# Contents

## 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 $0\in \mathrm{Obj}\left(\alpha \right)$ be the smallest element (when $\alpha$ is inhabited).

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

$X\to Y≔\mathrm{colim}\left(F\right)$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

$\begin{array}{ccccccc}X& \stackrel{F\left(0\le 1\right)}{\to }& {F}_{1}& \stackrel{F\left(1\le 2\right)}{\to }& {F}_{2}& \to & \cdots \\ & ↘& ↓& ↙& \cdots \\ & & Y\end{array}\phantom{\rule{thinmathspace}{0ex}},$\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

• $F\left(0\right)=X$ if $\alpha >0$,

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

$F\left(\beta \to \beta +1\right)\in I,$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 $\left\{\gamma \phantom{\rule{thickmathspace}{0ex}}\mid \phantom{\rule{thickmathspace}{0ex}}\gamma \le \beta \right\}$ is a colimiting cocone in $C$ for $F$ restricted to $\left\{\gamma \phantom{\rule{thickmathspace}{0ex}}\mid \phantom{\rule{thickmathspace}{0ex}}\gamma <\beta \right\}$.

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 $\left\{\gamma \phantom{\rule{thickmathspace}{0ex}}\mid \phantom{\rule{thickmathspace}{0ex}}\gamma \le \beta \right\}$ is a colimiting cone in $C$ for the disjoint union of $\left\{X\right\}$ and the restriction of $F$ to $\left\{\gamma +1\phantom{\rule{thickmathspace}{0ex}}\mid \phantom{\rule{thickmathspace}{0ex}}\gamma <\beta \right\}$.

This actually includes $F\left(0\right)=X$ 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

## References

The above formulation is taken from page 6 of

• Tibor Beke, Sheafifiable homotopy model categories (arXiv)

Revised on December 6, 2012 08:38:56 by Mike Shulman (192.16.204.218)