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.
Transfinite composition is indexed by ordinals. For convenience we first recall the definition of these assuming excluded middle in the ambient set theory (for definitions not assuming this see at ordinal and pointers given there):
1) (reflexivity) ;
2) (transitivity) if and then ;
3) (antisymmetry) if a and then .
This we may and will equivalently think of as a category with objects the elements of and a unique morphism precisely if . In particular an order-preserving function between partially ordered sets is equivalently a functor between their corresponding categories.
A partial order is a total order if in addition
4) (totality) either or .
A total order is a well order if in addition
5) (well-foundedness) every non-empty subset has a least element.
A limit ordinal is one that is not a successor.
The finite ordinals are labeled by , corresponding to the well-orders . Here is the successor of . The first non-empty limit ordinal is .
takes all successor morphisms in to elements in
The corresponding transfinite composition is the induced morphism
into the colimit of the diagram, schematically:
For purposes of constructive mathematics, the continuity condition should be stated as follows:
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