nLab distributivity of limits over colimits

Idea

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

$$a \cdot (b + c) \;=\; (a\cdot b) + (a\cdot c)$$

and dually. Since multiplication and addition are a decategorifications of products and coproducts in category theory, it is suggestive to ask for products to distribute over coproducts in any category.

More generally, one may consider arbitrary limits distributing over colimits. Note that this is not the same notion as limits commuting with colimits, although in some cases they are related (see at permuting limits and colimits for more).

Definitions

Explicitly using limits and colimits

Given small categories $I$ and $K$, for any diagram

$$D \,\colon\, I \times K \longrightarrow C$$

in a category $C$ that admits all necessary limits and colimits we have a morphism

$$f \,\colon\, colim_K lim_I D \longrightarrow lim_I colim_K D$$

induced using the universal property of $colim_K$ and $lim_I$ by the family of composites

$$lim_I D(-,k) \longrightarrow D(i,k) \longrightarrow colim_K D(i,-)$$

of the limit projection for $i\in I$ and the colimit injection for $k\in K$.

Recall that $I$-limits are said to commute with $K$-colimits if and only if $f$ is an isomorphism for all $D$.

Now we note that $f$ factors as a composition

$$colim_K lim_I D \xrightarrow{g} colim_{K^I} lim_I D' \xrightarrow{h} lim_I colim_K D.$$

Here $K^I$ is the functor category and $D'\colon I\times K^I\to C$ is obtained from $D$ by precomposition with the functor $(\pi_1,ev) : I\times K^I \to I\times K$ that sends a pair $(i,s)$ to $(i,s(i))$. The map $g$ is induced by the diagonal functor $K\to K^I$ sending each $k$ to the corresponding constant functor, while $h$ is defined by the universal properties of $colim_{K^I}$ and $lim_I$ from the family of composites

$$lim_I D'(-,s) \to D'(i,s) = D(i,s(i)) \to colim_K D(i,-)$$

of the limit projection for $i\in I$ and the colimit injection for $c(i)\in K$. We say that $I$-limits distribute over $K$-colimits if $h$ is an isomorphism for all $D$.

More generally, we can allow $K$ to vary with $i$, becoming a functor $K\colon I\to Cat$. Now instead of $K^I$ we use the limit category $lim_I K$, and the diagram $D$ is defined on the Grothendieck construction (lax colimit) $\int^I K$. We recover the above definition when $K$ is a constant functor.

Example

In the simple example $A \times (B + C) \cong (A \times B) + (A \times C)$ alluded to in the introduction,

• $I$ is a two-element discrete category,

• $K(0)$ is the terminal category

• $K(1)$ is again a two-element discrete category.

The Grothendieck construction $\int^I K$ is a three-element discrete category, and the diagram $D \colon \int^I K \to \mathcal{C}$ picks the objects $A$, $B$ and $C$.

If $\mathcal{K}$ is a class of small categories rather than a single one, we say that $I$-limits distribute over $\mathcal{K}$-colimits in a category $C$ if the corresponding map is an isomorphism for any $K \,\colon\, I\to Cat$ taking values in $\mathcal{K}$.
For instance, if $\mathcal{K}$ is the class of all discrete categories, we obtain the notion of $I$-limits distributing over coproducts, and so on.

If instead $I$-limits distribute over $K$-colimits for any single $K\in \mathcal{K}$, we may say instead that “$I$-limits distribute over uniform $K$-limits.”

The author of the previous paragraph doesn’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 $u : I\to J$ be any Grothendieck fibration and $v : K\to I$ a Grothendieck opfibration. Since fibrations are exponentiable functors, we can form the distributivity pullback

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

Thus $Y$ is the dependent product $\Pi_I K$, and $X = I\times_J \Pi_I K$, while the functor $p:X\to K$ is “evaluation”. Now in any sufficiently complete category, there is a Beck-Chevalley isomorphism

$$r^* \circ Ran_u \cong Ran_q \circ p^* \circ v^*.$$

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

$$Lan_r \circ Ran_q \circ p^* \to Ran_u \circ Lan_v$$

and we say that right Kan extensions along $u$ distribute over left Kan extensions along $v$ if this mate is an isomorphism.

If $J=1$ then this reduces to the above non-uniform notion, where $K\to I$ is the Grothendieck construction of a functor $I\to Cat$. Moreover, since isomorphisms of $J$-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 $u$ distribute over left Kan extensions along $v$ if and only if the above non-uniform distributivity condition holds for the fibers of $v$ over each object of $J$.

