A concrete category is a pair $(C,U)$ where $C$ is a category and $U$ is a faithful functor $C \to Set$. An abelian category is an additive category in which every morphism admits a kernel, a cokernel and its respective parallel morphism is iso.

I'm interested in generalizing the concept of free resolutions for abelian categories which are also concretizable. Which kind of the coherence condition between the concretization functor and the abelian structure is the most "natural"?

Slightly related question: Definition of a *Monoidal Abelian Category*


To construct free resolutions what you want is a functor $G : A \to \text{Set}$ ($A$ an abelian category) which

  • has a left adjoint $F : \text{Set} \to A$ and
  • preserves epimorphisms.

These conditions are satisfied, for example, by the usual forgetful functor from $R$-modules to sets. Note that we do not need to assume faithfulness. Now we can prove the following:

Lemma ("free objects are projective"): With the above hypotheses, $F$ preserves projective objects.

Here we need to define "projective" to mean $\text{Hom}(P, -)$ preserves epimorphisms, in order to correctly apply to nonabelian categories. Now the proof is very short: if $P \in \text{Set}$ is a projective object (every set is projective assuming the axiom of choice), then

$$\text{Hom}(F(P), -) \cong \text{Hom}(P, G(-))$$

so $\text{Hom}(F(P), -)$ is a composite of two functors which preserve epimorphisms and hence preserves epimorphisms.

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