Compact object (mathematics)

In mathematics, compact objects, also referred to as finitely presented objects, or objects of finite presentation, are objects in a category satisfying a certain finiteness condition.
Definition.
An object "X" in a category "C" which admits all filtered colimits (also known as direct limits) is called compact if the functor
commutes with filtered colimits, i.e., if the natural map
is a bijection for any filtered system of objects formula_3 in "C". Since elements in the filtered colimit at the left are represented by maps formula_4, for some "i", the surjectivity of the above map amounts to requiring that a map formula_5 factors over some formula_3.
The terminology is motivated by an example arising from topology mentioned below. Several authors also use a terminology which is more closely related to algebraic categories: use the terminology "finitely presented object" instead of compact object. call these the "objects of finite presentation".
Compactness in ∞-categories.
The same definition also applies if "C" is an ∞-category, provided that the above set of morphisms gets replaced by the mapping space in "C" (and the filtered colimits are understood in the ∞-categorical sense, sometimes also referred to as filtered homotopy colimits).
Compactness in triangulated categories.
For a triangulated category "C" which admits all coproducts, defines an object to be compact if
commutes with coproducts. The relation of this notion and the above is as follows: suppose "C" arises as the homotopy category of a stable ∞-category admitting all filtered colimits. (This condition is widely satisfied, but not automatic.) Then an object in "C" is compact in Neeman's sense if and only if it is compact in the ∞-categorical sense. The reason is that in a stable ∞-category, formula_8 always commutes with finite colimits since these are limits. Then, one uses a presentation of filtered colimits as a coequalizer (which is a finite colimit) of an infinite coproduct.
Examples.
The compact objects in the category of sets are precisely the finite sets.
For a ring "R", the compact objects in the category of "R"-modules are precisely the finitely presented "R"-modules. In particular, if "R" is a field, then compact objects are finite-dimensional vector spaces.
Similar results hold for any category of algebraic structures given by operations on a set obeying equational laws. Such categories, called varieties, can be studied systematically using Lawvere theories. For any Lawvere theory "T", there is a category Mod("T") of models of "T", and the compact objects in Mod("T") are precisely the finitely presented models. For example: suppose "T" is the theory of groups. Then Mod("T") is the category of groups, and the compact objects in Mod("T") are the finitely presented groups.
The compact objects in the derived category formula_9 "R"-modules are precisely the perfect complexes.
Compact topological spaces are "not" the compact objects in the category of topological spaces. Instead these are precisely the finite sets endowed with the discrete topology. The link between compactness in topology and the above categorical notion of compactness is as follows: for a fixed topological space formula_10, there is the category formula_11 whose objects are the open subsets of formula_10 (and inclusions as morphisms). Then, formula_10 is a compact topological space if and only if formula_10 is compact as an object in formula_11.
If formula_16 is any category, the category of presheaves formula_17 (i.e., the category of functors from formula_18 to sets) has all colimits. The original category formula_16 is connected to formula_17 by the Yoneda embedding formula_21. For "any" object formula_10 of formula_16, formula_24 is a compact object (of formula_17).
In a similar vein, any category formula_16 can be regarded as a full subcategory of the category formula_27 of ind-objects in formula_16. Regarded as an object of this larger category, "any" object of formula_16 is compact. In fact, the compact objects of formula_27 are precisely the objects of formula_16 (or, more precisely, their images in formula_27).
Non-examples.
Derived category of sheaves of Abelian groups on a noncompact X.
In the unbounded derived category of sheaves of Abelian groups formula_33 for a non-compact topological space formula_10, it is generally not a compactly generated category. Some evidence for this can be found by considering an open cover formula_35 (which can never be refined to a finite subcover using the non-compactness of formula_10) and taking a mapformula_37for some formula_38. Then, for this map formula_39 to lift to an elementformula_40it would have to factor through some formula_41, which is not guaranteed. Proving this requires showing that any compact object has support in some compact subset of formula_10, and then showing this subset must be empty.
Derived category of quasi-coherent sheaves on an Artin stack.
For algebraic stacks formula_43 over positive characteristic, the unbounded derived category formula_44 of quasi-coherent sheaves is in general not compactly generated, even if formula_43 is quasi-compact and quasi-separated. In fact, for the algebraic stack formula_46, there are no compact objects other than the zero object. This observation can be generalized to the following theorem: if the stack formula_43 has a stabilizer group formula_48 such that
then the only compact object in formula_44 is the zero object. In particular, the category is not compactly generated.
This theorem applies, for example, to formula_54 by means of the embedding formula_55 sending a point formula_56 to the identity matrix plus formula_57 at the formula_58-th column in the first row.
Compactly generated categories.
In most categories, the condition of being compact is quite strong, so that most objects are not compact. A category formula_16 is compactly generated if any object can be expressed as a filtered colimit of compact objects in formula_16. For example, any vector space "V" is the filtered colimit of its finite-dimensional (i.e., compact) subspaces. Hence the category of vector spaces (over a fixed field) is compactly generated.
Categories which are compactly generated and also admit all colimits are called accessible categories.
Relation to dualizable objects.
For categories "C" with a well-behaved tensor product (more formally, "C" is required to be a monoidal category), there is another condition imposing some kind of finiteness, namely the condition that an object is "dualizable". If the monoidal unit in "C" is compact, then any dualizable object is compact as well. For example, "R" is compact as an "R"-module, so this observation can be applied. Indeed, in the category of "R"-modules the dualizable objects are the finitely presented projective modules, which are in particular compact. In the context of ∞-categories, dualizable and compact objects tend to be more closely linked, for example in the ∞-category of complexes of "R"-modules, compact and dualizable objects agree. This and more general example where dualizable and compact objects agree are discussed in .