nLab
absolute pushout

Contents

Contents

Idea

An absolute pushout is a pushout which is preserved by any functor whatsoever. In general this happens because the pushout is a pushout for purely “diagrammatic” reasons. See absolute colimit for more.

Definitions

Definition

A particular pushout diagram in a particular category CC is an absolute pushout if it is preserved by every functor with domain CC.

Equivalently, since the Yoneda embedding is the free cocompletion of CC:

Definition

A particular pushout diagram in a particular category CC is an absolute pushout if it is preserved by the Yoneda embedding C[C op,Set]C \hookrightarrow [C^{op},Set].

Split pushouts

We propose the following notion of split pushout.

Definition

A commutative square

defines a split pushout if there exist sections ps=1p s = 1, qt=1q t = 1, mu=1m u = 1

so that pt=unp t = u n.

Split pushouts are absolute pushouts

Proposition

Split pushouts are absolute pushouts.

Proof

Note that split pushouts are preserved by arbitrary functors, so it suffices to show that a split pushout is a pushout in the category in which it lives. To that end consider, a cone under the span (p,q)(p,q):

Upon composing with the sections to qq and mm

we see that cc factors through the claimed pushout PP as c=(bu)nc = (b u) n. We must verify that bb also factors as b=(bu)mb = (b u) m. Since pp is an epimorphism, it suffices to prove that bp=bumpb p = b u m p, which follows easily:

bp=cq=bunq=bump. b p = c q = b u n q = b u m p.

This produces the desired factorization. Finally, since mm is an epimorphism, such factorizations are unique.

Note the proof that a split pushout defines a pushout square in the category in which it lives did not require pp to be a split epimorphism. However, arbitrary functors do not preserve epimorphisms. They do however preserve split epimorphisms, and thus the section guarantees that the image of pp will define an epimorphism in any category.

General characterization

Theorem

A commutative square

is an absolute pushout if and only if either there exist

  1. A section u:PBu:P\to B, such that mu=1 Pm u = 1_P.
  2. Morphisms r 1,,r k:BAr_1,\dots,r_k : B \to A and s 1,,s k:BAs_1,\dots,s_k : B\to A, for some k1k\ge 1, such that ps 1=1 Bp s_1 = 1_B, qs i=qr iq s_i = q r_i for all ii, pr i=ps i+1p r_i = p s_{i+1} for all i<ki\lt k, and pr k=ump r_k = u m.
  3. Morphisms t 1,,t +1:CAt_1,\dots,t_{\ell+1} : C \to A and v 1,,v :CAv_1,\dots,v_{\ell} : C\to A, for some 0\ell \ge 0, such that qt 1=1 Cq t_1 = 1_C, pt i=pv ip t_i = p v_i for all i<i\lt \ell, qv i=qt i+1q v_i = q t_{i+1} for all ii\le \ell, and pt +1=unp t_{\ell+1} = u n.

or the transpose thereof (i.e. interchanging BB with CC and so on).

Proof

For “if”, suppose given a commutative square

Since mm is a (split) epimorphism (by uu), any factorization of this square through the given one will be unique, so it suffices to show that such a factorization exists. Define x=bu:PXx = b u:P\to X. Then we have

xm=bum=bpr k=cqr k=cqs k=bps k=btr k1==bps 1=b x m = b u m = b p r_k = c q r_k = c q s_k = b p s_k = b t r_{k-1} = \dots = b p s_1 = b

and

xn=bun=bpt =cqt =cqv 1=bpv 1=bpt 1==cqt 1=c. x n = b u n = b p t_\ell = c q t_\ell = c q v_{\ell-1} = b p v_{\ell-1} = b p t_{\ell-1} = \dots = c q t_1 = c.

The transposed case is of course dual.

Conversely, suppose the given square is an absolute pushout. Thus, in particular the induced square

is a pushout in SetSet. Thus, in particular, the function hom(P,B)+hom(P,C)hom(P,P)\hom(P,B)+\hom(P,C) \to \hom(P,P) is surjective, and thus for 1 Phom(P,P)1_P\in \hom(P,P) there must be either a u:PBu:P\to B such that mu=1 Pm u = 1_P or a u:PCu':P\to C such that nu=1 Pn u' = 1_P. WLOG assume the former.

Now the induced square

is also a pushout in SetSet. We have two elements 1 B,umhom(B,B)1_B, u m \in \hom(B,B) that become equal in hom(B,P)\hom(B,P) (since m1 B=m=(1 P)m=mumm 1_B = m = (1_P) m = m u m), and in a pushout in SetSet this means they must be related by a zigzag of elements of the vertex hom(B,A)\hom(B,A). Unraveling this explicitly produces the morphisms r i,s ir_i,s_i.

Similarly, from the induced square

and the elements 1 Chom(C,C)1_C\in\hom(C,C) and snhom(C,B)s n \in \hom(C,B), we obtain the morphisms t i,v it_i,v_i.

In particular, when k=1k=1 and =0\ell=0, the above data reduces to

  1. A section u:PBu:P\to B, such that mu=1 Pm u = 1_P.
  2. Morphisms r,s:BAr,s : B \to A such that ps=1 Bp s = 1_B, qs=qrq s = q r, and pr=ump r = u m.
  3. A morphism t:CAt : C \to A such that qt=1 Cq t = 1_C and pt=unp t = u n.

This is precisely the data of the above-defined notion of split pushout, together with the additional morphism r:BAr:B\to A such that qr=qsq r = q s and pr=ump r = u m. However, given a split pushout as above we can define r=tqsr = t q s and check qr=qtqs=qsq r = q t q s = q s and pr=ptqs=unqs=umps=ump r = p t q s = u n q s = u m p s = u m. This gives another proof that any split pushout is an absolute pushout.

Examples

In their study of generalized Reedy categories, Berger and Moerdijk introduce the notion of an Eilenberg-Zilber category, one of the axioms of which demands that spans of split epimorphisms admit absolute pushouts. In practice, this seems to be the case because the pushout of these split epimorphisms is a split epimorphism as above, often with an additional section vv of nn satisfying the additional equation that vm=qsv m = q s.

References

The general characterization of absolute pushouts appears as Proposition 5.5 in:

  • Robert Paré, Robert On absolute colimits. J. Algebra 19 (1971), 80–95.

The Berger-Moerdijk definition of an Eilenberg-Zilber category appears in:

Last revised on April 13, 2021 at 16:12:54. See the history of this page for a list of all contributions to it.