transfinite composition



The axioms of a category ensure that every finite number of composable morphisms has a (unique) composite.

Transfinite composition is a means to talk about morphisms in a category that behave as if they were the result of composing infinitely many morphisms.


Let α\alpha denote in the following some ordinal number regarded as a poset, hence as a category itself. Let 0Obj(α)0 \in Obj(\alpha) be the smallest element (when α\alpha is inhabited).

Let CC be a category, XX an object of CC, and IMor(C)I \subset Mor(C) a class of morphisms in CC. A transfinite composition of morphisms in II is the morphism

XYcolim(F) X \to Y \coloneqq colim (F)

from a diagram

F:αC F\colon \alpha \to C

into its colimit in the coslice category X/CX/C, schematically

X F(01) F 1 F(12) F 2 Y, \array{ X &\stackrel{F(0 \leq 1)}{\to}& F_1 &\stackrel{F(1 \leq 2)}{\to}& F_2 &\to& \cdots \\ & \searrow & \downarrow & \swarrow & \cdots \\ && Y } \,,

where the diagram is such that

  • F(0)=XF(0) = X if α>0\alpha \gt 0,

  • FF takes all successor morphisms ββ+1\beta \stackrel{\leq}{\to} \beta + 1 in α\alpha to morphisms in II

    F(ββ+1)I, F(\beta \to \beta + 1) \in I ,
  • FF is continuous in that for every nonzero limit ordinal β<α\beta \lt \alpha, FF restricted to the full diagram {γγβ}\{\gamma \;|\; \gamma \leq \beta\} is a colimiting cocone in CC for FF restricted to {γγ<β}\{\gamma \;|\; \gamma \lt \beta\}.

Because of the first clause, we really do not need to mention XX in the data except to cover the possibility that α=0\alpha = 0. (In that case, the composite is just XX.)

For purposes of constructive mathematics, the continuity condition should be stated as follows:

  • For every ordinal β<α\beta \lt \alpha, FF restricted to {γγβ}\{\gamma \;|\; \gamma \leq \beta\} is a colimiting cone in CC for the disjoint union of {X}\{X\} and the restriction of FF to {γ+1γ<β}\{\gamma + 1 \;|\; \gamma \lt \beta\}.

This actually includes F(0)=XF(0) = X as a special case but says nothing when β\beta is a successor (so the successor clause is still required).


Transfinite composition plays a role in


The above formulation is taken from page 6 of

  • Tibor Beke, Sheafifiable homotopy model categories (arXiv)

Revised on October 10, 2013 19:15:51 by Urs Schreiber (