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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.
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.)
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 $A \to B$.
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.
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” (-: ).
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