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Sunday, April 13, 2014

Factorization systems

If (a category E) has (equalizers and coequalizers, kernel pairs and cokernel pairs)
then for each (arrow f in E) we have (the commutative diagram in E) which appears in (the center of the display below);
to its left and right are (2-diagrams in CAT) showing parts of (the adjunctions determined by those limit and colimits in E) : CATECATEYEYfKpfkpXregular epi.fkpcoequfkpCoequfCoimEXcokpηequfηffϵkpϵcoequYEfImfcokpEqufcokpequregular mono.YfcokpfCokpXEXE Further, (“the diagonal fill-in property”) there is one and only one
[(diagonal arrow from fCoim to fIm) which makes (both of the triangles of which it is an edge) commute].

All of this follows easily from (the properties of the limits and colimits which are mentioned in the diagram).

In many cases the diagonal fill-in is an isomorphism.
The paradigmatic example is when E=Set.
Then (the unique diagonal fill-in) is
[the canonical bijection between (the set of blocks in (the partition of X determined by f), i.e., the fibers of f) and (the image of f as a subset of Y)].
Concrete examples illustrating how this works, in the familiar case E=Set,
are given in the posts "The parts of a function" and "Classifying functions by their parts",
using a slightly different language aimed at readers more familiar with set theory than category theory.

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