# Compact object (mathematics)

__: Mathematical concept__

**Short description**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

- [math]\displaystyle{ \operatorname{Hom}_C(X, \cdot) : C \to \mathrm{Sets}, Y \mapsto \operatorname{Hom}_C(X, Y) }[/math]

commutes with filtered colimits, i.e., if the natural map

- [math]\displaystyle{ \operatorname{colim} \operatorname{Hom}_C(X, Y_i) \to \operatorname{Hom}_C(X, \operatorname{colim}_i Y_i) }[/math]

is a bijection for any filtered system of objects [math]\displaystyle{ Y_i }[/math] in *C*.^{[1]} Since elements in the filtered colimit at the left are represented by maps [math]\displaystyle{ X \to Y_i }[/math], for some *i*, the surjectivity of the above map amounts to requiring that a map [math]\displaystyle{ X \to \operatorname{colim}_i Y_i }[/math] factors over some [math]\displaystyle{ Y_i }[/math].

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: (Adámek Rosický) use the terminology *finitely presented object* instead of compact object. (Kashiwara Schapira) 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, (Neeman 2001) defines an object to be compact if

- [math]\displaystyle{ \operatorname{Hom}_C(X, \cdot) : C \to \mathrm{Ab}, Y \mapsto \operatorname{Hom}_C(X, Y) }[/math]

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, [math]\displaystyle{ \operatorname{Hom}_C(X, -) }[/math] 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 [math]\displaystyle{ D(R-\text{Mod}) }[/math] of *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.^{[2]} The link between compactness in topology and the above categorical notion of compactness is as follows: for a fixed topological space [math]\displaystyle{ X }[/math], there is the category [math]\displaystyle{ \text{Open}(X) }[/math] whose objects are the open subsets of [math]\displaystyle{ X }[/math] (and inclusions as morphisms). Then, [math]\displaystyle{ X }[/math] is a compact topological space if and only if [math]\displaystyle{ X }[/math] is compact as an object in [math]\displaystyle{ \text{Open}(X) }[/math].

If [math]\displaystyle{ C }[/math] is any category, the category of presheaves [math]\displaystyle{ \text{PreShv}(C) }[/math] (i.e., the category of functors from [math]\displaystyle{ C^{op} }[/math] to sets) has all colimits. The original category [math]\displaystyle{ C }[/math] is connected to [math]\displaystyle{ \text{PreShv}(C) }[/math] by the Yoneda embedding [math]\displaystyle{ h_{(-)}: C \to \text{PreShv}(C), X \mapsto h_{X} := \operatorname{Hom}(-, X) }[/math]. For *any* object [math]\displaystyle{ X }[/math] of [math]\displaystyle{ C }[/math], [math]\displaystyle{ h_X }[/math] is a compact object (of [math]\displaystyle{ \text{PreShv}(C) }[/math]).

In a similar vein, any category [math]\displaystyle{ C }[/math] can be regarded as a full subcategory of the category [math]\displaystyle{ \text{Ind}(C) }[/math] of ind-objects in [math]\displaystyle{ C }[/math]. Regarded as an object of this larger category, *any* object of [math]\displaystyle{ C }[/math] is compact. In fact, the compact objects of [math]\displaystyle{ \text{Ind}(C) }[/math] are precisely the objects of [math]\displaystyle{ C }[/math] (or, more precisely, their images in [math]\displaystyle{ \text{Ind}(C) }[/math]).

### Non-examples

#### Derived category of sheaves of Abelian groups on a noncompact X

In the unbounded derived category of sheaves of Abelian groups [math]\displaystyle{ D(\text{Sh}(X;\text{Ab})) }[/math] for a non-compact topological space [math]\displaystyle{ X }[/math], it is generally not a compactly generated category. Some evidence for this can be found by considering an open cover [math]\displaystyle{ \mathcal{U} = \{U_i \}_{i \in I} }[/math] (which can never be refined to a finite subcover using the non-compactness of [math]\displaystyle{ X }[/math]) and taking a map

