nLab
transfinite composition

Idea

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.

Definition

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

Let $C$ be a category, $X$ an object of $C$, and $I\subset \mathrm{Mor}(C)$ a class of morphisms in $C$. A transfinite composition of morphisms in $I$ is the morphism

$$X\to Y≔\mathrm{colim}(F)$$X \to Y \coloneqq colim (F)

from a diagram

$$F:\alpha \to C$$F\colon \alpha \to C

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

$$\begin{array}{ccccccc}X& \stackrel{F(0\le 1)}{\to }& F_1& \stackrel{F(1\le 2)}{\to }& F_2& \to & \cdots \\ & \searrow & \downarrow & \swarrow & \cdots \\ & & Y\end{array},$$\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

• $F(0)=X$ if $\alpha >0$,

• $F$ takes all successor morphisms $\beta \stackrel{\le }{\to }\beta +1$ in $\alpha$ to morphisms in $I$

  $$F(\beta \to \beta +1)\in I,$$F(\beta \to \beta + 1) \in I ,

• $F$ is continuous in that for every nonzero limit ordinal $\beta <\alpha$, $F$ restricted to the full diagram $\{\gamma \mid \gamma \le \beta \}$ is a colimiting cocone in $C$ for $F$ restricted to $\{\gamma \mid \gamma <\beta \}$.

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

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

• For every ordinal $\beta <\alpha$, $F$ restricted to $\{\gamma \mid \gamma \le \beta \}$ is a colimiting cone in $C$ for the disjoint union of $\{X\}$ and the restriction of $F$ to $\{\gamma +1\mid \gamma <\beta \}$.

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

Applications

Transfinite composition plays a role in

• the small object argument;

• cofibrantly generated model categories.

• the transfinite construction of free algebras

References

The above formulation is taken from page 6 of

• Tibor Beke, Sheafifiable homotopy model categories (arXiv)