nLab comprehensive factorization system

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Contents

Contents

Idea

Every functor F:CDF:C\to D factors as a final functor followed by a discrete fibration.

Statement

Theorem

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.

Proof

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.

Remark on the terminology

Under the functorial reformulation of the axiom of comprehension by Lawvere (1970) the comprehensive factorization can be viewed as a generalization of the epi-mono factorization of a function f:XYf:X\to Y occurring in the context of set-theoretic comprehension: the best approximation of ff by a property in 2 Y2^Y (i.e. χ imf\chi_{im f}) has extension imfYim f\hookrightarrow Y. Hence (in the notation of the above proof) the discrete fibration m:KDm:\int K\to D given by the factorization F=meF=m{}e can be viewed as the “extension” of the approximation of FF by the “property” KK in Set DSet^{D}.

More generally, instances of the comprehension scheme correspond to factorization systems (cf. Berger-Kaufmann).

References

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:

A generalisation to a bicategorical factorization system on CATCAT is considered in:

Last revised on November 1, 2024 at 11:01:17. See the history of this page for a list of all contributions to it.