small object

Small objects


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

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


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

Let CC be a category and F:JCF : J \to C a diagram, called a κ\kappa-filtered diagram. Let XCX \in C be any object.

Then the condition that XX commutes with the colimit over FF means that the map of hom-sets

lim jHom C(X,F(j))Hom C(X,lim jF(j)) \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 jHom C(X,F(j)) \lim_{\to^j} Hom_C(X,F(j))

may be expressed as a coequalizer

jJHom C(X,F(j))lim jHom C(X,F(j)) \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 CC from XX into one of the objects F(j)F(j). Being a quotient set, every element of it is represented by one of the original elements in jHom C(X,F(j))\coprod_j Hom_C(X,F(j)).

This means that we have


The map of hom-sets

lim jHom C(X,F(j))Hom C(X,lim jF(j)) \lim_{\to^j} Hom_C(X, F(j)) \to Hom_C(X,\lim_{\to^j} F(j))

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

F(j1) F(j) F(j+1) f^ X f lim F. \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 } \,.


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

Examples and applications

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

  • object classifier

Revised on October 21, 2017 11:12:28 by Dnl Grgk? (2003:e5:13cb:1701:149d:4d6:c583:e4)