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factorization system in a 2-category

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Definition

In a (weak) 2-category, the appropriate notion of an orthogonal factorization system is suitably weakened up to isomorphism. Specifically, a factorization system in a 2-category KK consists of two classes (E,M)(E,M) of 1-morphisms in KK such that:

  1. Every 1-morphism f:xyf:x\to y in KK is isomorphic to a composite eme\circ m where eEe\in E and mMm\in M, and

  2. For any e:abe:a\to b in EE and m:xym:x\to y in MM, the following square

    K(b,x) K(b,y) K(a,x) K(a,y)\array{ K(b,x) & \to & K(b,y) \\ \downarrow & \cong & \downarrow \\ K(a,x) & \to & K(a,y)}

    (which commutes up to isomorphism) is a 2-pullback in CatCat.

This second property is a “2-categorical orthogonality.” In particular, it implies that any square

a x e m b y\array{a & \to & x \\ ^e\downarrow & \cong & \downarrow^m \\ b & \to & y}

which commutes up to specified isomorphism, where eEe\in E and mMm\in M, has a diagonal filler bxb\to x making both triangles commute up to isomorphisms that are coherent with the given one. It also implies an additional factorization property for 2-cells.

Examples

Cat-enriched factorization systems

If instead KK is a strict 2-category and we require that

  1. Every 1-morphism in KK is equal to a composite of a morphism in EE and a morphism in MM, and

  2. The above square (which commutes strictly when KK is a strict 2-category) is a strict 2-pullback (i.e. a CatCat-enriched pullback).

then we obtain the notion of a CatCat-enriched, or strict 2-categorical, factorization system.

It is important to note that in general, the strict and weak notions of 2-categorical factorization system are incomparable; neither is a special case of the other. For example, on CatCat there is a weak 2-categorical factorization system where E=E= essentially surjective functors and M=M= fully faithful functors, and a strict 2-categorical factorization system where E=E= bijective on objects functors and M=M= fully faithful functors.

References

For instance

Revised on November 21, 2011 00:15:46 by Urs Schreiber (89.204.154.71)