category theory

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 $0 \in Obj(\alpha)$ be the smallest element (when $\alpha$ is inhabited).

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

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

from a diagram

$F\colon \alpha \to C$

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

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

where the diagram is such that

• $F(0) = X$ if $\alpha \gt 0$,

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

$F(\beta \to \beta + 1) \in I ,$
• $F$ is continuous in that for every nonzero limit ordinal $\beta \lt \alpha$, $F$ restricted to the full diagram $\{\gamma \;|\; \gamma \leq \beta\}$ is a colimiting cocone in $C$ for $F$ restricted to $\{\gamma \;|\; \gamma \lt \beta\}$.

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 $\beta \lt \alpha$, $F$ restricted to $\{\gamma \;|\; \gamma \leq \beta\}$ is a colimiting cone in $C$ for the disjoint union of $\{X\}$ and the restriction of $F$ to $\{\gamma + 1 \;|\; \gamma \lt \beta\}$.

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

References

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 (82.113.99.160)