nLab
compact object

Idea

An object of a category is often called compact if it is “finite” or “small” in some precise sense.

Definition

Let $C$ be a locally small category that admits filtered colimits. Then an object $X\in C$ is compact (or sometimes called finitely presented) if the corepresentable functor

$${\mathrm{Hom}}_C(X,-):C\to \mathrm{Set}$$Hom_C(X,-) : C \to Set

commutes with these filtered colimits.

So if for every filtered category $D$ and every functor $F:D\to C$ the canonical morphism

$$\mathrm{colim}C(X,F(-))\stackrel{\simeq }{\to }C(X,\mathrm{colim}F)$$colim C(X,F(-)) \stackrel{\simeq}{\to} C(X, colim F)

is an isomorphism.

Small objects

There is a slight variant of this definition.

If $\kappa$ is a regular cardinal, then objects $X$ such that $C(X,-)$ commutes with such $\kappa$-filtered colimits are also called $\kappa$-compact objects. An object which is $\kappa$-compact for some regular $\kappa$ is called a small object.

Examples

In $\mathrm{Set}$

In $C=$ Set an object is compact precisely if it is a (Kuratowski) finite set.

In $\mathrm{Grp}$

In $C=$ Grp an object is compact precisely if it is finitely presented as a group.

In a topos

For $C$ a topos, $X$ is compact if it is $K$-finite.

In $\mathrm{Top}$

Let $X$ be a topological space and let $C=\mathrm{Op}(X)$ be the category of open subsets of $X$. The an open subset $U\in C$ is a compact object in $C$ precisely if it is a compact topological space.

See the discussion below for variations of this theme.

Generalizations

This definition has an obvious generalization to compact object in an (infinity,1)-category.

Subtleties and different meanings

One has to be careful about the following variations on this theme

Compactness in additive categories

When $C$ is an additive category (often a triangulated category), an object $x$ in $C$ is called compact if for every set $S$ of objects of $C$ such that the coproduct ${\coprod }_{s\in S}s$ exists, the canonical map

$$\coprod _{s\in S}C(x,s)\to C(x,\coprod _{s\in S}s)$$\coprod_{s\in S} C(x,s)\to C(x,\coprod_{s\in S}s)

is an isomorphism of sets (a bijection).

Here is an application of this concept to characterize which abelian categories are categories of modules of some ring:

Theorem

Let $C$ be an abelian category. If $C$ has all small coproducts and has a compact projective generator, then $C\simeq R\mathrm{Mod}$ for some ring $R$. In fact, in this situation we can take $R=C(x,x{)}^{\mathrm{op}}$ where $x$ is any compact projective generator. Conversely, if $C\simeq R\mathrm{Mod}$, then $C$ has all small coproducts and $x=R$ is a compact projective generator.

Proof

This theorem, minus the explicit description of $R$, 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 $R$ is a compact projective generator of $R\mathrm{Mod}$.

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.

The notion can be modified for categories enriched over a closed monoidal category (compare to the notions of finite and/or rigid objects in various contexts).

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 $\kappa$-compact objects in the setup of $(\mathrm{\infty }\mathrm{,1})$-categories.

Compactness in non-additive 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 preserve filtered colimits (see below for discussion). For $C$ any category and $X\in C$, the condition that $C(X,-):C\to C$ preserves filtered colimits imposes some kind of finiteness condition on $X$. For instance

Compact objects in Top

Recall the above example of compact topological spaces. Notice that the statement which one might expect, that a topological space $X$ is compact if it is a compact object in Top is not quite right in general.

A counterexample is given for instance on page 49 of Hovey’s Model Categories, which itself was corrected by Don Stanley (see the errata of that book). See also the blog discussion here.

Namely, the two-element set with the indiscrete topology is a compact space $X$ for which

(1)$$\mathrm{Hom}(X,-):\mathrm{Top}\to \mathrm{Top}$$Hom(X, -): Top \rightarrow Top

doesn’t preserve filtered colimits, in fact not even colimits of sequences (functors out of the ordered set of natural numbers).

For example, consider the sequence of spaces

(2)$$X_n=[n,\mathrm{\infty })\times \{\mathrm{0,1}\}$$X_n=[n,\infty) \times \{0,1\}

where the open sets are of the form

(3)$$[n,\mathrm{\infty }]\cup [m,\mathrm{\infty })\times \{1\}$$[n, \infty]\cup [m,\infty) \times \{1\}

(where $m\ge n$), plus the empty set. Define $X_n\to X_{n+1}$ so that it sends a pair $(k,ϵ)$ to itself if $k>n$, and $(n,ϵ)$ to $(n+1,ϵ)$. This defines a functor

(4)$$F:\mathbb{N}\to \mathrm{Top}$$F: \mathbb{N} \rightarrow Top

The colimit $X_{\mathrm{\infty }}$ of this sequence is the two-element set $\{\mathrm{0,1}\}$ with the indiscrete topology. However, the identity map on this space does not factor through any of the canonical maps $X_n\to X_{\mathrm{\infty }}$. It follows that the comparison map

(5)$${\mathrm{colim}}_n\mathrm{Hom}(X_{\mathrm{\infty }},X_n)\to \mathrm{Hom}(X_{\mathrm{\infty }},X_{\mathrm{\infty }})$$colim_n Hom(X_\infty, X_n) \rightarrow Hom(X_\infty, X_\infty)

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 $X$ compact Hausdorff, but it could be worth considering.

But with a bit care on the assumptions, similar results do hold:

If $Y$ is compact, then $\mathrm{hom}(Y,-)$ preserves colimits of functors mapping out of limit ordinals, provided that the arrows of the cocone diagram,

(6)$$X_{\alpha }\to X_{\beta },$$X_\alpha \rightarrow X_\beta,

are closed inclusions of $T_1$ spaces. (This applies for example to the sequence of inclusions of n-skeleta in a CW-complex. Taking $Y=S_k$, this has obvious desirable consequences for the functor ${\pi }_k$.)

This example is discussed on page 50 of Hovey’s book.

Hovey wants this result in view of a small object argument on the way to proving that $\mathrm{Top}$ is a model category.

References

For the pages quoted in the context of the discussion of compact objects in Top see

• Mark Hovey, Model categories.

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

• Jacob Lurie, Higher Topos Theory