation to "initial = limit of identity functor" <a href="https://ncatlab.org/nlab/revision/diff/initial+object/24">diff</a>, <a href="https://ncatlab.org/nlab/revision/initial+object/24">v24</a>, <a href="https://ncatlab.org/nlab/show/initial+object">current</a>

   Add a bit more explanation to “initial = limit of identity functor”

   diff, v24, current

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeJun 23rd 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThe 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?

   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?

7. • CommentRowNumber7.
   • CommentAuthorvarkor
   • CommentTimeJan 8th 2019
   • PermaLink
   Author: varkor
   Format: MarkdownItexAdd "coterminator" to list of possible names. <a href="https://ncatlab.org/nlab/revision/diff/initial+object/28">diff</a>, <a href="https://ncatlab.org/nlab/revision/initial+object/28">v28</a>, <a href="https://ncatlab.org/nlab/show/initial+object">current</a>

   Add “coterminator” to list of possible names.

   diff, v28, current

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeSep 4th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded 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](https://doi.org/10.1017/CBO9780511525858)) <a href="https://ncatlab.org/nlab/revision/diff/initial+object/31">diff</a>, <a href="https://ncatlab.org/nlab/revision/initial+object/31">v31</a>, <a href="https://ncatlab.org/nlab/show/initial+object">current</a>

   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)

   diff, v31, current

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeOct 15th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexProdded by a comment [here](https://nforum.ncatlab.org/discussion/13006/natural-number/?Focus=96050#Comment_96050) I have clarified the wording ([here](https://ncatlab.org/nlab/show/initial+object#Strict)) 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]]*. <a href="https://ncatlab.org/nlab/revision/diff/initial+object/34">diff</a>, <a href="https://ncatlab.org/nlab/revision/initial+object/34">v34</a>, <a href="https://ncatlab.org/nlab/show/initial+object">current</a>

   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.

   diff, v34, current