[math]\displaystyle{ \phi\in\text{Hom}(\mathcal{F}^\bullet,\underset{i\in I}{\text{colim}} \mathbb{Z}_{U_i}) }[/math]

for some [math]\displaystyle{ \mathcal{F}^\bullet \in \text{Ob}(D(\text{Sh}(X;\text{Ab}))) }[/math]. Then, for this map [math]\displaystyle{ \phi }[/math] to lift to an element

[math]\displaystyle{ \psi \in \underset{i \in I}{\text{colim}} \text{ Hom}(\mathcal{F}^\bullet, \mathbb{Z}_{U_i}) }[/math]

it would have to factor through some [math]\displaystyle{ \mathbb{Z}_{U_i} }[/math], which is not guaranteed. Proving this requires showing that any compact object has support in some compact subset of [math]\displaystyle{ X }[/math], and then showing this subset must be empty.^{[3]}

#### Derived category of quasi-coherent sheaves on an Artin stack

For algebraic stacks [math]\displaystyle{ \mathfrak{X} }[/math] over positive characteristic, the unbounded derived category [math]\displaystyle{ D_{qc}(\mathfrak{X}) }[/math] of quasi-coherent sheaves is in general not compactly generated, even if [math]\displaystyle{ \mathfrak{X} }[/math] is quasi-compact and quasi-separated.^{[4]} In fact, for the algebraic stack [math]\displaystyle{ B\mathbb{G}_a }[/math], there are no compact objects other than the zero object. This observation can be generalized to the following theorem: if the stack [math]\displaystyle{ \mathfrak{X} }[/math] has a stabilizer group [math]\displaystyle{ G }[/math] such that

- [math]\displaystyle{ G }[/math] is defined over a field [math]\displaystyle{ k }[/math] of positive characteristic
- [math]\displaystyle{ \overline{G} = G\otimes_k\overline{k} }[/math] has a subgroup isomorphic to [math]\displaystyle{ \mathbb{G}_a }[/math]

then the only compact object in [math]\displaystyle{ D_{qc}(\mathfrak{X}) }[/math] is the zero object. In particular, the category is not compactly generated.

This theorem applies, for example, to [math]\displaystyle{ G=GL_n }[/math] by means of the embedding [math]\displaystyle{ \mathbb{G}_a \to GL_n }[/math] sending a point [math]\displaystyle{ x \in \mathbb{G}_a(S) }[/math] to the identity matrix plus [math]\displaystyle{ x }[/math] at the [math]\displaystyle{ n }[/math]-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 [math]\displaystyle{ C }[/math] is * compactly generated* if any object can be expressed as a filtered colimit of compact objects in [math]\displaystyle{ C }[/math]. 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 (Ben-Zvi Francis).

## References

- ↑ (Lurie 2009)
- ↑ (Adámek Rosický)
- ↑ Neeman, Amnon. "On the derived category of sheaves on a manifold.".
*Documenta Mathematica***6**: 483–488. https://eudml.org/doc/123048. - ↑ Hall, Jack; Neeman, Amnon; Rydh, David (2015-12-03). "One positive and two negative results for derived categories of algebraic stacks". arXiv:1405.1888 [math.AG].

- Adámek, Jiří; Rosický, Jiří (1994),
*Locally presentable and accessible categories*, Cambridge University Press, doi:10.1017/CBO9780511600579, ISBN 0-521-42261-2 - Ben-Zvi, David; Francis, John; Nadler, David (2010), "Integral transforms and Drinfeld centers in derived algebraic geometry",
*Journal of the American Mathematical Society***23**(4): 909–966, doi:10.1090/S0894-0347-10-00669-7

- Kashiwara, Masaki; Schapira, Pierre (2006),
*Categories and sheaves*, Springer Verlag, doi:10.1007/3-540-27950-4, ISBN 978-3-540-27949-5 - Lurie, Jacob (2009),
*Higher topos theory*, Annals of Mathematics Studies,**170**, Princeton University Press, ISBN 978-0-691-14049-0 - Neeman, Amnon (2001),
*Triangulated Categories*, Annals of Mathematics Studies,**148**, Princeton University Press

Original source: https://en.wikipedia.org/wiki/Compact object (mathematics).
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