Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality education elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory infinity integration integration-theory k-theory lie lie-theory limit limits linear linear-algebra locale localization logic mathematics measure-theory modal-logic model model-category-theory monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 4th 2010

    took the liberty of changing at ind-object the links that previously pointed to finitely presentable object directly to compact object.

    It would be nice if we could eventually expand on the query-box discussion at finitely presentable object, but currently there seems to be no point in directing to this entry instead of “compact object” if just commutitivity with colimits matters.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeMay 5th 2010

    Thanks for bringing up this issue; the pages compact object, finitely presentable object, and finitely generated object are currently a mess. I would suggest that finitely presentable object be changed into a redirect to the current page compact object, with whatever discussion there is to be had about the relationship between that and the more “concrete” algebraic notions (including that of “finitely presented object,” currently discussed at finitely generated object) placed somewhere on that page. Then the page finitely generated object can in turn focus on the analogous pair of notions (“generated” in the usual sense of algebra, and “generated” (“generable?”) in the categorical sense of preserving directed colimits of monomorphisms) and their relationship.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 5th 2010

    okay, I removed the former “finitely presented object”, made its name redirect to compact object and merged its content into compact object in the sections finitely presentable objects (mainly your, Mike’s, query box remark on what should be true) and References (the two references that do use these terms for compact objects).

    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeFeb 5th 2014
    • (edited Feb 5th 2014)

    I have one query about the ind-object entry. We say

    This is only vaguely motivated by the discussion given further up about some of the examples. Is there a more crucial well motivated reason for imposing this? Can we say why this is needed. As it is written it more or less looks like just a condition to get a nice description of the set of morphisms, and whilst that is fine, I am wondering if we cannot put something a bit more convincing here. (I have looked at SGA4 and its handling of IndCInd\!-\!C, but have got a bit confused over the motivations that it presents as well.)

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 5th 2014

    I can see Tim’s point. Might it be more convincing if the sections were rearranged, so that the description as filtered colimits of representables came first? One gets to the same direct description of the homs either way.

    There’s a general principle that for any small category CC, the free cocompletion is Set C opSet^{C^{op}} (this is the free cocompletion with respect to all colimits), and then to take the free cocompletion with respect to some class of colimits, one takes the closure of CC with respect to that class inside Set C opSet^{C^{op}}. This is covered in some paper by Max Kelly (possibly co-authored); would have to look it up. Then one can notice (by the Yoneda lemma) that the hom-functor Set C op(yc,)Set^{C^{op}}(y c, -), where y:CSet C opy: C \to Set^{C^{op}} is the Yoneda embedding, preserves all colimits. So whatever class of colimits one closes up under, homming out of ycy c will preserve those colimits.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeAug 29th 2016
    • (edited Aug 29th 2016)

    I have added statement of some more of the basic facts to the Properties-section at ind-object.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeAug 29th 2016

    I think that the statement introducing what is accessible category in the idea section of the ind-object is not entirely correct: it says that the category is accessible if it is (equivalent to) the ind-category of some small category. But it needs to be equivalent to κ\kappa-ind-category for some regular κ\kappa what is stronger, as Ind(C)Ind(C) is not necessarily equivalent to Ind κ(C)Ind_\kappa(C) for any κ\kappa. At least when by small filtered colimit one takes small in the sense of ZF and not in the sense of a Grothendieck universe. Right ?

    Another thing: do we have in nnLab somewhere treatment of the category of strict Ind-objects (colimits of small filtered diagrams where all morphisms are monic). For practice this is very imporant and it is much harder to find the statements in the literature. Moreover, the colimits in Ind s(C)Ind^s(C) there are often not inherited from Ind(C)Ind(C).

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeAug 29th 2016

    I created an entry strict ind-object with redirect strict pro-object, so far having only the definition. I believe there might be more scattered material somewhere in the nnLab, if so, please let me know.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeAug 30th 2016

    Thanks, Zoran. I have added to strict ind-object references, missing plural redirects, and cross-link from ind-object.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeAug 30th 2016
    • (edited Aug 30th 2016)

    I am trying to understand how to express Pro-morphisms between sequential diagrams (towers) in terms of components.

