# nLab commutativity of limits and colimits

### Context

#### Limits and colimits

limits and colimits

# Contents

## Idea

In general, limits and colimits do not commute.

It is therefore of interest to list the special conditions under which certain limits do commute with certain colimits.

## Filtered colimits commute with finite limits

For $C$ a small filtered category, the functor $colim_C : [C,Set] \to Set$ commutes with finite limits.

More in detail, let

• $C$ be a small filtered category

• $D$ be a finite category (or more generally an L-finite category);

• $F : C \times D^{op} \to Set$ a functor;

then the canonical morphism

$colim_C lim_D F \stackrel{\simeq}{\to} lim_D colim_C F$

is an isomorphism.

In fact, $C$ is a filtered category if and only if this is true for all finite $D$ and all functors $F : C \times D^{op} \to Set$.

## Sifted colimits commute with finite products

Similarly to the example of filtered limits, for $C$ a small sifted category, the functor $colim_C : [C,Set] \to Set$ commutes with finite products. In fact, this is usually taken to be the definition of a sifted category, and then a theorem of Gabriel and Ulmer characterizes sifted categories as those for which the diagonal functor $C \to C \times C$ is a final functor.

For more on this see at distributivity of products and colimits.

## Taking orbits under the action of a finite group commutes with cofiltered limits

More precisely, if $G$ is a finite group, $C$ is a small cofiltered category and $F : C \to G-Set$ is a functor, the canonical map

$(\lim F)/G \to \lim_{j \in F} (F(j)/G)$

is an isomorphism. This fact is mentioned by André Joyal in Foncteurs analytiques et espèces de structures; a proof can be found here.

## Coproducts commute with connected limits

###### Proposition

Let $A$ be a set, $C$ a connected category, and $F \colon C\times A \longrightarrow Set$ a functor. Then the canonical morphism

$\underset{a\in A}{\coprod} \underset{\longleftarrow}{\lim}_{c\in C} F(c,a) \longrightarrow \underset{\longleftarrow}{\lim}_{c\in C} \coprod_{a\in A}F(c,a)$

is an isomorphism. This remains true if Set is replaced by any Grothendieck topos.

More generally, if $\mathbf{H}$ is an (∞,1)-topos, $A$ is an n-groupoid, and $C$ is a small (∞,1)-category whose classifying space is n-connected, then $C$-limits commute with $A$-colimits in $\mathbf{H}$. This follows from the fact that the colimit functor $\mathbf{H}^A\to\mathbf{H}$ induces an equivalence of (∞,1)-topoi $\mathbf{H}^A\simeq \mathbf{H}_{/A}$. For example, if $C$ is a cofiltered (∞,1)-category or even a cosifted (∞,1)-category, then the classifying space of $C$ is weakly contractible and hence $C$-limits commute with $A$-colimits in $\mathbf{H}$ for any ∞-groupoid $A$.

## Classes of limits and sound doctrines

In general, for any class of limits $\Phi$, one may consider the class of all colimits that commute with $\Phi$-limits and dually. These classes of limits and colimits share many of the properties of the above examples, especially when $\Phi$ is a sound doctrine.

## Certain colimits are stable by base change

Let $C$ be a category with pullbacks and colimits of shape $D$.

We say that colimits of shape $D$ are stable by base change or stable under pullback if for every functor $F : D \to C$ and for all pullback diagrams of the form

$\array{ (colim_D F) \times_Z Y &\to& colim_D F \\ \downarrow && \downarrow \\ Y &\to & Z }$

the canonical morphism

$colim_{d \in D} (F(d) \times_Z Y) \stackrel{\simeq}{\to} (colim_D F) \times_Z Y$

is an isomorphism.

All colimits are stable under base change in for instance

• $C =$ Set;
• hence for $C =$ a presheaf category $[S^{op},Set]$ (since colimits in such $C$ are computed objectwise in $Set$), see limits and colimits by example);
• more generally, any topos;

but not in for instance

• $C =$ Ab.

Remark

In topos theory and (∞,1)-topos theory one says that colimits are universal if they are preserved under pullback.

Revised on February 19, 2017 06:38:07 by Mike Shulman (76.167.222.204)