Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I created cylinder on a presheaf and will fill it in more as I read through Ast308. I plan on adding more stuff as I get to it (things about test categories and localisers, etc.).
This is similar but not the same as cylinder object, since even though it is specialized to presheaf categories, we don’t require any notion of a weak equivalence a priori.
Hmm.. Now that I think about it, perhaps it should be “cylinder for a presheaf” or something like that.
I took the liberty of changing instances of IX to $I X$ (if you don’t want your cylinder notation to look like the Roman numeral for nine, you should put a space between I and X).
We’ve had this discussion here at the nForum: it’s improper notation to write $f \coprod g$ for the universal mapping $X \coprod Y \to Z$ induced from the pair of maps $f: X \to Z$, $g: Y \to Z$, because it clashes with the meaning of $\coprod$ as a functor. For the same reason as that it’s not proper to write
$f \times g: Z \to X \times Y$(even if many people do!) but it is proper to write $\langle f, g \rangle: Z \to X \times Y$ and it is proper to write $f \times g: W \times Z \to X \times Y$. So you may want to consider an alternative.
The simplest thing might be to say: “let $h: X \coprod Y \to Z$ be the map universally determined by the pair of maps $f: X \to Z$, $g: Y \to Z$” and continue using $h$ as needed. (For what it’s worth, I often use $+$ for coproducts because $\coprod$ looks heavy to me, and I often use $(f, g): X + Y \to Z$ to denote this $h$, so that the notation $(f, g)$ is the coproduct counterpart to the product notation $\langle f, g\rangle$.)
I think that $(f,g)$ is actually pretty common for the product map $Z\to X\times Y$, since it generalizes the common notation for ordered pairs. I prefer $[f,g]$ for the coproduct map $X\sqcup Y \to Z$.
@Mike: it is indeed common – I don’t deny that. I was merely saying what I use.
I have no objection to $[f, g]$, but some people who reserve $[-, -]$ for hom-notation might not like it. To each his own – but I do remember discussion about notation like $f \coprod g$ or $f + g$ for what we’re discussing and how it’s not right.
@Harry As I understand the original motivation of this, a cylinder functor gives a notion of homotopy equivalence. The condition of weak equivalence is external and provides a way of comparing an externally given notion of weak equivalence with the notion of homotopy specified by the cylinder. One good point about cylinder objects if they are not required to be functorial is that they are nearly functorial (if I remember Ken Brown’s paper).
Another thing is that as yet no one has provided cylinder object as an entry. Have you any thoughts about what should go there?
Cylinder functors are stronger than cylinder objects, yes. We consider the presheaf category as a module for the strict monoidal category [Psh(A),Psh(A)] (with the product given by composition). Then a cylinder functor is a functor $\mathfrak{I}$ and three natural transformations assigning a cylinder object to every presheaf on $A$.
We say that two maps are elementarily $\mathfrak{I}$-homotopic if there exists a lift from $\partial I\otimes X:=\partial I(X)=X\coprod X$ (since the taking the coproduct with yourself is (endo-) functorial) to $IX\to Y$ (just as in the model category case). However, to get honest $\mathfrak{I}$-homotopy, we take the equivalence relation generated by elementary $\mathfrak{I}$-homotopy and take the quotient.
Now we have some axioms:
A presheaf category is said to have a donnée homotopique elementaire if it is equipped with a cylinder functor that
a.) commutes with all small colimits and monomorphisms
b.) The natural transformation $\partial^j_{(-)}:1_{Psh(A)}\to I$ sends arrows of Psh(A) to commutative squares in Psh(A) in the obvious way. We require that it sends all monomorphisms to cartesian squares. (I really don’t feel like drawing the diagram right now).
These give us a nicer homotopy relation (and sets us up via the adjoint functor theorem for the existence of a right adjoint). We now come up with the axioms for an anodyne structure compatible with a cylinder functor:
a.) All maps are generated as LLP(RLP(S)) where S is an honest set of monomorphisms.
b.) (I’m busy. I’ll type these out later)
c.) “
Note that all presheaves (on a small category) are accessible (|Arr(A/X)|-accessible), so we do not need to do anything like require that S admits the small object argument, since this is automatic (by a fairly straightforward lemma. It’s not immediate.).
Another thing is that as yet no one has provided cylinder object as an entry. Have you any thoughts about what should go there?
I added content to cylinder object.
Harry,
I think it would be good if the entry you write would make clear the distinction between the general notion of cylinder object and cylinder functor and the particular notion of a presheaf category with donnée homotopique elementaire ,
I think a good way to begin this entry would be to says:
a category of presheaves is said to have/to be xyz if it is equipped with cylinder objects/with a cylinder functor such that abc.
Another thing: is it intentionally that the labels in your diagram are themselves matrix entries and not small labels sitting on top/below or next to an arrow.
I think it would be good if the entry you write would make clear the distinction between the general notion of cylinder object and cylinder functor and the particular notion of a presheaf category with donnée homotopique elementaire ,
Yes, I realized that there was some inconsistent labeling on my part. I need to change the whole page name. I just haven’t gotten around to fixing it yet.
Another thing: is it intentionally that the labels in your diagram are themselves matrix entries and not small labels sitting on top/below or next to an arrow.
No. That’s because I don’t know how to use iTex.
Also, with respect to what you wrote about cylinder objects and cylinder functors, I think we should do the following.
Change the definition so that it says something like “in the context of model categories, one requires that the map $\sigma_X$: $I X\to X$ (or whatever you called it) is a weak equivalence. However, the notion of a cylinder object makes sense in general.”
That’s because I don’t know how to use iTex.
See the square diagram at template page (or at any other page with diagrams, for that matter :-)
“in the context of model categories, one requires
Well, not just there. I tried to make the difference between cylinders in cats with weak equivalences and in model categories. But you are right of course that one can also consider factorizations of the codiagonal without saying anything about weak equivalences. If that’s really a useful notion, we should add that to the page on cylinder objects.
That is the notion used in Ast308 if we require the left part of the factorization to be a “cellular cofibration”, i.e., monomorphic. The notion of a weak equivalence can be derived from “elementary homotopy data” (donnée homotopique elementaire) and a class of monomorphisms $S$ that generate a class of anodyne maps for the cylinder I or just a class of anodyne maps for the cylinder I. We take cofibrations to be all monomorphisms.
Note further that when I say anodyne morphism, I do not mean a weak cofibration. I’d rather not type out the details right now, but I’m thinking of renaming the page cylinder on a presheaf as “Notes on Astérisque 308”, where I’ll put down all of the definitions/state theorems, and fill in the parts of the proofs that I thought were confusing.
1 to 12 of 12