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I edited the question in view of several helpful replies (thanks).

When we define ind-objects in a category, we use in general filtered diagrams in a category, not just sequences $A_1 \rightarrow A_2 \rightarrow A_3 \dots$ indexed by the integers. More generally, if one wants a "bigger" limit, we could look at diagrams indexed by an ordinal.

Is there a simple example of an ind-object that can't be indexed by an ordinal?

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    $\begingroup$ Positive integers with the divisibility order. $\endgroup$
    – David Roberts ♦
    Commented Jul 10, 2015 at 0:21
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    $\begingroup$ Please describe $\mathbf{C}$ as a direct limit of a countably-indexed directed system of finitely generated $\mathbf{Q}$-subalgebras. Lots of interesting "ind"-constructions cannot be expressed with a countable index set. $\endgroup$
    – grghxy
    Commented Jul 10, 2015 at 0:40
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    $\begingroup$ For countable filtered diagrams, you can always find a cofinal sequence, but for uncountable filtered diagrams you usually cannot. $\endgroup$
    – Eric Wofsey
    Commented Jul 10, 2015 at 0:48
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    $\begingroup$ It's like sequences versus nets. $\endgroup$
    – Benjamin Steinberg
    Commented Jul 10, 2015 at 0:58
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    $\begingroup$ There is also some discussion of this early in Adámek and Rosicky's book "Locally presentable and accessible categories" (the section on directed and filtered colimits). $\endgroup$
    – Tyler Lawson
    Commented Jul 10, 2015 at 5:59

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As far as I can tell, the idea to use filtered categories instead of just ordinal-indexed diagrams is due to Grothendieck. In fact, let me use this opportunity to advertise my favorite text on abstract category theory: Exposé I of SGA 4. It rocks.

Anyway: Proposition 8.1.6 (loc. cit.), which Grothendieck attributes to Deligne, says that every filtered category receives a cofinal map from an ordered set.

Grothendieck remarks that, while this result says that the two points of view on filtered objects (general or ordered) are essentially equivalent, filtered objects arise more naturally.

A nice example is Grothendieck's Theorem 8.3.3 on ind-representability. Aside from some set-theoretic issues, this basically boils down to the statement that exactness of a presheaf of sets $F$ on a category $C$ is equivalent to the category $C_{/F}$ being filtered.

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  • $\begingroup$ Thanks a lot. That's exactly what I was looking for. Perhaps this is folklore, but as an outsider trying to learn - it is quite strange that is not discussed in more references on the topic, which leave you just with the definition of filtered category and a sense of puzzlement. $\endgroup$
    – category_student
    Commented Jul 10, 2015 at 5:48
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    $\begingroup$ Note that ordinal-indexed diagrams (as opposed to directed set-indexed diagrams) are not sufficient to get all ind-objects. However, all filtered colimits can be computed as iterated ordinal-indexed colimits (assuming the latter exist). The point is that if you have something like the poset of finite subsets of an uncountable set, you can only turn it into an ordinal-indexed colimit if you are able to take intermediate colimits along the way (corresponding to initial segments of some well-ordering of your uncountable set). $\endgroup$
    – Eric Wofsey
    Commented Jul 10, 2015 at 6:32
  • $\begingroup$ @DylanWilson Do you read it in French or has someone translated it to English? $\endgroup$
    – David White
    Commented Jul 10, 2015 at 17:54
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    $\begingroup$ French, but math french isn't difficult $\endgroup$
    – Dylan Wilson
    Commented Jul 10, 2015 at 19:01

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