nLab small object

Definition

An object $X$ of a category is small if it is $\kappa$-compact for some regular cardinal $\kappa$ (and therefore also for all greater regular cardinals).

Here, $X$ is called $\kappa$-compact if the corepresentable functor $hom(X,\cdot)$ preserves $\kappa$-directed colimits.

Details

We unwrap the definition further. Let $J$ be a $\kappa$-filtered poset, i.e. one in which every sub-poset $J' \subset J$ of cardinality $|J'| \lt \kappa$ has an upper bound in $J$.

Let $C$ be a category and $F : J \to C$ a diagram, called a $\kappa$-filtered diagram. Let $X \in C$ be any object.

Then the condition that $X$ commutes with the colimit over $F$ means that the map of hom-sets

$$\lim_{\to^j} Hom_C(X, F(j)) \to Hom_C(X,\lim_{\to^j} F(j))$$

is an isomorphism, i.e. a bijection.

By the general properties of colimit (recalled at limits and colimits by example), the colimit

$$\lim_{\to^j} Hom_C(X,F(j))$$

may be expressed as a coequalizer

$$\stackrel{\to}{\to} \coprod_{j \in J} Hom_C(X,F(j)) \to \lim_{\to^j} Hom_C(X,F(j))$$

hence as a quotient set of the the set of morphism in $C$ from $X$ into one of the objects $F(j)$. Being a quotient set, every element of it is represented by one of the original elements in $\coprod_j Hom_C(X,F(j))$.

This means that we have

Restatement

The map of hom-sets

$$\lim_{\to^j} Hom_C(X, F(j)) \to Hom_C(X,\lim_{\to^j} F(j))$$

is onto precisely if every morphism $X \to \lim_\to F$ lifts to a morphism $X \to F(j)$ into one of the $F(j)$, schematically:

$$\array{ \cdots&\to&F(j-1) &\to& F(j) &\to& F(j+1) &\to& \cdots \\ &&&{}^{\mathllap{\exists \hat f}}\nearrow&\downarrow & \swarrow \\ &&X& \stackrel{f}{\to} &\lim_\to F } \,.$$

Properties

Let $\lambda \gt \kappa$ be a regular cardinal greater than $\kappa$. Then any $\lambda$-filtered category $D$ is also $\kappa$-filtered. For being $\lambda$-filtered means that any diagram in $D$ of size $\lt\lambda$ has a cocone; but any diagram of size $\lt\kappa$ is of course also $\lt\lambda$. Thus, any $\lambda$-filtered colimit is also a $\kappa$-filtered colimit, so any functor which preserves $\kappa$-filtered colimits must in particular preserve $\lambda$-filtered colimits. It follows that any $\kappa$-compact object is also $\lambda$-compact.

Examples and applications

• By definition, in a locally presentable category every object is a colimit over small objects.

• Smallness of objects plays a crucial role in the small object argument.

Related concepts

• cosmall object?, which is just the dual concept, but is interesting in its own right.

• object classifier