Derivatives on Functors I'm not even sure if this question makes pedantic sense but is there any way to rigorously define the notion of taking the derivative of a functor? 
 A: I know at least two, maybe even three contexts where "derivatives" (in a loose sense) exist:


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*If $F : A \to B$ is a (covariant) functor between abelian categories which is left exact, and the categories in question are well behaved, then we can talk about the right derived functors $R^iF$ of $F$. These are functors such that whenever $0 \to A \to B \to C \to 0$ is exact, then you get a long exact sequence:
$$0 \to F(A) \to F(B) \to F(C) \to R^1F(A) \to R^1F(B) \to R^1F(C) \to R^2F(A) \to \dots$$

*As a loose generalization, is $F : A \leftrightarrows B : G$ is an adjoint pair of functors between model categories that satisfies some properties (namely that it is a Quillen adjunction) then one can form the (total) derived functors of $F$ and $G$:
$$\mathbb{L}F : \mathrm{Ho}(A) \leftrightarrows \mathrm{Ho}(B) : \mathbb{R}G$$
When $A$ and $B$ are appropriate categories, this is basically the same notion as before. For example in dg-modules, the left derived functor of $- \otimes_R M$ evaluated at $K(N,0)$ is basically $\mathrm{Tor}_*^R(M,N)$.

*In a completely different context, you've got calculus of functors. I'm not as knowledgeable about the subject, but you can view the functor of immersions $\mathrm{Imm}(M,N)$ as the first derivative of the functor of embeddings $\mathrm{Emb}(M,N)$ between two manifolds. As you can see an immersion is an embedding around each point of $M$, but not globally an immersion -- this is a "polynomial of degree", because we're concerned about the behavior around a single point at a time and this is enough to retrieve the whole functor, just like affine maps. As you go higher and higher you can compute some sort of Taylor series to eventually retrieve the initial functor.
A: A categorical presentation of a notion of derivative for polynomial functors (aka containers) generalizing Huet's zipper is given in ∂ for Data: Differentiating Data Structures. An explanation of this work in term of programming constructs is given by McBride in Clowns to the left of me, jokers to the right.
