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I would like to pick out small objects from a category. I would like to find such a notion which

Dream 1. Picks out the schemes of finite type over $k$ from the category of $k$-schemes. Or at least picks out something relevant.

Dream 2. Picks out the top spaces homotopic to finite CW-complexes from the category of topological spaces (equipped with the Quillen model structure). (or something relevant)

Question: What notions exist for "small" objects (other than the compact objects)?

References or comments are welcomed.:) Thanks!

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    $\begingroup$ Too many questions ? Some ideas regarding 4. are here. $\endgroup$
    – Dietrich Burde
    Jan 2, 2014 at 19:52
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    $\begingroup$ The compact objects in the category of $k$-algebras are exactly the finitely presented $k$-algebras, but $k$-schemes are related to these contravariantly, so I think schemes of finite type is not the right conjecture (these should be the "cocompact" objects). $\endgroup$
    – Qiaochu Yuan
    Jan 2, 2014 at 20:13
  • $\begingroup$ Concerning 3, model structures are irrelevant for compactness, and the only compact topological spaces, if compact = $\aleph_0$-presentable, are the finite discrete spaces, if I remember well. $\endgroup$
    – Fernando Muro
    Jan 2, 2014 at 22:56
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    $\begingroup$ @Marci The derived hom might change, but "smallness" is usually defined in terms of the ordinary hom. $\endgroup$
    – Zhen Lin
    Jan 3, 2014 at 1:15
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    $\begingroup$ This might help. The first step in defining the Grothendieck group of a scheme is to restrict the objects to small ones (i.e. to coherent sheaves), otherwise the group will be trivial. Now, I would like to define the Grothendieck group of some categories. So the first step would be to define small objects in it. For example, for the category of k-schemes, I would like to obtain Dream 1. $\endgroup$
    – Marci
    Jan 3, 2014 at 17:43

2 Answers 2

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The subcategory of spaces equivalent to finite CW-complexes is the smallest one containing a point and closed under finite homotopy colimits. (In fact, this is the universal homotopy theory generated under finite hocolims by a single object).

Someone more patient than I could might fill in the details to make the following sentence both (i) sensical and (ii) correct: If $G$ is finite, the subcategory of $G$-spaces equivalent to a finite $G$-CW-complex is the smallest containing a point and closed under finite 'enriched' (weighted is probably the right word) homotopy colimits. (Here I'm enriched in $G$-spaces instead of spaces like above).

The category of schemes is probably too ugly of a place to try and make a similar statement, but perhaps one of the many colimit-populated enlargements would allow you to make the statement that some reasonable enlargement of the category of schemes of finite type over $k$ is generated by a point under some allowed collection of colimits.

(By the way, in the first two examples, honest compact objects are obtained by idempotent completion, i.e. adding retracts.)

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  • $\begingroup$ Thank you! My only problem is the following. Point in a general category should be either the initial object or the terminal object. In the category of $R$-mod or in the category of (co)chain complexes, however, it is the trivial object (0), and any colimit of that is the trivial object, so I won't get any interesting objects. $\endgroup$
    – Marci
    Jan 3, 2014 at 20:44
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    $\begingroup$ I think the fact that we had the initial/final object was incidental. That a point is a compact generator is what we're after. In the category of chain complexes of $R$-modules you'd start with $R$; closing under finite homotopy colimits would give finite complexes of free modules. Again, idempotent completing gives perfect complexes. $\endgroup$
    – Dylan Wilson
    Jan 3, 2014 at 20:54
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    $\begingroup$ "Point" can also mean "monoidal unit for some natural monoidal structure" (not necessarily the product or coproduct, as in the case of the terminal or initial object respectively). $\endgroup$
    – Qiaochu Yuan
    Jan 3, 2014 at 23:54
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I'm not really sure what you're asking, so this may not qualify as an answer. But it is too long to be a comment. When I see the word compact, I think of it as $\aleph_0$-small (a.k.a. $\aleph_0$-presentable). Of course, there is also the notion of $\lambda$-small (a.k.a. $\lambda$-presentable or $\lambda$-compact) for any ordinal $\lambda$. Is this what you're asking about? If so, I can give you a number of references to learn more about this.

One small point of confusion is that for many years people studying triangulated categories used the words compact and small interchangeably. With Amnon Neeman's book Triangulated Categories this seems to have changed. That book is a great source to learn about notions related to smallness, e.g. the notion of a $\lambda$-well-generated triangulated category.

I don't think the theory of small objects is intrinsically bound to that of model categories, but model category theorists have worked out much of the theory of smallness because they need it (e.g. for the small object argument). I learned everything I know about smallness from Mark Hovey's book Model Categories and I think the nLab article is very good also. For question 1, a model category theorist might consider the category of motivic spaces. This category is compactly generated, as is proven in the appendix to Po Hu's article S-modules in the category of schemes

Question 2 seems like it should be asked as a separate question, because I think it's doubtful that the answer will be very related to a general discussion of smallness. However, it does remind me to mention $\Delta$-generated spaces, which form a combinatorial model category Quillen equivalent to the usual model category of topological spaces (i.e. all objects are small). Dan Dugger has a survey about this topic, and I also like the treatment given in this paper by Fajstrup and Rosicky.

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  • $\begingroup$ There are also some facts about $\Delta$-generated spaces not in your references in Section 2 of my paper Homotopical interpretation of globular complex by multipointed d-space. $\endgroup$
    – Philippe Gaucher
    Jan 3, 2014 at 18:24
  • $\begingroup$ @PhilippeGaucher: Thanks for posting, and for including that section in your paper. I was not aware of it till now. $\endgroup$
    – David White
    Jan 3, 2014 at 19:21
  • $\begingroup$ Thank you! I would like to see some kind of a universal way to pick out "small" objects (=finitely presented or finitely built or things like that) from a category. $\endgroup$
    – Marci
    Jan 3, 2014 at 20:45
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    $\begingroup$ This question is about an intuitive notion of smallness rather than a technical notion, e.g. the intuition that schemes of finite types are the "small" objects in schemes. The OP wants a notion that makes this intuition precise. $\endgroup$
    – Qiaochu Yuan
    Jan 3, 2014 at 23:51
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    $\begingroup$ @DavidWhite It is rather unfair to say that the theory of smallness was developed by model categorists. There is the work of Grothendieck et al. [SGA4], Gabriel and Ulmer, Makkai and Paré, Adámek and Rosický... $\endgroup$
    – Zhen Lin
    Jan 4, 2014 at 1:36

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