restricted product




Given a set SS and a collection {K sι sX s} sS\{K_s \stackrel{\iota_s}{\hookrightarrow} X_s\}_{s \in S} of morphisms in some category (typically monomorphisms), then the restricted product sS X s\underset{s\in S}{\prod}^\prime X_s is vaguely the like the actual product sSX s\underset{s\in S}{\prod} X_s of all the X sX_s, but subject to the restriction that for each element (x s) sS(x_s)_{s \in S} all but a finite number of components x sx_s are in the image of ι s\iota_s.


Assume that the ambient category 𝒞\mathcal{C} has ordinary products. Write 𝒫 finS\mathcal{P}_{fin} S for the poset of finite subsets of SS and consider the functor

X K:𝒫 fin𝒞 X^K \colon \mathcal{P}_{fin}\longrightarrow \mathcal{C}

which is given on a finite subset USU \subset S by the products

UX U K(uUX u)×(sSUK s) U \mapsto X^K_U \coloneqq \left(\underset{u \in U}{\prod} X_u\right) \times \left( \underset{s \in S-U}{\prod} K_s \right)

and which sends an inclusion U 1U 2U_1 \hookrightarrow U_2 of subsets to the evident morphism between these products whose components f sf_s are the identity for sS(U 2U 1)s \in S-(U_2-U_1) and are ι s\iota_s for sU 2U 1s \in U_2-U_1.

Then the restricted product is the (filtered) colimit over this functor

sS X slimU𝒫 finSX U K. \underset{s\in S}{\prod}^\prime X_s \coloneqq \underset{\underset{U \in \mathcal{P}_{fin}S}{\longrightarrow}}{\lim} X^K_U \,.



Last revised on January 14, 2021 at 04:59:51. See the history of this page for a list of all contributions to it.