I asked [the same question](https://math.stackexchange.com/q/2181688/660) a week ago on Mathematics Stackexchange but got no answer.


What would be a simple example of an additive functor $F:\mathcal C\to\mathcal C'$ of abelian categories such that the right derived functor
$$
RF:\text D(\mathcal C)\to\text D(\mathcal C')
$$ 
does **not** exist?


My reference for the notions involved in this post is the book **Categories and Sheaves** by Kashiwara and Schapira.


Here is a reminder of the definition of a right derived functor $RF$ in the above setting:


Let $\text K(F):\text K(\mathcal C)\to\text K(\mathcal C')$ be the triangulated functor induced by $F$ between the homotopy categories, let $X$ be in $\text K(\mathcal C)$, assume that the colimit
$$
\operatorname\*{colim}\_{X\to Y}\ \text K(F)(Y),\tag1
$$ 
where $X\to Y$ runs over all the quasi-isomorphisms out of $X$ in $\text K(\mathcal C)$, exists in $\text D(\mathcal C')$, and denote this colimit by $RF(X)$. The right derived functor $RF$ of $F$ is *defined at* $X\in\text D(\mathcal C)$ if, for any functor $G:\text D(\mathcal C')\to\mathcal A$, the colimit
$$
\operatorname\*{colim}\_{X\to Y}\ G(\text K(F)(Y))
$$ 
exists in $\mathcal A$ and coincides with $G(RF(X))$. The right derived functor $RF$ of $F$ *exists* if $RF$ is defined at $X$ for all $X$ in $\text D(\mathcal C)$. 


The ideal would be to have an example of a pair $(F,X)$, where $F:\mathcal C\to\mathcal C'$ is an additive functor of abelian categories and $X$ is an object of $\text D(\mathcal C)$, such that (1) does **not** exist in $\text D(\mathcal C')$.



Example of an additive functor admitting no right derived functor
Let ${\cal C}$ be the category of finite dimensional ${\bf Z}/2$-vector spaces equipped with a ${\bf Z}/2$ action, let ${\cal C'}$ be the category of finite dimensional ${\bf Z}/2$-vector spaces and let $F: {\cal C} \to {\cal C'}$ be the fixed subspace functor $V \mapsto V^{{\bf Z}/2}$. Consider the chain complex $X$ such that $X\_n = {\bf Z}/2$ with trivial action in each degree $n \in {\bf Z}$ and with trivial differentials. Then the colimit $RF(X) = {\rm colim}\_{X \to Y}Y^{{\bf Z}/2}$ does exist as a complex of vector spaces (since this category has all limits and colimits), and $RF(X)$ will be the derived functor there, but it will not be degreewise finite dimensional (and from here a simple argument shows that the colimit cannot exist in ${\cal D}({\cal C'})$). In fact, since ${\rm Ext^n\_{{\cal C}}({\bf Z}/2,{\bf Z}/2)} \cong {\bf Z}/2$ for all $n \geq 0$ a spectral sequence argument shows that $RF(X)$ has infinite dimensional homologies in all degrees. 


