Observation 1: I have often read statements like "a contravariant functor $F : C \rightarrow D$ is actually a covariant functor $F: C^{op} \rightarrow D$.

Observation 2 In the context of left-exact functors, some authors define a contravariant left-exact (additive) functor by the property that if $$ A \rightarrow B \rightarrow C \rightarrow 0 $$ is exact in C, then $$ 0 \rightarrow F(C) \rightarrow F(B) \rightarrow F(A) $$ is exact in D.

My issue is now in the reconciliation of observations (1) and (2). In particular, let's consider $C = R-mod$. If we follow the adage from (1) about contravariant functors being covariant functors from the opposite category, then the definition in (2) should be reworded to say:

A contravariant functor $F : C \rightarrow D$ is left-exact if whenever

$$ 0 \rightarrow C \rightarrow B \rightarrow A $$ is exact in $C^{op}$, then $$ 0 \rightarrow F(C) \rightarrow F(B) \rightarrow F(A) $$ is exact in D.

However, as noted in the accepted answer to this question: https://mathoverflow.net/questions/29442/what-is-the-opposite-category-of-the-category-of-modules-or-hopf-algebra-repres, $R-mod^{op}$ is not a category of modules and hence it seems unlikely, even if it made sense to talk about exact sequences and homology in this opposite category (of which I am not sure it does), that a sequence would be exact in this opposite category if and only if it was exact in our original category of R-modules. This seems to make the remark in observation (1) seem to be complete nonsense.

Question Is there a way to formally define left/right-exact contravariant in terms of left/right-exact sequences in the opposite of the domain category? If not, what does it really mean for a "contravariant functor to be a covariant functor from the opposite category"? If yes, then how do we even know how to define homology in an arbitrary opposite category?


The fact you reference, that the opposite of modules is not a module category, is actually rather esoteric, compared with the fact that it is an abelian category. This is precisely a category in which the notions of homological algebra make sense: the homs are abelian groups, there are biproducts, kernels, and cokernels, every mono is a kernel and every epi a cokernel. With this formalism in place, it is in fact true that $A \to^f B \to^g C$ is exact in any abelian category if and only if $C\to^{g^*}B\to^{f^*}A$ is exact in its opposite, as is shown (unless it's taken for granted) in any book on homological algebra.


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