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 denote in the following some ordinal number regarded as a poset, hence as a category itself. Let be the smallest element (when is inhabited).
Let be a category, an object of , and a class of morphisms in . A transfinite composition of morphisms in is the morphism
from a diagram
into its colimit in the coslice category , schematically
where the diagram is such that
takes all successor morphisms in to morphisms in
is continuous in that for every nonzero limit ordinal , restricted to the full diagram is a colimiting cocone in for restricted to .
Because of the first clause, we really do not need to mention in the data except to cover the possibility that . (In that case, the composite is just .)
For purposes of constructive mathematics, the continuity condition should be stated as follows:
- For every ordinal , restricted to is a colimiting cone in for the disjoint union of and the restriction of to .
This actually includes as a special case but says nothing when 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