    Generally for cofiltered diagrams X:K 1𝒞X \colon K_1 \to \mathcal{C} and Y:K 2𝒞Y \colon K_2 \to \mathcal{C}, there are initial functors KK 1K \to K_1 and KK 2K \to K_2 such that Pro-morphisms between the two diagrams are represented by a system of KK-indexed morphisms (e.g. Kashiwara-Schapira 06, prop. 6.1.13).

    Now in the special case that K 1=K 2= K_1 = K_2 = \mathbb{N}_{\geq} are both the tower diagram, how may we improve this statement? May we assume that KK may be taken to be \simeq \mathbb{N}_{\geq}, too?

    In (Blanc 96, p. 6) it says that in this case every pro-morphism from XX to YY is represented by a sequence of components

    f s:X n sY s f_s \;\colon\; X_{n_s} \longrightarrow Y_s

    which are compatible in that – while the squares

    X n s+1 f s+1 Y s+1 X n s f s Y s \array{ X_{n_s+1} &\overset{f_{s+1}}{\longrightarrow}& Y_{s+1} \\ \downarrow && \downarrow^{} \\ X_{n_s} &\underset{f_{s}}{\longrightarrow}& Y_{s} }

    need not commute – there is m sn s+1,n sm_s \geq n_{s+1}, n_s such that the whiskering

    X m s X n s+1 f s+1 Y s+1 X n s f s Y s \array{ X_{m_s} \\ \downarrow \\ X_{n_s+1} &\overset{f_{s+1}}{\longrightarrow}& Y_{s+1} \\ \downarrow && \downarrow^{} \\ X_{n_s} &\underset{f_{s}}{\longrightarrow}& Y_{s} }

    does commute.

    I realize that this must be elementary. But right now I am confused as to why this is.

    But if true, I suppose I could choose a sequence of m sm_s as above, define inductively

    f˜ s+1:X m sX s+1f n s+1Y s+1 \tilde f_{s+1} \colon X_{m_s} \to X_{s+1} \overset{f_{n_s+1}}{\longrightarrow} Y_{s+1}

    to get commuting diagrams

    X m s f˜ s+1 Y s+1 X m s1 f˜ s Y s \array{ X_{m_s} &\overset{\tilde f_{s+1}}{\longrightarrow}& Y_{s+1} \\ \downarrow && \downarrow^{} \\ X_{m_{s-1}} &\underset{\tilde f_{s}}{\longrightarrow}& Y_{s} }

    and thus answer my above question on whether we may choose K= K = \mathbb{N}_{\geq} positively.

    Is this right?

    • CommentRowNumber11.
    • CommentAuthorTim_Porter
    • CommentTimeAug 30th 2016
    • (edited Aug 30th 2016)

    There are various methods. The important result is the reindexing lemma which can be found in Artin and Mazur LN 100. Here is a version in latex (I am rather rushed so will not attempt to do it in the correct form for here.) Let X:𝒞\xymatrix{ X\ar[r]^f\ar[d]_\cong&X:\mathcal{I}\to \mathcal{C}, and Y:𝒥𝒞Y :\mathcal{J}\to \mathcal{C} be two objects in Pro𝒞Pro\!-\!\mathcal{C}, and f:XYf: X\to Y be a morphism between them. There is a cofiltering category, = f\mathcal{H} = \mathcal{H}_f, with cofinal functors, φ:\varphi : \mathcal{H}\to \mathcal{I}, ψ:𝒥\psi : \mathcal{H}\to \mathcal{J}, and a natural transformation, f :XφYψf^\prime : X\varphi \to Y\psi such that the diagram (IN XYMATRIX IN THE SOURCE)

    commutes in Pro𝒞Pro\!-\!\mathcal{C}, (where the vertical arrows are the natural isomorphisms induced by the cofinal reindexing functors, φ\varphi and ψ\psi respectively).

    (This means that the KK you are looking at depends on the morphism you start with.)

