nLab limits commute with limits

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.

Related statements

• preserved limit

• adjoints preserve (co-)limits

• hom-functor preserves limits

• commutativity of limits and colimits

References

e.g.

• Saunders MacLane, §IX.2 in: Categories for the Working Mathematician, Graduate texts in mathematics, Springer (1971) [doi:10.1007/978-1-4757-4721-8]