Michael Shulman
coherent 2-category

Definition

A 2-category is called coherent if

  1. it has finite limits,
  2. finite jointly-eso families are stable under pullback, and
  3. every finitary 2-polycongruence which is a kernel can be completed to an exact 2-polyfork.

Here a family {f i:A iB} is said to be jointly-eso if whenever m:CB is ff and every f i factors through m (up to isomorphism), then m is an equivalence.

Likewise, we have infinitary coherent 2-categories in which “finite” in the second two conditions is replaced by “small.”

Examples

  • Cat is coherent.

  • A 1-category is coherent as a 2-category iff it is coherent as a 1-category.

  • A (0,1)-category (= poset) is coherent iff it is a distributive lattice, and infinitary-coherent iff it is a frame.

Factorizations

The following are proven just like their unary analogues in a regular 2-category.

Lemma

(Street’s Lemma) In a finitely complete 2-category where finite jointly-eso families are stable under pullback, if {e i:A iB} is finite and jointly-eso and n:BC is such that the induced functor ker(e i)ker(ne i) is an equivalence, then n is ff.

Theorem

A 2-category is coherent if and only if

  1. it has finite limits,
  2. finite jointly-eso families are stable under pullback,
  3. every finite family {f i} factors as f i=me i where m is ff and {e i} is jointly-eso, and
  4. every jointly-eso family is the quotient of its kernel.

Of course, there are infinitary versions. In particular, we conclude that in a coherent (resp. infinitary-coherent) 2-category, the posets Sub(X) have finite (resp. small) unions that are preserved by pullback.

Colimits

Lemma

A coherent 2-category has a strict initial object; that is an initial object 0 such that any morphism X0 is an equivalence.

Proof

The empty 2-congruence is the kernel of the empty family (over any object), so it must have a quotient 0, which is clearly an initial object. The empty family over 0 is jointly-eso, so for any X0 the empty family over X is also jointly-eso; but this clearly makes X initial as well.

Two ffs m:AX and n:BX are said to be disjoint if the comma objects (m/n) and (n/m) are initial objects. If initial objects are strict, then this implies that the pullback A× XB is also initial, but it is strictly stronger already in Pos.

Lemma

In a coherent 2-category, if AX and BX are disjoint subobjects, then their union AB in Sub(X) is also their coproduct A+B.

Proof

If A and B are disjoint subobjects of X, then the kernel of {AX,BX} is the disjoint union of ker(A) and ker(B). Therefore, a quotient of it (which is a union of A and B in Sub(X)) will be a coproduct of A and B.

A coproduct A+B in a 2-category is disjoint if A and B are disjoint subobjects of A+B. We say a coherent 2-category is positive if any two objects have a disjoint coproduct. By Lemma 3, this is equivalent to saying that any two objects can be embedded as disjoint subobjects of some other object. Disjoint coproducts in a coherent 2-category are automatically stable under pullback, so a positive coherent 2-category is necessarily extensive. Conversely, we have:

Lemma

A regular and extensive 2-category is coherent (and positive).

In the presence of finite coproducts, a family {e i:A iB} is jointly-eso iff iA iB is eso; thus regularity and universal coproducts imply that finite jointly-eso families are stable under pullback. And assuming extensivity, any 2-polycongruence {C ij}{C i} gives rise to an ordinary 2-congruence ijC ij iC i. Likewise, 2-polyforks {C ij}{C i}X correspond to 2-forks ijC ij iC iX, and the property of being a kernel or a quotient is preserved; thus regularity implies coherency.

Preservation

If K is coherent, then easily so are K co, disc(K), gpd(K), pos(K), and Sub(1). Moreover, we have:

Theorem

If K is a coherent 2-category, so are the fibrational slices Opf(X) and Fib(X) for any XK.

Revised on February 17, 2009 17:58:52 by Mike Shulman (75.3.140.11)