nLab comprehensive factorization system





Let EE be the class of final functors and MM be the class of discrete fibrations. Then (E,M)(E,M) is an orthogonal factorization system of Cat, called the comprehensive factorization system.


Let F:CDF:C\to D be a functor. Define K:DSetK:D\to Set as the left Kan extension of the constant presheaf CSetC\to Set at the singleton along FF. Explicitly, K(d)K(d) is the set of connected components of F/dF/d. Let E=KE=\int K, so an object of EE is an ordered pair (d,[α:Fcd])(d, [\alpha:Fc\to d]) where [α][\alpha] denotes the connected component of (c,α)(c,\alpha). Then it is not hard to verify that e:CEe:C\to E mapping c(Fc,[id fc])c\mapsto (Fc,[id_{fc}]) is final, the canonical m:EDm:E\to D is a discrete fibration, and F=meF=me.

Now we show that EE and MM are replete subcategories of CatCat. Clearly they include all isomorphisms.

If functors F:CDF:C\to D and G:DEG: D\to E are final, then we show that GFG\circ F is final. For eEe\in E, there is an element (d,α:eGd)(d,\alpha:e\to Gd) of e/Ge/G, and thence an element (c,β:dFc)(c,\beta:d\to Fc) of d/Fd/F, so we obtain an element (c,eαGdGβGFc)(c, e \stackrel{\alpha}{\to} Gd \stackrel{G\beta}{\to} GFc) of e/GFe/GF. Now we must show that any two elements (c,γ:eGFc),(c,γ:eGFc)(c,\gamma:e\to GFc),(c',\gamma':e\to GFc') are connected. Since GG is final, elements (Fc,γ)(Fc,\gamma) and (Fc,γ)(Fc',\gamma') of e/Ge/G are connected. It suffices to consider the case of a zig-zag of length one: a morphism f:FcFcf:Fc\to Fc' such that

By finality of FF, the elements (c,id:FcFc)(c,id:Fc\to Fc) and (c,f:FcFc)(c', f:Fc\to Fc') of Fc/FFc/F are connected. A zig-zag path between them, by precomposition with γ\gamma, becomes a zig-zag path between (c,γ)(c,\gamma) and (c,γ)(c',\gamma'). So GFG\circ F is final.

The proof that discrete fibrations form a subcategory is omitted.

Now we must show that the lifting problem

has a unique solution hh when eEe\in E and mMm\in M.

We prove uniqueness first. For bBb\in B, let (a,α:be(a))b/e(a,\alpha:b\to e(a))\in b/e. Then h(α)h(\alpha) must be the unique lifting of g(α)g(\alpha), and h(b)h(b) the domain of this lifting, proving uniqueness of hh on objects. For β:bb\beta:b\to b' in BB, h(β)h(\beta) must be the unique lifting of g(β)g(\beta), so hh is unique (if it exists).

Now we must show that this hh is well-defined, functorial, and a solution to the lifting problem. If (a,α:be(a))(a',\alpha':b\to e(a')) is another element of b/eb/e, then WLOG let u:aau:a\to a' such that

Lifting this diagram, we see that g(α)g(\alpha) and g(α)g(\alpha') must lift to morphisms with identical domain, so hh is well-defined on objects.

For β:bb\beta:b\to b' in BB, let α:be(a)\alpha:b'\in e(a), and by the diagram

we see that g(β)g(\beta) and g(αβ)g(\alpha\circ\beta) must lift to morphisms with identical domains, so h(β)h(\beta) has domain h(b)h(b).

Functoriality now follows easily from uniqueness of lifting for a discrete fibration, and it is not hard to show that hh is a solution to the lifting problem.

Dually, there is an orthogonal factorisation system (E,M)(E,M) on CatCat for which EE is the class of initial functors and MM is the class of discrete opfibrations.


Note there is a mistake in the proof of the main theorem of the paper above, as noted on page 74 of:

Internal comprehensive factorisations (and torsors) are considered in:

Last revised on January 10, 2024 at 16:27:39. See the history of this page for a list of all contributions to it.