nLab
elegant Reedy category

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Homotopy theory

Contents

Definition

Definition

An elegant Reedy category is a Reedy category RR such that the following equivalent conditions hold

  • For every monomorphism ABA\hookrightarrow B of presheaves on RR and every xRx\in R, the induced map A x⨿ L xAL xBB xA_x \amalg_{L_x A} L_x B \to B_x is a monomorphism.

  • Every span of codegeneracy maps in R R_- has an absolute pushout in R R_-.

  • Both the following conditions hold:

    1. For every monomorphism ABA\hookrightarrow B of presheaves on RR, every nondegenerate element of AA remains nondegenerate in BB.

    2. Every element of a presheaf RR is a degeneracy of some nondegenerate element in a unique way.

Properties

The principal theorem about elegant Reedy categories is that the Reedy model structure on presheaves (i.e. contravariant diagrams) over an elegant Reedy category coincides with the injective model structure. This is not true for presheaves valued in any model category, only well-behaved ones. We clarify the necessary conditions by building up to this theorem in stages, adding hypotheses on the codomain of the presheaves as necessary.

Degeneracies are split

Lemma

If RR is elegant, then every codegeneracy map (i.e. morphism in R R_-) is a split epimorphism.

Proof

Let f:xyf:x\to y be a codegeneracy map; then the span yfxfyy \xleftarrow{f} x \xrightarrow{f} y has an absolute pushout, consisting of say g:yzg:y\to z and h:yzh:y\to z with gf=hfg f = h f. This absolute pushout is preserved by R(z,)R(z,-), so 1 zR(z,z)1_z\in R(z,z) must be the image under gg or hh of some map s:zys:z\to y; WLOG say it is gg, so we hav 1 z=gs1_z = g s. Now we have sh:yys h:y\to y and 1 y1_y such that gsh=h=h1 yg s h = h = h 1_y, and our absolute pushout is preserved by R(y,)R(y,-), so there must be a zigzag of elements in R(y,z)R(y,z) relating shs h to 1 y1_y. At one end of that zigzag, there must be a t:yxt:y\to x such that ft=1 yf t = 1_y; hence ff is split epi.

Degeneracies in subobjects

Lemma

Let RR be elegant and f:xyf:x\to y a codegeneracy in RR. Let MM be any category, and μ:AB\mu:A\to B a monomorphism in M R opM^{R^{\mathrm{op}}}. Then the following naturality square is a pullback:

A y A f A x μ y μ x B y B f B x\array{ A_y & \xrightarrow{A_f} & A_x \\ {}^{\mu_y}\downarrow & & \downarrow^{\mu_x} \\ B_y & \xrightarrow{B_f} & B_x }
Proof

This depends only on the fact that ff is split epi in RR. Let s:yxs:y\to x be a section of it, and let PP be the pullback of B fB_f and μ x\mu_x, with projections p:PA xp:P\to A_x and q:PB yq:P\to B_y with μ xp=B fq\mu_x p = B_f q, and an induced map ϕ:A yP\phi:A_y \to P such that pϕ=A fp \phi = A_f and qϕ=μ yq\phi = \mu_y.

We claim that A sp:PA yA_s p : P \to A_y is an inverse of ϕ\phi, making it an isomorphism. On the one hand we have A spϕ=A sA f=A fs=1A_s p \phi = A_s A_f = A_{f s} = 1. On the other, to show that ϕA sp=1\phi A_s p = 1 it suffices to show that pϕA sp=pp \phi A_s p = p and qϕA sp=qq \phi A_s p = q. For the first, since μ x\mu_x is monic, it suffices to show μ xpϕA sp=μ xp\mu_x p \phi A_s p = \mu_x p, and for that we have

μ xpϕA sp=B fqϕA sp=B fμ yA sp=B fB sμ xp=B fsμ xp=μ xp. \mu_x p \phi A_s p = B_f q \phi A_s p = B_f \mu_y A_s p = B_f B_s \mu_x p = B_{f s} \mu_x p = \mu_x p.

And for the second, we have

qϕA sp=μ yA sp=B sμ xp=B sB fq=B fsq=q.q \phi A_s p = \mu_y A_s p = B_s \mu_x p = B_s B_f q = B_{f s} q = q.
Lemma

Let RR be elegant, MM a category with pullback-stable colimits, and μ:AB\mu:A\to B a monomorphism in M R opM^{R^{\mathrm{op}}}. Then for any object xRx\in R, the following square is a pullback, where L xL_x denotes the Reedy latching object at xx:

L xA A x L xμ μ x L xB B x.\array{ L_x A & \to & A_x \\ {}^{L_x \mu}\downarrow & & \downarrow^{\mu_x} \\ L_x B & \to & B_x. }
Proof

The map L xBB xL_x B \to B_x is by definition the colimit in M/B xM/B_x of a diagram whose objects are morphisms of the form B f:B yB xB_f : B_y \to B_x, for ff a codegeneracy. By the Lemma 2, each of these pulls back along μ x\mu_x to A f:A yA xA_f : A_y \to A_x, forming the corresponding diagram whose colimit is L xAA xL_x A \to A_x, and by assumption the pullback preserves the colimit.

