The Stacks project

Remark 4.22.4. Let $\mathcal{C}$ be a category. There exists a big category $\text{Ind-}\mathcal{C}$ of ind-objects of $\mathcal{C}$. Namely, if $F : \mathcal{I} \to \mathcal{C}$ and $G : \mathcal{J} \to \mathcal{C}$ are filtered diagrams in $\mathcal{C}$, then we can define

\[ \mathop{\mathrm{Mor}}\nolimits _{\text{Ind-}\mathcal{C}}(F, G) = \mathop{\mathrm{lim}}\nolimits _ i \mathop{\mathrm{colim}}\nolimits _ j \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(F(i), G(j)). \]

There is a canonical functor $\mathcal{C} \to \text{Ind-}\mathcal{C}$ which maps $X$ to the constant system on $X$. This is a fully faithful embedding. In this language one sees that a diagram $F$ is essentially constant if and only if $F$ is isomorphic to a constant system. If we ever need this material, then we will formulate this into a lemma and prove it here.

Comments (2)

Comment #9768 by on

For the interested reader, a reference for these ideas is [KS, 2.6 (specially eq. (2.6.4)), 6.1, 6.2]. An older reference is [SGA4, Exposé I, 8.4.4].

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  • 10 comment(s) on Section 4.22: Essentially constant systems

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