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 $D$ of limits, we have the associated notions of $D$-limits, $D$-filtered colimits, and $D$-filtered cocompletion $D-Ind(C)$ of a category $C$ together with the associated colimit functor $colim\colon D-Ind(C)\to C$.

We say that some class of limits distributes over $D$-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 in the special case of distributivity of filtered colimits over small limits in locally finitely presentable categories and distributivity of sifted colimits over small limits in varieties of algebras.

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 $g$ is an isomorphism, which is to say that the diagonal map $K\to K^I$ 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.

Distributivity of (∞,1)-limits over (∞,1)-colimits

The condition that small (∞,1)-limits distribute over (∞,1)-filtered (∞,1)-colimits precisely characterizes compactly assembled (∞,1)-categories among all presentable (∞,1)-categories.

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

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

According to Theorem 2.1 in ARVloc, a category is precontinuous if and only if it has small limits and filtered colimits, filtered colimits commute with finite limits, and small products distribute over filtered colimits.

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

According to Definition 4.5 in ALRalg, a category is algebraically exact if it admits small limits and sifted colimits, and the functor $colim: Sind(C)\to C$ 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 $K\colon I\to Cat$ case, let $I$ be the walking cospan $0 \to 1 \leftarrow 2$, and let $K(0) = J$ for some category $J$ while $K(1)=K(2)=\ast$, the terminal category. Then $\int^I K$ is $J$ together with a new terminal object $1$ and a new object $2$ having only a map to $1$, and so a diagram $D: \int^I K \to C$ consists of a $J$-diagram $D_0$ in a slice category $C/Z$ over some object $Z$ along with a morphism $f:Z\to Y$. The limit category $\lim_I K$ is isomorphic to $J$, and the functor $D'\colon I\times \lim_I K\to C$ has $D'(0,j) = D_0(j)$, with $D'(1,j) =Z$ and $D'(2,j)=Y$. Therefore, $\colim_{\lim_I K} \lim_I D'$ is the $J$-colimit of the pullback of $D_0$ along $f$, while $\lim_{i\in I} \colim_{K(i)} D$ is the pullback along $f$ of the colimit of $D_0$. Thus, the distributivity condition for this $K$ holds if and only if $J$-colimits are universal (stable under pullback).

For formal filtered limits and colimits

In the case of presentation of ind-objects and pro-objects as formal small filtered colimits and cofiltered limits of (co)representables, a corollary is that there is a functor

$$Ind(Pro(C))\to Pro(Ind(C)).$$

In a number of interesting cases this functor is inclusion.

Related entries

• commutativity of limits and colimits, permuting limits and colimits
• distributivity pullback
• distributive law
• distributivity for monoidal structures
• universal colimit

References

The original definition is due to Jan-Erik Roos, see Definition 1 in

• Jan-Erik Roos, Introduction à l’étude de la distributivité des foncteurs lim par rapport aux lim dans les catégories des faisceaux (topos), Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences 259 (1964), 969–972 (PDF),

which is the first in a series of 3 papers, the others being

• Jan-Erik Roos, Sur la distributivité des foncteurs lim par rapport aux lim dans les catégories des faisceaux (topos), Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences 259 (1964), 1605–1608

• Jan-Erik Roos, Complément à l’étude de la distributivité des foncteurs lim par rapport aux lim dans les catégories des faisceaux (topos), Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences 259 (1964), 1801–1804.

Additional references:

• Jiří Adámek, Jiří Rosický, Enrico Vitale, Algebraic theories. A categorical introduction to general algebra CUP 2011.

• Jiří Adámek, William Lawvere, Jiří Rosický, Continuous Categories Revisited TAC 11, 2003.

• Jiří Adámek, William Lawvere, Jiří Rosický, How algebraic is algebra TAC 8, 2001.

• Jiří Adámek, Francis Borceux, Stephen Lack, Jiří Rosický, A classification of accessible categories JPAA 175, 2002.

• Jiří Adámek, Jiří Rosický, Algebra and local presentability: how algebraic are they? (A survey).

• Jiří Adámek, Jiří Rosický, Enrico Vitale,

  On Algebraically Exact Categories and Essential Localizations of Varieties

• Richard Garner and Stephen Lack, Lex colimits, JPAA 216.6, 2012.