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distributivity of limits over colimits

Contents

Contents

Idea

In elementary algebra (or more generally in a ring), multiplication distributes over addition, meaning that

a(b+c)=(ab)+(ac) a\cdot (b+c) = (a\cdot b) + (a\cdot c)

and dually. Since multiplication and addition are a decategorification of products and coproducts in a category, we can categorify this relationship to say what it means for products to distribute over coproducts in any category. More generally, we can talk about arbitrary limits distributing over colimits. Note that this is not the same as limits commuting with colimits, although in some cases they are related.

Definitions

Explicitly using limits and colimits

Given small categories II and KK, for any diagram D:I×KCD\colon I\times K\to C in a category CC that admits all necessary limits and colimits we have a morphism

f:colim Klim IDlim Icolim KDf\colon colim_K lim_I D \to lim_I colim_K D

induced using the universal property of colim Kcolim_K and lim Ilim_I by the family of composites

lim ID(,k)D(i,k)colim KD(i,) lim_I D(-,k) \to D(i,k) \to colim_K D(i,-)

of the limit projection for iIi\in I and the colimit injection for kKk\in K. Recall that II-limits are said to commute with KK-colimits if and only if ff is an isomorphism for any DD.

Now we note that ff factors as a composition

colim Klim IDgcolim K Ilim IDhlim Icolim KD.colim_K lim_I D \xrightarrow{g} colim_{K^I} lim_I D' \xrightarrow{h} lim_I colim_K D.

Here K IK^I is the functor category and D:I×K ICD'\colon I\times K^I\to C is obtained from DD by precomposition with the functor (π 1,ev):I×K II×K(\pi_1,ev) : I\times K^I \to I\times K that sends a pair (i,s)(i,s) to (i,s(i))(i,s(i)). The map gg is induced by the diagonal functor KK IK\to K^I sending each kk to the corresponding constant functor, while hh is defined by the universal properties of colim K Icolim_{K^I} and lim Ilim_I from the family of composites

lim ID(,s)D(i,s)=D(i,s(i))colim KD(i,) lim_I D'(-,s) \to D'(i,s) = D(i,s(i)) \to colim_K D(i,-)

of the limit projection for iIi\in I and the colimit injection for c(i)Kc(i)\in K. We say that II-limits distribute over KK-colimits if hh is an isomorphism for any DD.

More generally, we can allow KK to vary with ii, becoming a functor K:ICatK\colon I\to Cat. Now instead of K IK^I we use the limit category lim IKlim_I K, and the diagram DD is defined on the Grothendieck construction (lax colimit) IK\int^I K. We recover the above definition when KK is a constant functor.

If 𝒦\mathcal{K} is a class of small categories rather than a single one, we say that II-limits distribute over 𝒦\mathcal{K}-colimits in a category CC if the corresponding map is an isomorphism for any K:ICatK:I\to Cat taking values in 𝒦\mathcal{K}. For instance, if 𝒦\mathcal{K} is the class of all discrete categories, we obtain the notion of II-limits distributing over coproducts, and so on. If instead II-limits distribute over KK-colimits for any single K𝒦K\in \mathcal{K}, we may say instead that “II-limits distribute over uniform KK-limits.”

I don’t know whether distributivity over uniform colimits is actually any weaker, but if it is, then the non-uniform version seems more correct.

For Kan extensions

Yet more generally, let u:IJu : I\to J be any Grothendieck fibration and v:KIv : K\to I a Grothendieck opfibration. Since fibrations are exponentiable functors, we can form the distributivity pullback

X p K v I q u Y r J \array{ X & \xrightarrow{p} & K & \xrightarrow{v} & I \\ ^q\downarrow &&&& \downarrow^u \\ Y && \xrightarrow{r} && J }

Thus YY is the dependent product Π IK\Pi_I K, and X=I× JΠ IKX = I\times_J \Pi_I K, while the functor p:XKp:X\to K is “evaluation”. Now in any sufficiently complete category, there is a Beck-Chevalley? isomorphism

r *Ran uRan qp *v *. r^* \circ Ran_u \cong Ran_q \circ p^* \circ v^*.

This is because any pullback of a fibration is an exact square. This isomorphism has a mate

Lan rRan qp *Ran uLan v Lan_r \circ Ran_q \circ p^* \to Ran_u \circ Lan_v

and we say that right Kan extensions along uu distribute over left Kan extensions along vv if this mate is an isomorphism.

If J=1J=1 then this reduces to the above non-uniform notion, where KIK\to I is the Grothendieck construction of a functor ICatI\to Cat. Moreover, since isomorphisms of JJ-diagrams are pointwise, right and left Kan extensions along fibrations and opfibrations respectively are given by fiberwise limits and colimits respectively, and dependent exponentials are preserved by pullback (the Beck-Chevalley condition), we can say that right Kan extensions along uu distribute over left Kan extensions along vv if and only if the above non-uniform distributivity condition holds for the fibers of vv over each object of JJ.

