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I have seen filtered object also used in settings where there may not be an $\emptyset$ (or initial object) and where the filtration can be doubly infinite. In the current entry both seem not to be allowed. Is this deliberate with another term being used for the more general concept?
just add a comment that there are varying evident ways to generalize or restrict the definition.
The filtering can be indexed in any ordered abelian group. This is the natural setup. Complete filtering can be defined only if the abelian group is ordered as a directed set. Filtered object can be filtered by inclusions or reverse inclusions (ascending and descending filtrations).
I have made some changes, but perhaps positive and negative grading need defining explicitly.
Of course, I wrote it wrong! A filtered algebra is not a filtered object in the category of algebras. Recall the basic $S_i \cdot S_j \subset S_{i+j}$. This needs a monoid object in the symmetric monoidal category of filtered vector spaces. We have the entry on $G$-graded vector spaces, along with the tensor product defined when $G$ is a monoid, but not yet for the filtered vector spaces. The latter are of course filtered objects in the category of vector spaces, but the monoidal product should be discussed somewhere.
The filtration in dual category is called the cofiltration. Further discussion on cofiltrations is moved here (to math phys philosophy section of nForum) as I have some research interest there.
I have added to filtered object the statement of the conditions “exhaustive filtration”, “Hausdorff filtration” and “complete filtration”, as in Boardman 99.
I also edited the definition slightly, making it a little more general (the previous version assumed that the filtration ends in $X_0$ and that $X = X_0$).
Also added the statement (here) that an abelian group may be reconstructed from the subquotients of an exhaustive and complete Hausdorff filtering by subgroups.
Couple little faults about the page: for some reason “def. 3” in Definition 3 refers back to Definition 2, and also the Hausdorff condition as stated doesn’t really make sense for abelian groups since an abelian group cannot be empty.
Thanks. I have fixed the doubled labels of the definition. Also I replaced
is [[initial object|empty]] ([[zero object|zero]] in the case of an [[abelian category]])
with
is [[initial object|initial]] ([[zero object|zero]] in the case of an [[abelian category]])
if you think that’s better.
I am going to remove the $\emptyset$ when I get a moment, since clearly that is the offender.
Hm, I always use $\emptyset$ as the symbol for the initial object.
Urs, I don’t know if it’s just me, but in my opinion that notation should not be used except in certain situations: initial objects are typically far from being empty, so it’s confusing or at least looks like a typo. I more often use $0$ as a generic symbol for the initial object.
Maybe I should try to make “certain situations” more precise. I’d think a necessary condition here would be that we are dealing with strict initial objects (as would be the case for say cartesian closed categories, or extensive categories).
If you have a concrete category $U: C \to Set$ (so $U$ is faithful) and if $I$ is initial with $U(I) \cong \emptyset$, then I think I’d be okay with writing $\emptyset$ in place of $I$. Let me prove that in this case $I$ is strict. If $e: X \to I$ is any map, then $U(X) = \emptyset$ since $\emptyset$ is strict initial in $Set$. Then it follows from faithfulness of $U$ that the only endomorphism $X \to X$ is the identity, so the composite $e: X \to I$ followed by $!: I \to X$ must be the identity, so $X \cong I$.
Similarly, if $U: C \to Set^J$ is a faithful functor into a power of $Set$, and $I$ is initial with $U(I) \cong 0$, then I think I’d be okay with writing $\emptyset$ in place of $I$. This applies for example to any Grothendieck topos, or any category $C$ with a separating family $c_i$ such that $\hom(c_i, I) = \emptyset$.
initial objects are typically far from being empty, so it’s confusing or at least looks like a typo. I more often use 0 as a generic symbol for the initial object.
Initial objects are also typically far from being zero objects. That’s how it goes with notation, it’s never perfect and rarely a worthwhile target for being dogmatic.
Maybe I should try to make “certain situations” more precise
I certainly never suggested that initial objects are necessarily “empty” in any sense. Just as you never suggested that initial objects are neceesarily zero objects in any sense.
There are plenty of $n$Lab pages where “$\emptyset$” is the notation used generically for the initial object. To start with, there is initial object.
Urs, can you tell me where else you have seen people use $\emptyset$ used as the generic symbol for the initial object where it is later instantiated to say abelian groups?
On the other hand, I am reasonably certain that people do use $0$ as a generic symbol, whether or not it is not a zero object.
The trouble as I see it with $\emptyset$ is that in cases where the objects are reasonably construed as sets with structure, $\emptyset$ in most people’s experience will suggest the empty set endowed with the at-most one structure, if one is there. So as I said, this will definitely look like a typo in certain cases.
There are plenty of $n$Lab pages where “$\emptyset$” is the notation used generically for the initial object. To start with, there is initial object.
Yeah, that’s because you changed it, for some reason. From $0$ I might add, which was introduced by Mike. Why did you feel an urge to change it?
I think that it is rather dangerous and nonstandard to say “filtered object” for a sequence which is not a sequence of subobjects. It is rather standard that all connecting morphisms of a filtration on an object are monomorphisms. Urs put the reference by Boardman which abides by that. Rognes in few sentences in his notes considered spectra and filtrations in the same statement so those couple were more general but elsewhere also has monomorphisms. All systematics treatises of filtrations per se I know (for example in algebra) take monomorphisms.
On the other hand, at present the entry has almost no categorical point of view which I would like to change as follows. The usual filtered objects are ascending as they correspond to functors from a filtered or directed category to a category such that every bonding morphism is monic. Exhaustive is the same as cocomplete. The descending case is either used in the case when you can invert the order (say if you use integers) so it is secretly ascending or on the contrary when you really want a functor from a cofiltered category but the values of the functor will be quotients at every level (you then see in the formulas things like $F_s/F_k$ or $X/F_k X$ and in fact one deals secretly not with sequence of ideals but associated inverse sequence of quotients. The fact that one deals with topology here is the fact that formal topologies come by taking limits of associated cofiltrations. Thus from the true $n$POV there are only ascending filtrations by subobjects and cofiltrations by quotient objects. Everything else reduces to those two dual cases clearly seen in the theory of filtered limits and cofiltered colimits: having monos or epis means that one has strict filtered objects or strict cofiltered objects. Noncomplete (nonexhaustive) cases are rather procedures: one takes an arbitrary object, a system of subobjects of it and this will make a filtration of the true colimit of that system which is the completion of the original one. This colimit and cofiltration have nothing to do with the original object except that they were constructed within it. By the universal property then one has a map from the original object into the completion, that is the colimit of the associated filtered functor. All the other lingo is just historical accident not very compatible with nPOV (including exhaustive which stays for cocomplete and which should be part of the definition).
I see arguments for often restricting to monomorphisms in the 1-categorical case, but the current version of the page does already say that the morphisms are “often required to be monomorphisms”. We could even change that to “usually”, with a pointer to the $\infty$-case where that condition definitely does have to be discarded. Or we could add monomorphy to the definition but remark afterwards that this can be omitted; I don’t feel strongly either way.
However, I do think we should keep the current bi-infinite definition of filtration. The discussion of subobjects vs quotient objects is interesting and important and should be included on the page, but I don’t think it’s strong enough that we shouldn’t even define the general notion of bi-infinite filtration, especially since the latter appears frequently in the literature.
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