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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.
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.
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).
I have one query about the ind-object entry. We say
- the third by the assumption that each object is a compact object;
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 $Ind\!-\!C$, but have got a bit confused over the motivations that it presents as well.)
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 $C$, the free cocompletion is $Set^{C^{op}}$ (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 $C$ with respect to that class inside $Set^{C^{op}}$. 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 $Set^{C^{op}}(y c, -)$, where $y: C \to Set^{C^{op}}$ is the Yoneda embedding, preserves all colimits. So whatever class of colimits one closes up under, homming out of $y c$ will preserve those colimits.
I have added statement of some more of the basic facts to the Properties-section at ind-object.
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 $\kappa$-ind-category for some regular $\kappa$ what is stronger, as $Ind(C)$ is not necessarily equivalent to $Ind_\kappa(C)$ for any $\kappa$. 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 $n$Lab 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 $Ind^s(C)$ there are often not inherited from $Ind(C)$.
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 $n$Lab, if so, please let me know.
Thanks, Zoran. I have added to strict ind-object references, missing plural redirects, and cross-link from ind-object.
I am trying to understand how to express Pro-morphisms between sequential diagrams (towers) in terms of components.
Generally for cofiltered diagrams $X \colon K_1 \to \mathcal{C}$ and $Y \colon K_2 \to \mathcal{C}$, there are initial functors $K \to K_1$ and $K \to K_2$ such that Pro-morphisms between the two diagrams are represented by a system of $K$-indexed morphisms (e.g. Kashiwara-Schapira 06, prop. 6.1.13).
Now in the special case that $K_1 = K_2 = \mathbb{N}_{\geq}$ are both the tower diagram, how may we improve this statement? May we assume that $K$ 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 $X$ to $Y$ is represented by a sequence of components
$f_s \;\colon\; X_{n_s} \longrightarrow Y_s$which are compatible in that – while the squares
$\array{ X_{n_s+1} &\overset{f_{s+1}}{\longrightarrow}& Y_{s+1} \\ \downarrow && \downarrow^{} \\ X_{n_s} &\underset{f_{s}}{\longrightarrow}& Y_{s} }$need not commute – there is $m_s \geq n_{s+1}, n_s$ such that the whiskering
$\array{ X_{m_s} \\ \downarrow \\ X_{n_s+1} &\overset{f_{s+1}}{\longrightarrow}& Y_{s+1} \\ \downarrow && \downarrow^{} \\ X_{n_s} &\underset{f_{s}}{\longrightarrow}& Y_{s} }$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 $m_s$ as above, define inductively
$\tilde f_{s+1} \colon X_{m_s} \to X_{s+1} \overset{f_{n_s+1}}{\longrightarrow} Y_{s+1}$to get commuting diagrams
$\array{ X_{m_s} &\overset{\tilde f_{s+1}}{\longrightarrow}& Y_{s+1} \\ \downarrow && \downarrow^{} \\ X_{m_{s-1}} &\underset{\tilde f_{s}}{\longrightarrow}& Y_{s} }$and thus answer my above question on whether we may choose $K = \mathbb{N}_{\geq}$ positively.
Is this right?
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 $\xymatrix{ X\ar[r]^f\ar[d]_\cong&X:\mathcal{I}\to \mathcal{C}$, and $Y :\mathcal{J}\to \mathcal{C}$ be two objects in $Pro\!-\!\mathcal{C}$, and $f: X\to Y$ be a morphism between them. There is a cofiltering category, $\mathcal{H} = \mathcal{H}_f$, with cofinal functors, $\varphi : \mathcal{H}\to \mathcal{I}$, $\psi : \mathcal{H}\to \mathcal{J}$, and a natural transformation, $f^\prime : X\varphi \to Y\psi$ such that the diagram (IN XYMATRIX IN THE SOURCE)
commutes in $Pro\!-\!\mathcal{C}$, (where the vertical arrows are the natural isomorphisms induced by the cofinal reindexing functors, $\varphi$ and $\psi$ respectively).
(This means that the $K$ 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.
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?
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.
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 https://ncatlab.org/nlab/show/pro-object 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 $k$ there is at least one $s$ with morphism from $X_s$ to $Y_k$ and one looks at totality of all such rather than choosing the index function and having the strange notion of premorphism.
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.
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).
There were other accents (e.g. on Cech) which fail to work in links. As you say it is not first priority.
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.
Thanks, Tim
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