The formulation in terms of Kan extensions generalizes naturally to derivators, and can also be internalized to internal categories and fibrations in any locally cartesian closed category.

For sound doctrines

(See Section 6 in ABLR.)

Given a sound doctrine DD of limits, we have the associated notions of DD-limits, DD-filtered colimits, and DD-filtered cocompletion DInd(C)D-Ind(C) of a category CC together with the associated colimit functor colim:DInd(C)Ccolim\colon D-Ind(C)\to C.

We say that some class of limits distributes over DD-filtered colimits if the functor colim preserves these limits.

See Appendix A in AR for a comparison of this definition to the above explicit definition.

Relation to commutativity of limits and colimits

Observe that distributivity is asymmetric with respect to limits and colimits, whereas commutativity of limits and colimits is a symmetric notion. In some cases, however, distributivity and commutativity are equivalent. This happens exactly when the above functor gg is an isomorphism, which is to say that the diagonal map KK IK\to K^I is a final functor.

Thus, for instance, finite limits distribute over (uniform) filtered colimits if and only if finite limits commute with filtered colimits. The same is true for finite products and sifted colimits.

Examples

Finite products

The distributivity of finite products over arbitrary coproducts is the most classical version. See distributivity for monoidal structures for various generalizations.

More generally, finite products distribute over arbitrary colimits in any cartesian closed category, such as Set.

Filtered colimits

If DD is the doctrine of finite limits, then DD-filtered colimits are precisely filtered colimits and the DD-filtered cocompletion of CC is Ind(C)Ind(C), the category of Ind-objects in CC.

According to Definition 5.11 in ALR, a category is precontinuous if it admits small limits and filtered colimits, and small limits distribute over filtered colimits, i.e., the functor colim:Ind(C)Ccolim: Ind(C)\to C is continuous.

In particular, any locally finitely presentable category, equivalently the category of algebras over some finitary essentially algebraic theory, is precontinuous. See Theorem 5.13 and Lemma 5.14 in ALR.

In fact, as shown in ALR, precontinuous categories with continuous functors that preserve filtered colimits form the accessible equational hull? of the bicategory of locally finitely presentable categories and continuous functors that preserve filtered colimits.

Sifted colimits

If DD is the doctrine of finite products, then DD-filtered colimits are precisely sifted colimits and the DD-filtered cocompletion of CC is Sind(C)Sind(C), the nonabelian derived category? of CC.

According to Definition 4.5 in ALRalg, a category is algebraically exact if it admits small limits and sifted colimits, and small limits distribute over sifted colimits, i.e., the functor colim:Sind(C)Ccolim: Sind(C)\to C is continuous. In particular (Example 5.3 in ALRalg), any variety of algebras is an algebraically exact category. In particular, this includes Set.

In fact, as shown in ALRalg, algebraically exact categories with continuous functors that preserve sifted colimits form the accessible equational hull? of the bicategory of varieties of algebras and continuous functors that preserve sifted colimits.

Indexed products and coproducts

When the definition for Kan extensions is internalized to internal categories and fibrations, and then specialized to the case of discrete categories, the notion of distributivity of dependent products over dependent sums encoded by a distributivity pullback becomes the statement that in the codomain fibration, indexed products? distribute over indexed coproducts? in this sense.

Pullback-stable colimits

As a nontrivial example of the K:ICatK\colon I\to Cat case, let II be the walking cospan 0120 \to 1 \leftarrow 2, and let K(0)=JK(0) = J for some category JJ while K(1)=K(2)=*K(1)=K(2)=\ast, the terminal category. Then IK\int^I K is JJ together with a new terminal object 11 and a new object 22 having only a map to 11, and so a diagram D: IKCD: \int^I K \to C consists of a JJ-diagram D 0D_0 in a slice category C/ZC/Z over some object ZZ along with a morphism f:ZYf:Z\to Y. The limit category lim IK\lim_I K is isomorphic to JJ, and the functor D:I×lim IKCD'\colon I\times \lim_I K\to C has D(0,j)=D 0(j)D'(0,j) = D_0(j), with D(1,j)=ZD'(1,j) =Z and D(2,j)=YD'(2,j)=Y. Therefore, colim lim IKlim ID\colim_{\lim_I K} \lim_I D' is the JJ-colimit of the pullback of D 0D_0 along ff, while lim iIcolim K(i)D\lim_{i\in I} \colim_{K(i)} D is the pullback along ff of the colimit of D 0D_0. Thus, the distributivity condition for this KK holds if and only if JJ-colimits are universal (stable under pullback).

References

Last revised on January 15, 2018 at 17:59:42. See the history of this page for a list of all contributions to it.