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
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.
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).
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.)
always commutes with finite colimits since these are limits.
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.
of R-modules are precisely the perfect complexes.
Instead these are precisely the finite sets endowed with the discrete topology.
can be regarded as a full subcategory of the category
In the unbounded derived category of sheaves of Abelian groups
Some evidence for this can be found by considering an open cover
(which can never be refined to a finite subcover using the non-compactness of
over positive characteristic, the unbounded derived category
of quasi-coherent sheaves is in general not compactly generated, even if
This observation can be generalized to the following theorem: if the stack
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.
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.
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).