# 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 in *C*.[1] Since elements in the filtered colimit at the left are represented by maps , for some *i*, the surjectivity of the above map amounts to requiring that a map factors over some .

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 (1994) use the terminology *finitely presented object* instead of compact object. Kashiwara & Schapira (2006) 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

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, 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 *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 *X*, there is the category O(*X*) whose objects are the open subsets of *X* (and inclusions as morphisms). Then, *X* is a compact topological space if and only if *X* is compact as an object in O(*X*).

If *C* is any category, the category of presheaves (i.e., the category of functors from to sets) has all colimits. The original category *C* is connected to P(*C*) by the Yoneda embedding . For *any* object *X* of *C*, *j*(*X*) is a compact object (of P(*C*)).

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

## Compactly generated categories

In most categories, the condition of being compact is quite strong, so that most objects are not compact. A category *C* is *compactly generated* if any object can be expressed as a filtered colimit of compact objects in *C*. 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 & Nadler (2010).

## References

- Lurie (2009, §5.3.4)
- Adámek & Rosický (1994, Chapter 1.A)

- Adámek, Jiří; Rosický, Jiří (1994),
*Locally presentable and accessible categories*, Cambridge University Press, doi:10.1017/CBO9780511600579, ISBN 0-521-42261-2, MR 1294136 - 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, arXiv:0805.0157, doi:10.1090/S0894-0347-10-00669-7, MR 2669705

- Kashiwara, Masaki; Schapira, Pierre (2006),
*Categories and sheaves*, Springer Verlag, doi:10.1007/3-540-27950-4, ISBN 978-3-540-27949-5, MR 2182076

- Lurie, Jacob (2009),
*Higher topos theory*, Annals of Mathematics Studies,**170**, Princeton University Press, arXiv:math.CT/0608040, ISBN 978-0-691-14049-0, MR 2522659