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for the Café-discussion I added to zero object the details of the proof that in a $Set_*$-enriched category every terminal or initial object is zero.
In the course of this I did a bit of brushing-up of a bunch of related entries. For instance at pointed set I made the closed monoidal structure on $Set_*$ manifest, etc.
In entry kernel, the kernel is defined as a pullback from the zero object, and it says "This definition actually makes sense in any category with an initial object, giving a more general notion". Why leaving this to the remark and not to define from the very beginning in the minimal, nonsymmetric setup ? I understand not doing the most general definition when the general one is more difficult, but the more general definition here is less difficult because the noton of initial object is yet simpler then of zero object. I prefer that the definition of the kernel is the one which is general, rather than leaving for a remark. Should I change it ?
I thought the same. Please change it.
OK, I changed it and added a new paragraph discussing the representing object definition. One has to be careful with Coker though. I am not sure how much that definition could extend beyond the abelian case where it is often used (I learned it years ago from Gelfand-Manin Methods...book, where a counterexample against representability of naive coker functor is given as well).
Okay, thanks. We should eventually add more discussion on how and when these different definitions are equivalent.
I added the proposition that the kernel of a kernel is 0 in 1-category theory. And that this crucially is no longer true in higher category theory.
I hope this doesn’t sound offensive, but I’m surprised that someone familiar with category theory would even think of asking whether coker represents the functor $Z\mapsto coker(hom(Z,X) \to hom(Z,Y))$. That seems to me like asking whether the coproduct $A\sqcup B$ represents the functor $Z\mapsto hom(Z,A) \sqcup hom(Z,B)$. Both are a colimit construction, so “obviously” the question to ask is about the functor they corepresent. Am I missing something?
Thank you Mike of emphasising the general principles; one learns them in general case, while somebody else in a particular case like this one. Indeed, one can always restate any colimit construction via corepresentation of some functor. But $coker f$, for $f:X\to Y$, is by no means corepresenting $Z\mapsto coker(hom(Z,X) \to hom(Z,Y))$ (this is a contravariant functor, so only representing it makes sense, though it is a wrong idea as we both pointed out). Instead one should corepresent the covariant functor $Z\mapsto ker(hom(Y,Z) \to hom(X,Z))$ involving $ker$ of a morphism of abelian groups.
I mean if a category $A$ is enriched in $Ab$, its opposite category is also enriched in $Ab$, not $Ab^{op}$, so one wants to represent or corepresent by some functors into $Ab$. For $f:X\to Y$ the $ker f$ is represented by a $ker$-construction in $Ab$, while, $coker$ is corepresented by another $ker$ construction in $Ab$ ! Of course, one can view that $ker$-construction in $Ab$ as a $coker$ construction in $Ab^{op}$ bit it is strange to talk about $Ab^{op}$ while we have enrichment of opposite category also in $Ab$.
When I said “the functor they corepresent,” I assumed it would be obvious that I was thinking of the functor $Z \mapsto ker(hom(Y,Z) \to hom(X,Z))$, since as you say that is, in fact, the functor which coker corepresents.
For f:X→Y the ker f is represented by a ker-construction in Ab, while, coker is corepresented by another ker construction in Ab !
Of course. Is that surprising? The product $X\times Y$ in any category is also represented by a product-construction in Set, while the coproduct $X\sqcup Y$ is corepresented by another product-construction in Set.
Again: somebody learns that in general case, somebody first in the case of products, somebody first in the case of kernels. I learned it years ago from Gelfand-Manin with the case of kernels first. To me it was surprising :)
By the way how does one say shortly a quotient object in the category of functors ? I mean a subobject in that category is a subfunctor, what about a quotient object. Sintagm quotient functor is used in many other senses, for example in localization theory, for certain class of localization functors.
@9, I see. Yes, I see how that could be surprising to someone who hasn’t encountered any such idea before.
@10, In a context where other things called “quotient functors” are not appearing, I would just say “quotient functor” with a remark at first usage reminding the reader that this phrase is also used in other ways. In a context where there is danger of confusion, I don’t know. You could use a different name for your functors, like diagram or presheaf and talk about quotient diagrams or quotient presheaves.
But what about an nlab entry on this topic. We do have subfunctor, while we do hesitate to have an entry for quotient functor. Maybe to write an intro on disambiguation, make passages about each of the several competent ones ? What is the wisdom of the foundations people ? I’d like to address this ones soon.
I agree with Zoran #9. It’s not surprising any more; indeed, the converse would now surprise me, since I see moving to the representing functor as a reduction to logic, and limits in $Set$ are logically more fundamental than colimits in $Set$. (Not so much in $Ab$, however, so I still want to understand this example better, although it doesn’t surprise me.)
I think that a page quotient functor with a discussion of the uses of that term would be a very good idea.
Sure, quotient functor sounds fine to me.
I have touched zero object: worked on the formatting, moved some of the paragraphs to different subsections, added basic examples.
Looking back on a discussion about two years old, I see there is little or nothing on “quotient functor” at quotient category (to which quotient functor redirects) that considers the concept dual to subfunctor; all I see is discussion of localizations. (Yaron makes a similar pertinent comment in a query box.) It would be good to clear this up.
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