All presheaves are “Reedy monomorphic”

Lemma

Let RR be elegant and let MM be an infinitary-coherent category. Then for any xRx\in R and AM R opA\in M^{R^{\mathrm{op}}}, the map L xAA xL_x A \to A_x is a monomorphism.

Proof

We use the terminology from the page κ-ary exact category. Consider the sink with target A xA_x consisting of all morphisms A f:A yA xA_f : A_y \to A_x indexed by nonidentity codegeneracies ff with domain xx. By assumption, for any two such f:xyf:x\to y and f:xyf':x\to y' there is an absolute pushout g:yzg:y\to z and g:yzg':y'\to z. By absoluteness, A zA_z is the pullback A y× A xA yA_y \times_{A_x} A_y. Thus, the images of these absolute pushouts form the kernel of this sink.

Now L xAL_x A is the colimit of the diagram whose objects are A yA_y indexed by such f:xyf:x\to y and whose morphisms are A g:A yA yA_g: A_{y'} \to A_{y} for g:yyg:y\to y' a codegeneracy with gf=fg f = f'. In this case, by the universal property of pullback, we have a unique map from A yA_{y'} to A zA_z, where zz is the absolute pushout of ff and ff'. Thus, a cocone under the above kernel is also a cocone under this diagram, and the converse is easy to see. Hence, L xAL_x A is the quotient of the above kernel.

However, in any infinitary-regular category, the quotient of the kernel of a sink is exactly the extremal-epic / monic factorization of that sink. Therefore, the induced map L xAA xL_x A \to A_x is monic.

Reedy = injective

Theorem

If RR is elegant and MM is a Grothendieck topos, then for any xRx\in R and monomorphism μ:AB\mu:A\to B in M R opM^{R^{\mathrm{op}}}, the induced map L xB L xAA xB xL_x B \sqcup_{L_x A} A_x \to B_x is monic.

Proof

Since Grothendieck toposes are infinitary-coherent, by Lemma 4 L xBB xL_x B\to B_x is monic. By assumption A xB xA_x \to B_x is monic. And since Grothendieck toposes have pullback-stable colimits, by Lemma 3 the square

L xA A x L xμ μ x L xB B x.\array{ L_x A & \to & A_x \\ {}^{L_x \mu}\downarrow & & \downarrow^{\mu_x} \\ L_x B & \to & B_x. }

is a pullback. In other words, L xAL_x A is the intersection of the subobjects L xBL_x B and A xA_x of B xB_x. But in any coherent category, the pushout of two subobjects over their intersection is their union, and hence in particular a subobject of their common codomain.

Corollary

If RR is elegant and MM is a Cisinski model category, then the Reedy model structure on M R opM^{R^{\mathrm{op}}} coincides with the injective model structure.

Proof

By definition, they have the same weak equivalences, so it suffices to show that their classes of cofibrations coincide. But every Reedy cofibration in any Reedy model structure is an injective (i.e. objectwise) cofibration, and the converse is Theorem 1.

The most common application is when M=SSetM = SSet. Thus, for instance, every simplicial presheaf on an elegant Reedy category is Reedy cofibrant.

Examples

  • The simplex category Δ\Delta is an elegant Reedy category.

  • Joyal’s disk categories Θ n\Theta_n are elegant Reedy categories.

  • Every direct category is a Reedy category with no degeneracies, hence trivially an elegant one.

  • If XX is any presheaf on an elegant Reedy category RR, then the opposite of its category of elements (elX) op(el X)^{op} is again an elegant Reedy category. This is fairly easy to see from the fact that Set elXSet^{el X} is equivalent to the slice category Set R op/XSet^{R^{op}}/X.

  • Every EZ-Reedy category? is elegant.

Note that unlike the notion of Reedy category, the notion of elegant Reedy category is not self-dual: if RR is elegant then R opR^{op} will not generally be elegant.

References

Elegant Reedy categories are useful to model homotopy type theory.

Revised on October 24, 2014 20:20:42 by Urs Schreiber (185.26.182.38)