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    • CommentRowNumber1.
    • CommentAuthorTim_Porter
    • CommentTimeOct 27th 2010

    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.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeOct 27th 2010

    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.

    • CommentRowNumber3.
    • CommentAuthorTim_Porter
    • CommentTimeOct 28th 2010

    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!

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeOct 28th 2010
    • (edited Oct 28th 2010)

    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.

    • CommentRowNumber5.
    • CommentAuthorTim_Porter
    • CommentTimeJan 9th 2020

    I removed ‘Tholen’ from the list of references as his name was not associated with any further information. Does anyone know which of Walter Tholen’s papers was being refered to here?

    I also found out where the stray info just above that had strayed from and fixed it.

    diff, v50, current

  1. 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 𝒞\mathcal{C} and a full subcategory of any category with cofiltered limits and cofiltered colimits which admits a fully faithful functor from 𝒞\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?

    diff, v51, current

    • CommentRowNumber7.
    • CommentAuthorTim_Porter
    • CommentTimeApr 22nd 2020

    I made a reference to myself ’live’!

    diff, v56, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeApr 22nd 2020

    Tim, the link to your name didn’t work. You need to type

      [[Tim Porter|Porter]]
    

    instead of

      [[Porter]].
    

    I have fixed it.

    • CommentRowNumber9.
    • CommentAuthorTim_Porter
    • CommentTimeApr 22nd 2020

    Thanks. I suspected that might happen but then forgot to check back!

    • CommentRowNumber10.
    • CommentAuthorTim_Porter
    • CommentTimeApr 22nd 2020

    Added some more links

    diff, v59, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 22nd 2020

    I have added the intended link to pro-homotopy theory.

    diff, v60, current

    • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeApr 25th 2020

    Added the reference Tholen, Canadian J. Math. 1984.

    diff, v62, current

    • CommentRowNumber13.
    • CommentAuthorJem Lord
    • CommentTimeNov 5th 2020

    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 limHom A(Rd 1(Unknown characterUnknown characterUnknown character),Rd 2(Unknown characterUnknown characterUnknown character))Hom A(limRd 1(Unknown characterUnknown characterUnknown character),limRd 2(Unknown characterUnknown characterUnknown character))\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:ProFinSetSetlim : ProFinSet \to Set was fully faithful.

    diff, v65, current

    • CommentRowNumber14.
    • CommentAuthorGuest
    • CommentTimeJun 19th 2021

    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

    • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 19th 2021

    I don’t see any problem; the definition given seems to be the standard one.

    • CommentRowNumber16.
    • CommentAuthorGuest
    • CommentTimeJun 20th 2021
    The set of arrows from a pro-object F :
    • CommentRowNumber17.
    • CommentAuthorGuest
    • CommentTimeJun 20th 2021
    The set of arrows from a pro-object F:
    • CommentRowNumber18.
    • CommentAuthorGuest
    • CommentTimeJun 20th 2021

    The set of arrows from a pro-object F:

    • CommentRowNumber19.
    • CommentAuthorGuest
    • CommentTimeJun 20th 2021
    It seems certain unicode characters break something when posting as either Text or Markdown.

    The set of arrows from a pro-object F:D→C to a pro-object G:E→C is not the limit of the functor D^op×E→Set.

    Christian
    • CommentRowNumber20.
    • CommentAuthorRichard Williamson
    • CommentTimeJun 20th 2021
    • (edited Jun 20th 2021)

    I think the definition is correct. Remember that this a limit with D opD^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.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeJun 20th 2021

    It’s true that a limit over 𝒟 opF opSet op\mathcal{D}^{op} \overset{F^{op}}{\longrightarrow} Set^{op} is the colimit of 𝒟FSet\mathcal{D} \overset{F}{\longrightarrow} Set, but since Set opSet^{op} does not appear in the present context, that’s not quite applicable here.

    Consider the case that 𝒟=\mathcal{D} = \varnothing is the empty category and =*\mathcal{E} = \ast the terminal category. Then lim(lim𝒟(anything))\underset{\underset{\mathcal{E}}{\longleftarrow}}{lim}\big( \underset{\underset{\mathcal{D}}{\longrightarrow}}{lim}(anything)\big) \simeq \varnothing , while lim𝒟 op×(anything)*\underset{\underset{\mathcal{D}^{op} \times \mathcal{E}}{\longleftarrow}}{\lim}(anything) \simeq \ast

    • CommentRowNumber22.
    • CommentAuthorRichard Williamson
    • CommentTimeJun 20th 2021
    • (edited Jun 20th 2021)

    What we have here is the limit of the functor Hom(F(),G())Hom(F(-), G(-)) for some (covariant) functors F:DCF : D \rightarrow C and G:ECG : E \rightarrow C. This is the same as taking the limit of the functor colim dDHom(F(d),G())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).

    • CommentRowNumber23.
    • CommentAuthorGuest
    • CommentTimeJun 20th 2021

    The limit of a functor D op×ESetD^\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 opSetD^\op \to \Set would be the same as the colimit of the same functor.

    Christian

    • CommentRowNumber24.
    • CommentAuthorRichard Williamson
    • CommentTimeJun 20th 2021
    • (edited Jun 20th 2021)

    As I mentioned, the specific form of the functor in question is essential here. (Edit: removed something confusing.)

    • CommentRowNumber25.
    • CommentAuthorGuest
    • CommentTimeJun 20th 2021

    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:DEF : D \to E, there is generally no family of elements f X,YHom E(F(X),F(Y))f_{X,Y} \in \Hom_E(F(X), F(Y)) natural in XD opX \in D^\op and YDY \in D.

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeJun 20th 2021

    re #24: No equivalent re-writing of the hom-functor changes the fact that it is a functor with values in SetsSets. Which means that Guest’s argument in #23 applies, of which my concrete counter-example in #21 is a specialization.

    • CommentRowNumber27.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 20th 2021
    • (edited Jun 20th 2021)

    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(Set^C)^{op}, a bit more illuminating. One is taking the free completion w.r.t. just cofiltered limits of representables.

    • CommentRowNumber28.
    • CommentAuthorRichard Williamson
    • CommentTimeJun 21st 2021
    • (edited Jun 21st 2021)

    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.

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2021

    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!).

    diff, v71, current

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