Michael Shulman
comprehensive factorization

In Cat there is a factorization system (E,M) where E is the class of initial functors and M is the class of discrete opfibrations; see

  • Street, Walters, The comprehensive factorization of a functor. Bull. Amer. Math. Soc. 79, 1973.

We can construct an analogous factorization system in any 1-exact and countably-coherent 2-category.

Definition

A morphism f:AB in a 2-category K is initial if it is left orthogonal to discrete opfibrations. That is, whenever g:CD is a discrete opfibration, the following square is a pullback in Cat:

K(B,C) K(B,D) K(A,C) K(A,D)\array{ K(B,C) & \to & K(B,D)\\ \downarrow & & \downarrow\\ K(A,C) & \to & K(A,D)}
Theorem

Every morphism in a 1-exact countably-coherent 2-category (in particular, in an n-pretopos for n1) factors, up to isomorphism, as an initial morphism followed by a discrete opfibration.

Proof

By the 2-categorical analogue of a standard theorem about factorization systems in 1-categories, it suffices to show that

  1. discrete opfibrations are closed under composition, and
  2. for every X, the discrete opfibrational slice DOpf(X) is a full reflective subcategory of the slice 2-category K/X.

The first point is true in any 2-category. For the second, we factor the inclusion as

DOpf(X)Opf(X)K/XDOpf(X) \to Opf(X) \to K/X

and observe that both forgetful functors have left adjoints. The left adjoint to DOpf(X)Opf(X) is discretization in the 1-exact and countably-coherent 2-category Opf(X). The left adjoint to Opf(X)K/X constructs the “free opfibration” on a functor f:AX, which is given by A× XX 2.

Corollary

In any 1-exact countably-coherent 2-category, (initial, discrete opfibration) is a (2-categorical) factorization system.

Corollary

In any 1-exact countably-coherent 2-category, each pullback functor f *:DOpf(Y)DOpf(X) has a left adjoint Lan f.

Proof

Lan f is given by composing with f, then factoring into an initial morphism followed by a discrete opfibration.

Examples

  • In a 1-category, every morphism is a discrete opfibration, so the only initial morphisms are isomorphisms. The factorization system (isomorphisms,all morphisms) is well-known.

  • In a (2,1)-category, the discrete opfibrations are precisely the faithful morphisms. Thus, in this case the comprehensive factorization system coincides with the (eso+full, faithful) factorization system.

Beck-Chevalley conditions

The appropriate Beck-Chevalley condition for the adjunctions Lan ff * refers not to pullback squares, but to comma squares. The easiest proof of this fact uses the internal logic. We start with an internal characterization of initial morphisms, analogous to the classical characterization in Cat as the functors f:AB for which each comma category (f/b), for bB, is nonempty and connected.

For the rest of this page we make the standing assumption that K is a 1-exact and countably-coherent 2-category.

Lemma

A morphism f:AB in K is initial iff the following are internally valid:

b:B(a:A)(α:hom B(f(a),b)) b:B,a 1:A,a 2:A,α 1:hom B(f(a 1),b),α 2:hom B(f(a 2),b) iφ i\begin{gathered} b:B | \top \vdash (\exists a:A)(\exists \alpha:hom_B(f(a),b))\\ b:B, a_1:A, a_2:A, \alpha_1:hom_B(f(a_1),b), \alpha_2:hom_B(f(a_2),b) | \top \vdash \bigvee_{i\in \mathbb{N}} \varphi_i \end{gathered}

where for each i, φ i expresses “there is a zigzag of length i connecting α 1 and α 2 over b.”

Thus, for instance, we have

φ 1 (γ:hom A(a 1,a 2))(α 2f(γ)=α 1) φ 2 (c:A)(β:hom B(f(c),b))(γ 1:hom A(a 1,c))(γ 2:hom A(a 2,c))(α 1=βf(γ 1)α 2=βf(γ 2))\begin{aligned} \varphi_1 \equiv& (\exists \gamma:hom_A(a_1,a_2)) (\alpha_2 \circ f(\gamma) = \alpha_1)\\ \varphi_2 \equiv& (\exists c:A)(\exists \beta:hom_B(f(c),b))(\exists \gamma_1:hom_A(a_1,c))(\exists \gamma_2:hom_A(a_2,c)) (\alpha_1 = \beta \circ f(\gamma_1) \wedge \alpha_2 = \beta \circ f(\gamma_2)) \end{aligned}

and so on.

Proof

f:AB will be initial iff the discrete-opfibration part of its factorization is an equivalence. By construction, this factorization is the discrete reflection of X=A× BB 2B in Opf(B), which is constructed as the quotient of the equivalence relation X 1 generated by X 2 (the power taken in Opf(B)). Therefore, this discrete reflection will be the terminal object 1 B iff

  1. XB is eso and
  2. the kernel of XB in Opf(B) is X 1.

It is fairly evident that XB is eso iff the first displayed sequent holds in K. For the second, note that the kernel of XB in Opf(B) (or equivalently, in K/B) is just the pullback X× BX, since B is discrete as an object of K/B. By definition of X, this pullback can be computed as the lax limit

Lax pullback X× BX A A B f f

and therefore it is precisely the context of the second displayed sequent. Since X 1 is the equivalence relation generated by X 2, it is necessarily contained in this kernel, so it suffices to show the converse implication. But X 1 is defined as the symmetric transitive closure of the image of X 2, which makes it essentially a countable union of “zigzags” from X 2, and this translates directly into the conclusion of the second sequent.

Lemma

Initial morphisms in K are stable under transfer across comma squares, in the sense that if

Comma Square (f/g) A B C f g q p

is a comma square with f initial, then q is also initial.

Proof

Given the characterization of initial morphisms in Lemma 1, we can simply observe that the usual proof of this fact in Cat takes place entirely in countably-coherent logic.

Theorem

The adjunctions Lan ff * for discrete opfibrations in K satisfy the Beck-Chevalley condition for comma squares. That is, if

Comma Square (f/g) A B C f g q p

is a comma square, then the canonical transformation Lan qp *g *Lan f between functors DOpf(A)DOpf(B) is an equivalence.

Note that unlike the case for a pullback square, there is no possible “handedness” ambiguity in saying that a comma square satisfies the Beck-Chevalley condition; there is no transformation Lan pq *f *Lan g at all.

The existence of the transformation in the correct direction also depends on the fact that opfibrational slices are functorial at the level of 2-cells. In particular, for a comma square there is no transformation Σ qp *g *Σ f between functors K/AK/B.

Proof

Given a discrete opfibration XA, let XLan fXC be the (initial,discrete-opfibration) factorization of the composite XAfC. Now by properties of comma squares and pullbacks, the pasting composite

Composite Square 1 p *X X (f/g) A B C f g q p

is also a comma square, and thus so is

Composite Square 2 p *X X g *Lan fX Lan fX B C g

where the upper 2-cell is opcartesian for the opfibration Lan fXC, and the map p *Xg *Lan fX is obtained from the universal property of the pullback square on the bottom. Again, it follows from general properties of pullbacks and comma squares that the top square in this latter diagram is also a comma square. Thus, by Lemma 2, its left-hand morphism is initial. But then the left-hand composite p *Xg *Lan fXB is an (initial, discrete-opfibration) factorization of p *X(f/g)B, and hence it exhibits the desired equivalence Lan qp *Xg *Lan fX.

Passing to 2-cell duals, we obtain:

Corollary

For a comma square as above, the canonical transformation Lan pq *f *Lan g between functors DFib(B)DFib(A) is an equivalence.

Revised on June 12, 2012 11:10:00 by Andrew Stacey? (129.241.15.200)