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 understood that the old terminology was ’projective system’, and ’projective limit’ refereed to the limit of a projective system. Can anyone confirm that? if I am right the present entry is slightly incorrect, but this needs checking first before changing it.
The meaning of words projective system and inverse system, projective limit and inverse limit really depends on author. Some limit such limits to (co)filtered case, some to directed case, some just mean small limits. I do not think there was ever a uniform terminology here. So nowdays we should not really distinguish projective from inverse limits in the generality. Who needs directed or (co)filtered case can specify.
That was not quite my point. The current entry derives pro-object from projective limit rather than projective system. Of course, projective was used as a synonym for inverse in this context…. and has far too many other uses as well!
Shakespeare’s Stratford upon Avon…Edit: comment in a wrong discussion. Sorry.
The treatment in pro-object anyway looks good and standard. I mean one can talk ladders and equivalences among ladders for morphisms but it is equivalent as talking about lim colim homs.
Edited this page quite substantially.
1) Added more structure to the page.
2) Gave more details on the definition of the category of pro-objects in a category, especially around the definition of the arrows.
3) Completely re-worked the explicit description of these arrows, as I found the old one too notation- and terminology heavy to be readable.
4) The main reason for my edit: I formulate and prove a proposition (obvious once formulated precisely) establishing an equivalence of categories between pro-objects in a category $\mathcal{C}$ and a full subcategory of any category with cofiltered limits and cofiltered colimits which admits a fully faithful functor from $\mathcal{C}$. As discussed in an example and remarks following the proposition, This recovers the usual equivalence between the definition of a profinite group as a pro-object in the category of finite groups and as a topological group obtained as a cofiltered limit of discrete topological groups, but also shows that one can replace topological spaces by many other categories here. This latter point, and the proposition itself, does not seem very well-known. Is that impression correct? If not, can anybody provide a reference?
Tim, the link to your name didn’t work. You need to type
[[Tim Porter|Porter]]
instead of
[[Porter]].
I have fixed it.
Thanks. I suspected that might happen but then forgot to check back!
I have added the intended link to pro-homotopy theory.
Fixed proposition 4.1 (by adding the assumption of cocompactness); the previous version of the proposition was false and depended on a non-existent natural isomorphism $\lim Hom_A(R \circ d_1(−),R \circ d_2(−)) \cong Hom_A(\lim R \circ d_1(−), \lim R \circ d_2(−))$ which allowed us to conclude, for instance, $lim : ProFinSet \to Set$ was fully faithful.
DefinitionCategoryOfProObjects is broken in the current version. It does not take any colimits. This was (introduced in v51)[https://ncatlab.org/nlab/revision/diff/pro-object/51]
Christian
I don’t see any problem; the definition given seems to be the standard one.
The set of arrows from a pro-object F:
I think the definition is correct. Remember that this a limit with $D^op$ in the source, the ’colimits’ that you are referring to come from there.
To write mathematics, one uses the Markdown+Itex option, and writes LaTeX within dollar signs as one usually would.
I missed it at the time, but I also don’t think there was anything wrong with the proposition that was changed in #13 except a typo (a missing ’cofiltered’ in one place I think), and in fact I prefer it as it was before, but I’ll leave it in its present form.
It’s true that a limit over $\mathcal{D}^{op} \overset{F^{op}}{\longrightarrow} Set^{op}$ is the colimit of $\mathcal{D} \overset{F}{\longrightarrow} Set$, but since $Set^{op}$ does not appear in the present context, that’s not quite applicable here.
Consider the case that $\mathcal{D} = \varnothing$ is the empty category and $\mathcal{E} = \ast$ the terminal category. Then $\underset{\underset{\mathcal{E}}{\longleftarrow}}{lim}\big( \underset{\underset{\mathcal{D}}{\longrightarrow}}{lim}(anything)\big) \simeq \varnothing$ , while $\underset{\underset{\mathcal{D}^{op} \times \mathcal{E}}{\longleftarrow}}{\lim}(anything) \simeq \ast$
What we have here is the limit of the functor $Hom(F(-), G(-))$ for some (covariant) functors $F : D \rightarrow C$ and $G : E \rightarrow C$. This is the same as taking the limit of the functor $colim_{d \in D}Hom(F(d), G(-))$ in the ’usual’ notation (which I’m not particularly fond of because it is imprecise, ignoring arrows).
The limit of a functor $D^\op \times E \to \Set$ is not the same as first taking the colimit in D and then the limit in E. If that was the case, then for E = 1 the limit of $D^\op \to \Set$ would be the same as the colimit of the same functor.
Christian
As I mentioned, the specific form of the functor in question is essential here. (Edit: removed something confusing.)
If what I said above does not convince you, maybe this does: with your definition, the stated identity morphisms do not make sense. For $F : D \to E$, there is generally no family of elements $f_{X,Y} \in \Hom_E(F(X), F(Y))$ natural in $X \in D^\op$ and $Y \in D$.
re #24: No equivalent re-writing of the hom-functor changes the fact that it is a functor with values in $Sets$. Which means that Guest’s argument in #23 applies, of which my concrete counter-example in #21 is a specialization.
Ah, you are quite right – thanks for bringing this to attention. (My eye glided right over item 2 to Remark 2.4, which is correct.) I actually find the alternative definition, where one computes homs in the free completion $(Set^C)^{op}$, a bit more illuminating. One is taking the free completion w.r.t. just cofiltered limits of representables.
Yes, thank you persevering, and apologies for getting my wires crossed here! I was mainly attempting to avoid the notation which mentions only objects, at the same time as avoiding something too ugly to write down succinctly, but obviously this cannot be at the price of correctness :-). Let’s fix it. I’m tied up myself at the moment, so if anyone else wishes to do it, please feel free! Otherwise I’ll get to it when I can.
Okay, I have re-written the definition in a new section (here).
I have kept the discussion in terms of spans (here) but haven’t looked through this. If this relied on the previous erroneous definition it may need attention.
In the course of this I removed the section “Alternative point of view via filtered limits of presheaves”. Because, first of all this is subsumed in the new beautified writeup I made, and second it’s not much of an “alternative” point of view anyways (this is Grothendieck’s original definition!).
1 to 29 of 29