# nLab limits commute with limits

Contents

### Context

#### Limits and colimits

limits and colimits

category theory

# Contents

## Idea

One of the basic facts of category theory is that the order of two limits (a kind of universal construction) does not matter, up to isomorphism.

## Statement

###### Proposition

(limits commute with limits)

Let $\mathcal{D}$ and $\mathcal{D}'$ be small categories and let $\mathcal{C}$ be a category which admits limits of shape $\mathcal{D}$ as well as limits of shape $\mathcal{D}'$. Then these limits “commute” with each other, in that for $F \;\colon\; \mathcal{D} \times {\mathcal{D}'} \to \mathcal{C}$ a functor (hence a diagram of shape the product category), with corresponding adjunct functors

${\mathcal{D}'} \overset{F_{\mathcal{D}}}{\longrightarrow} [\mathcal{D},\mathcal{C}] \phantom{AAA} {\mathcal{D}} \overset{F_{\mathcal{D}'}}{\longrightarrow} [{\mathcal{D}'}, \mathcal{C}]$

we have that the canonical comparison morphism

(1)$lim F \simeq lim_{\mathcal{D}} (lim_{\mathcal{D}'} F_{\mathcal{D}} ) \simeq lim_{\mathcal{D}'} (lim_{\mathcal{D}} F_{\mathcal{D}'} )$

is an isomorphism.

###### Proof

Since the limit-construction is the right adjoint functor to the constant diagram-functor, this is a special case of right adjoints preserve limits.

See limits and colimits by example for what formula (1) says for instance for the special case $\mathcal{C} =$ Set.

###### Remark

(general non-commutativity of limits with colimits)

In general limits do not commute with colimits. But under a number of special conditions of interest they do. Special cases and concrete examples are discussed at commutativity of limits and colimits.

Last revised on April 7, 2021 at 02:59:50. See the history of this page for a list of all contributions to it.