    There are useful generalisations of this: see D. C. Isaksen, Calculating limits and colimits in pro-categories, Fundamenta Mathematicae, 175, (2002), 175 – 194.

    I have not checked through in detail what you have written but the only you seem to be missing is in using the description of colimits in Sets. ( I think this was given in detail in the book on Shape Theory that Cordier and I wrote, but the treatment there could be improved I think.) Other nice sources are Edwards and Hastings lecture notes.

    Sorry this is a bit hurried as I have to rush off soon.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeAug 30th 2016
    • (edited Aug 30th 2016)

    Tim, that’s the statement that I did point to. What I am asking is how much control we have over the re-indexing shape in the case of pro-morphisms between tower diagrams. In this case, may we always assume that \mathcal{H} (in your notation) is again the tower shape?

    I found another preprint that claims this :

    Assaf Libman, “Tower techniques for cofacial resolutions”. (pdf)

    (p. 4) Why is it true?

    • CommentRowNumber13.
    • CommentAuthorTim_Porter
    • CommentTimeAug 30th 2016

    If Zoran has not answered this for you by Thursday when I will be back home with better reference books, I think I can give you the answer. YES to ’may we always assume that ℋ\mathcal{H} (in your notation) is again the tower shape?’ but I need to double check one or two things out first. I am travelling at the moment.

    • CommentRowNumber14.
    • CommentAuthorzskoda
    • CommentTimeAug 30th 2016
    • (edited Aug 30th 2016)

    Urs, 10

    But right now I am confused as to why this is.

    This is triviality nothing to do with towers. The formula for Hom in Pro-category via lim colim is computed in Set. We know how lim looks there – as threads, and colim – as germs (equivalence classes), see what I wrote at in section via formal co-filtered limits. So it is straightfoward to just write out what the equivalence class as germs gives – the agreement at common restriction gives this wiskering.

    Now the morphisms of towers can indeed be replaced by level morphisms (this is true for more general inverse systems han towers). Now depending on the formalism, this is a different premorphism but the same morphism. See Theorem 3, page 12 of Sibe Mardešić, Jack Segal, Foundations of shape theory, North-Holland 1982.

    I personally do not like this way with making choices (coming originally I think from a Graeme Segal’s equivariant K-theory paper) and like staying with quantifiers instead – so for every kk there is at least one ss with morphism from X sX_s to Y kY_k and one looks at totality of all such rather than choosing the index function and having the strange notion of premorphism.

    • CommentRowNumber15.
    • CommentAuthorTim_Porter
    • CommentTimeApr 12th 2018
    • (edited Apr 12th 2018)

    I noticed some grey links with characters not showing. Both involved people with first name Jiří and who have entries in the Lab. There may be other cases of this where accents are not being registered correctly. The fix I did was to use redirects without accents. This problem has occurred (and was fixed I thought) before but perhaps a change in the system has negated what was done before.

    diff, v62, current

  1. Thanks for fixing, Tim! The changes I have made in the recent months should not have affected this. I will take a look when I get the chance (I suppose it is not the highest priority).

    • CommentRowNumber17.
    • CommentAuthorTim_Porter
    • CommentTimeApr 12th 2018

    There were other accents (e.g. on Cech) which fail to work in links. As you say it is not first priority.

    • CommentRowNumber18.
    • CommentAuthorRichard Williamson
    • CommentTimeApr 12th 2018
    • (edited Apr 12th 2018)

    Hi Tim, I took a look at this after all, and as far as I can see everything is actually working fine now (without my having done anything). You can see that in the latest edit at ind-object, I have written Jiří Adámek as-is, without there being any problem with the redirect. In this particular case, the ? in the source before you fixed it had been present ever since the reference was first added to the page back in 2014, so I suppose that there was some bug at that time which has since been fixed, but that the page had not been fixed until now.

    If you find any examples where things do not seem to be working even now, I’ll be happy to take a look.

    • CommentRowNumber19.
    • CommentAuthorTim_Porter
    • CommentTimeApr 12th 2018

    Thanks, Tim

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)