nLab limits commute with limits



Limits and colimits

Category theory



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 π’Ÿ\mathcal{D} and π’Ÿβ€²\mathcal{D}' be small categories and let π’ž\mathcal{C} be a category which admits limits of shape π’Ÿ\mathcal{D} as well as limits of shape π’Ÿβ€²\mathcal{D}'. Then these limits β€œcommute” with each other, in that for F:π’ŸΓ—π’Ÿβ€²β†’π’žF \;\colon\; \mathcal{D} \times {\mathcal{D}'} \to \mathcal{C} a functor (hence a diagram of shape the product category), with corresponding adjunct functors

π’Ÿβ€²βŸΆF π’Ÿ[π’Ÿ,π’ž]AAAπ’ŸβŸΆF π’Ÿβ€²[π’Ÿβ€²,π’ž] {\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)limF≃lim π’Ÿ(lim π’Ÿβ€²F π’Ÿ)≃lim π’Ÿβ€²(lim π’ŸF π’Ÿβ€²) 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 π’ž=\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 May 1, 2023 at 06:11:51. See the history of this page for a list of all contributions to it.