I just wanted to take a second to post this paper: http://arxiv.org/abs/1211.0054

In it, the author defines zeroth derived functors in abelian categories that do not have enough projectives or injectives.

A brief piece of it: Let $C$ and $D$ be abelian categories. Take the functor category $(C,D)$ and consider any subcategory $S\subset (C, D)$ and then the full subcategory of $S$ consisting of left exact functors in $S$, $Lex(S)$. There is the canonical inclusion $s: Lex(S) \rightarrow S$. We say $S$ admits a zeroth right derived functor if there is a left adjoint (say $r^0: S \rightarrow Lex(S)$) satisfying two properties (we'll call the map $s\circ r^0$ the zeroth right derived functor and relabel it R^0):

1. The unit of adjunction $u: 1_s \rightarrow R^0$ is an isomorphism on injectives, and

2)The 'other'composition, $r^0\circ s$ is isomorphic to the the identity functors of the full subcategory of left exact functors of $S$. (The paper says something different, namely that $r^0\circ s \cong 1_S$. Based on the direction of the arrows, that composition should land in the $Lex(S)$.

In general, the unit of adjunction has a kernel and by evaluating the exact sequence

$0 \rightarrow \ker u \rightarrow 1_S \rightarrow R^0$

at $F$, a functor from $C$ to $D$, we recover a definition for the injective stabilization of a functor $F$, as defined by Auslander and Bridger in Stable Module Theory.

The author goes on to say that $r^0$ is a functor which takes functors $F$ and produces left exact functors $r^0F$. Then the inclusion $s(r^0F)=R^0F$ is the zeroth right derived functor of $F$, the result of making F left exact while changing as little as possible about the functor.

Now, all this being said, I did not write the paper, and I do not know much more about the details. I just think the paper is an interesting one.

I just wanted to take a second to post this paper: http://arxiv.org/abs/1211.0054

In it, the author defines zeroth derived functors in abelian categories that do not have enough projectives or injectives.

A brief piece of it: Let $C$ and $D$ be abelian categories. Take the functor category $(C,D)$ and consider any subcategory $S\subset (C, D)$ and then the full subcategory of $S$ consisting of left exact functors in $S$, $Lex(S)$. There is the canonical inclusion $s: Lex(S) \rightarrow S$. We say $S$ admits a zeroth right derived functor if there is a left adjoint (say $r^0: S \rightarrow Lex(S)$) satisfying two properties (we'll call the map $s\circ r^0$ the zeroth right derived functor and relabel it $R^0$):

1. The unit of adjunction $u: 1_s \rightarrow R^0$ is an isomorphism on injectives, and

2)The 'other'composition, $r^0\circ s$ is isomorphic to the the identity functor of the full subcategory of left exact functors of $S$. (The paper says something different, namely that $r^0\circ s \cong 1_S$. Based on the direction of the arrows, that composition should land in the $Lex(S)$.

In general, the unit of adjunction has a kernel and by evaluating the exact sequence

$0 \rightarrow \ker u \rightarrow 1_S \rightarrow R^0$

at $F$, a functor from $C$ to $D$, we recover a definition for the injective stabilization of a functor $F$, as defined by Auslander and Bridger in Stable Module Theory.

The author goes on to say that $r^0$ is a functor which takes functors $F$ and produces left exact functors $r^0F$. Then the inclusion $s(r^0F)=R^0F$ is the zeroth right derived functor of $F$, the result of making F left exact while changing as little as possible about the functor.

Now, all this being said, I did not write the paper, and I do not know much more about the details. I just think the paper is an interesting one.