objects such that commutes with certain colimits
An object of a category is called compact if it is “finite” or “small” in some precise sense. There are however different formalizations of this idea. Here discussed is the notion, usually going by this term, where an object is called compact if mapping out of it commutes with filtered colimits.
This means that if any other object is given as the colimit of a “suitably increasing” family of objects , then every morphism
out of the compact object into that colimit factors through one of the inclusions .
The notion of small object is essentially the same, with a bit more flexibility on when the family is taken to be “suitably increasing”. An important application of the above factorization property is accordingly named the small object argument. On the other hand, there is also the notion of finite object (in a topos) which, while closely related, is different. See also Subtleties and different meanings below.
is an isomorphism.
More generally, if is a regular cardinal, then an object such that commutes with -filtered colimits is called -compact, or -presented or -presentable. An object which is -compact for some regular is called a small object.
A -small colimit of -compact objects is again a -compact object.
by general properties of the hom functor. Now using that every is -compact and is -filtered this is
Since this (co)limit is taken in Set ,the -small limit over commutes with the -filtered colimit
We can take the limit again to a colimit in the first argument
which proves the claim.
In Set an object is compact precisely if it is a finite set. For this to hold constructively, filtered categories (appearing in the definition of filtered colimit) have to be understood as categories admitting cocones of every Bishop-finite diagram. (An object of Set is a Kuratowski-finite precisely if it is a finitely generated object, or equivalently if it is compact when regarded as a discrete topological space.)
For a topos, is compact if …
Let be a topological space and let be the category of open subsets of . Then an open subset is a compact object in precisely if it is a compact topological space. (It is not true that is a compact object of iff it is a compact topological space; see below.)
One has to be careful about the following variations of the theme of compactness.
(Some of these subtleties are resolved by noticing that there is a hierarchy of notions of compact objects that are secretly different but partly go by the same name. Some discussion of this is currently at compact topos, but more detailed discussion should eventually be somewhere…)
Here is an application of this concept to characterize which abelian categories are categories of modules of some ring:
Let be an abelian category. If has all small coproducts and has a compact projective generator, then for some ring . In fact, in this situation we can take where is any compact projective generator. Conversely, if , then has all small coproducts and is a compact projective generator.
This theorem, minus the explicit description of , can be found as Exercise F on page 103 of Peter Freyd’s book Abelian Categories. The first part of this theorem can also be found as Prop. 2.1.7. of Victor Ginzburg’s Lectures on noncommutative geometry. Conversely, it is easy to see that is a compact projective generator of .
Zoran: While Ginzburg’s reference is surely a worthy to look at, it would be better not to give false impression that this reconstruction theorem is due Ginzburg or at all new. It is rather a classical and well know fact probably from early 1960s, essentially small strengthening of a variant of a circle of abelian reconstruction theorems including the Gabriel-Popescu theorem(probably our variant could be read off from classical algera book by Faith for example, or Popescu’s book on abelian categories, in any case it is well known in noncommutative algebraic geometry). In fact for this fact, if I think better, the reconstruction belongs usually to expositions which treat classical Morita theory for rings.
A triangulated category is compactly generated if it is generated (see generator) by a set of compact objects.
Compact objects in the derived categories of quasicoherent sheaves over a scheme are called perfect complexes. Any compact object in the category of modules over a perfect ring is finitely generated as a module. Lurie uses -compact objects in the setup of -categories.
In non-additive contexts, the above definition is not right. For instance, with this definition a topological space would be compact iff it is connected. In general one should expect to instead preserve filtered colimits, as above.
Recall the above example of compact topological spaces. Notice that the statement which one might expect, that a topological space is compact if it is a compact object in Top, is not quite right in general.
Namely, the two-element set with the indiscrete topology is a compact space for which
For example, consider the sequence of spaces
where the open sets are of the form
(where ), plus the empty set. Define so that it sends a pair to itself if , and to . This defines a functor
The colimit of this sequence is the two-element set with the indiscrete topology. However, the identity map on this space does not factor through any of the canonical maps . It follows that the comparison map
is not surjective, and therefore not an isomorphism.
Todd (posted from n-category cafe): I don’t know if the story is any different for compact Hausdorff, but it could be worth considering.
But with a bit of care on the assumptions, similar results do hold:
If is compact, then preserves colimits of functors mapping out of limit ordinals, provided that the arrows of the cocone diagram,
are closed inclusions of spaces. (This applies for example to the sequence of inclusions of n-skeleta in a CW-complex. Taking , this has obvious desirable consequences for the functor .)
This example is discussed on page 50 of Hovey’s book.
Compact objects are discussed under the term “finitely presentable” or “finitely-presentable” objects for instance in
Kashiwara, Schapira, Categories and Sheaves, Definition 6.3.2;
For the pages quoted in the context of the discussion of compact objects in Top see
For the general definition with an eye towards the definition of compact object in an (infinity,1)-category see section A.1.1 section 5.3.4 of