limits commute with limits

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.

**(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 βcommuteβ 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.

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.

**(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.