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1. • CommentRowNumber1.
   • CommentAuthorPeter Heinig
   • CommentTimeJul 16th 2017
   • (edited Jul 16th 2017)
   • PermaLink
   Author: Peter Heinig
   Format: MarkdownItexAdded to [[global element]], which seems not to have had a Latest Changes-thread so far (hence this newly created one), a remark on a formalization of "name of a morphism" which I just stumbled upon and find a noteworthy thing. Perhaps this should go somewhere on the nLab, but to me [[global element]] seemed the most fitting place. I had always thought something like, "Well, if one really has to be careful and formal about the distinction between names or morphisms and morphisms per se, then the [[protocategories]] and [[protomorphisms]] in the sense of Freyd and Scedrov give one way to do so, and there is an introduction to this in the Elephant." I was surprised to find someone connecting this to internal homs, hence made this note in [[global element]]. If there are some "situating comments" on this that can conveniently be made, I would be happy to read them here.

   Added to global element, which seems not to have had a Latest Changes-thread so far (hence this newly created one), a remark on a formalization of “name of a morphism” which I just stumbled upon and find a noteworthy thing.

   Perhaps this should go somewhere on the nLab, but to me global element seemed the most fitting place.

   I had always thought something like, “Well, if one really has to be careful and formal about the distinction between names or morphisms and morphisms per se, then the protocategories and protomorphisms in the sense of Freyd and Scedrov give one way to do so, and there is an introduction to this in the Elephant.” I was surprised to find someone connecting this to internal homs, hence made this note in global element.

   If there are some “situating comments” on this that can conveniently be made, I would be happy to read them here.

2. • CommentRowNumber2.
   • CommentAuthorPeter Heinig
   • CommentTimeJul 16th 2017
   • (edited Jul 16th 2017)
   • PermaLink
   Author: Peter Heinig
   Format: MarkdownItexActually, I do not see the "point" of this particular usage of Baez', but find it some thing worthy of note and, perhaps, brief discussion. (There is also an evident issue of infinite regress, concerning the _name of the morphism which Baez calls the name of the morphism_, which I decided not to get into, at least now.)

   Actually, I do not see the “point” of this particular usage of Baez’, but find it some thing worthy of note and, perhaps, brief discussion.

   (There is also an evident issue of infinite regress, concerning the name of the morphism which Baez calls the name of the morphism, which I decided not to get into, at least now.)

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 16th 2017
   • (edited Jul 16th 2017)
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   Author: Todd_Trimble
   Format: MarkdownItexI entered some corrections. The point is that it is sometimes convenient to refer to the name of a morphism. For example, if you look at [[Lawvere's fixed point theorem]], you have to actually come up with a point $1 \to B$, and in order to do that, you have to *name* a certain morphism $A \to B$.

   I entered some corrections.

   The point is that it is sometimes convenient to refer to the name of a morphism. For example, if you look at Lawvere’s fixed point theorem, you have to actually come up with a point $1 \to B$, and in order to do that, you have to name a certain morphism $A \to B$.

4. • CommentRowNumber4.
   • CommentAuthorPeter Heinig
   • CommentTimeJul 17th 2017
   • (edited Jul 17th 2017)
   • PermaLink
   Author: Peter Heinig
   Format: MarkdownItexMany thanks for the corrections. I made a minuscule additionto [[global element]], explicitly...err...naming ? the naming-function. My initial rendition of the (handwritten) lecture notes (which do not make these systematizing attempts), using the ":=" only seemed not clear enough. I shied away from saying anything about what kind of function $\mathcal{C}(A,B)\rightarrow\mathcal{C}(I,[A,B])$ the function $\text{"}\cdot\text{"}$ should be, for reasons I do not know how to verbalize. Evidently, one would like to have $\text{"}\cdot\text{"}$ injective, so that no two morphisms get the same name. Moreover, currently, according to this point of view (which I find a valuable addition to the article already in its "discrete" version deplored in what follows), _naming morphisms_ is a _set-function between external hom-sets_. According to the philosophy that _verbs_ are _functors_, its being a mere function seems deplorable and it should become a _functor_ between enriched hom-sets. I do not see how to make _naming morphisms_ a functor in a meaningful way though.

   Many thanks for the corrections. I made a minuscule additionto global element, explicitly…err…naming ? the naming-function. My initial rendition of the (handwritten) lecture notes (which do not make these systematizing attempts), using the “:=” only seemed not clear enough.

   I shied away from saying anything about what kind of function

   $\mathcal{C}(A,B)\rightarrow\mathcal{C}(I,[A,B])$

   the function $\text{"}\cdot\text{"}$ should be, for reasons I do not know how to verbalize.

   Evidently, one would like to have $\text{"}\cdot\text{"}$ injective, so that no two morphisms get the same name.

   Moreover, currently, according to this point of view (which I find a valuable addition to the article already in its “discrete” version deplored in what follows), naming morphisms is a set-function between external hom-sets. According to the philosophy that verbs are functors, its being a mere function seems deplorable and it should become a functor between enriched hom-sets. I do not see how to make naming morphisms a functor in a meaningful way though.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 17th 2017
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexOkay, I expanded on this (and did some rearranging), giving the precise definition of the naming function, which is manifestly a natural bijection. (I suppose you won't find the idea of describing some verbs as *natural transformations* "deplorable" (-: ).

   Okay, I expanded on this (and did some rearranging), giving the precise definition of the naming function, which is manifestly a natural bijection. (I suppose you won’t find the idea of describing some verbs as natural transformations “deplorable” (-: ).

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeJul 6th 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexAdded more general definition that works for categories without a terminal object. <a href="https://ncatlab.org/nlab/revision/diff/global+element/19">diff</a>, <a href="https://ncatlab.org/nlab/revision/global+element/19">v19</a>, <a href="https://ncatlab.org/nlab/show/global+element">current</a>

   Added more general definition that works for categories without a terminal object.

   diff, v19, current