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2 change of order of axioms

One of the intuitions of (simplicial) T-complexes is that you can make the simplicial singular complex $SX$ of a space into a natural Kan complex, by making choices of the appropriate retractions in the models. But of course these choices are not unique, and what is unclear is what are the relations among these natural fillers! The idea has something to do with "composition" since in a Kan complex the filler of a horn makes the remaining face in some sense the composition of the other faces; the filler is a "program" for determining this composition.

Keith Dakin (1977) then axiomatised the idea that these natural fillers are in some sense "thin". So the axioms are:

1. Degenerate elements are thin.
2. Every horn has a unique thin filler.
3. Degenerate elements are thin.
4. If every face but one of a thin element is thin, then so also is the remaining face.

A T-complex is of rank $\leq n$ if every element of dimension $>n$ is thin.

T-complexes of rank $1$ are equivalent to groupoids, and (Dakin) those of rank $2$ are equivalent to crossed modules over groupoids. Nick Ashley (1978) completed the job by showing, among other things, that $T$-complexes are equivalent to crossed complexes, and giving the topological example, the fundamental $T$-complex of a filtered space.

These theses are available from here.

1

One of the intuitions of (simplicial) T-complexes is that you can make the simplicial singular complex $SX$ of a space into a natural Kan complex, by making choices of the appropriate retractions in the models. But of course these choices are not unique, and what is unclear is what are the relations among these natural fillers! The idea has something to do with "composition" since in a Kan complex the filler of a horn makes the remaining face in some sense the composition of the other faces; the filler is a "program" for determining this composition.

Keith Dakin (1977) then axiomatised the idea that these natural fillers are in some sense "thin". So the axioms are:

1. Every horn has a unique thin filler.
2. Degenerate elements are thin.
3. If every face but one of a thin element is thin, then so also is the remaining face.

A T-complex is of rank $\leq n$ if every element of dimension $>n$ is thin.

T-complexes of rank $1$ are equivalent to groupoids, and (Dakin) those of rank $2$ are equivalent to crossed modules over groupoids. Nick Ashley (1978) completed the job by showing, among other things, that $T$-complexes are equivalent to crossed complexes, and giving the topological example, the fundamental $T$-complex of a filtered space.

These theses are available from here.