Definition 13.14.2. Assumptions and notation as in Situation 13.14.1. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$.

1. we say the right derived functor $RF$ is defined at $X$ if the ind-object

$(X/S) \longrightarrow \mathcal{D}', \quad (s : X \to X') \longmapsto F(X')$

is essentially constant1; in this case the value $Y$ in $\mathcal{D}'$ is called the value of $RF$ at $X$.

2. we say the left derived functor $LF$ is defined at $X$ if the pro-object

$(S/X) \longrightarrow \mathcal{D}', \quad (s: X' \to X) \longmapsto F(X')$

is essentially constant; in this case the value $Y$ in $\mathcal{D}'$ is called the value of $LF$ at $X$.

By abuse of notation we often denote the values simply $RF(X)$ or $LF(X)$.

[1] For a discussion of when an ind-object or pro-object of a category is essentially constant we refer to Categories, Section 4.22.

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