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Note: I'm only addressing David's question how one one can "add cokernels" to an additive category and how one can use this in order to embed an additive category into an abelian one, even in a universal way.

Concerning the request on minimality in the question, it is unclear to me what exactly it is, Lev wants to achieve. What happens in the mentioned example of sheaves that one identifies an interesting category (vector bundles = finitely generated projective $\mathcal{O}_{X}$-modules) and ends up with the category coherent sheaves. The In this case, the category constructed below happens to embed embeds into the category of sheaves just because one started out with a class of projective objects in an abelian category already.

The idea is to embed the additive category $\mathcal{A}$ into the category of morphisms $\mathcal{A}^{\to}$. Note that a morphism in $\mathcal{A}^{\to}$ corresponds to a commutative square. More precisely, the idea is that a morphism $(A \to B)$ of $\mathcal{A}$ should represent its cokernel.

Let's pretend this works. Since the cokernel of $0 \to A$ in $\mathcal{A}$ is $A$, we see that we must embed $\mathcal{A}$ via $A \mapsto (0 \to A)$. This embedding of $\mathcal{A} \to \mathcal{A}^{\to}$ is clearly fully faithful. But we're not quite there, yet. There are nonzero morphisms of morphisms $(A^{-1} \to A^{0}) \to (B^{-1} \to B^{0})$ that should induce the zero map on the (putative) cokernels, namely precisely those for which the map $A^{0} \to B^{0}$ factors over $B^{-1}$:

It is easy to see that these morphisms form an ideal $\mathcal{J}$ in $\mathcal{A}^{\to}$ (they are closed under composition and sums), so we may factor this ideal out. In other words, a morphism of $\mathcal{A}^{\to}$ is identified with zero if and only if it lies in $\mathcal{J}$. The resulting category $\text{fp}(\mathcal{A}) = \mathcal{A}^{\to}/\mathcal{J}$ is what we want because we have:

Theorem (Freyd, 1965; Beligiannis, 2000)

1. The category $\text{fp}(\mathcal{A})$ has cokernels. The functor $A \mapsto (0 \to A)$ is fully faithful and universal among functors to additive categories with cokernels: more precisely, every functor $F: \mathcal{A} \to \mathcal{C}$ to a category with cokernels extends uniquely to a cokernel-preserving functor $\text{fp} (\mathcal{A}) \to \mathcal{C}$.
2. Moreover, $\text{fp}{(\mathcal{A})}$ is abelian if and only if $\mathcal{A}$ has weak kernels (a weak kernel has the factorization property of a kernel but uniqueness of the factorization is not required).

So if $\mathcal{A}$ has weak kernels, we're already done. If not, we may play the same game again, using this theorem. We first embed the category with kernels $(\text{fp} \mathcal{A})^{\text{op}}$ into the abelian category $\text{fp}\left( (\text{fp} \mathcal{A})^{\text{op}} \right)$ and then pass to the opposite category $\mathcal{F}{(\mathcal{A})} = \left(\text{fp}\left( (\text{fp} \mathcal{A})^{\text{op}} \right)\right)^{\text{op}}$. The diagrams get a bit unwieldy, but one may check that:

Theorem (Adelman, 1971) The embedding $\mathcal{A} \to \mathcal{F}(\mathcal{A})$ is fully faithful and universal among all functors to abelian categories (every functor to an abelian category extends to an exact functor on $\mathcal{F}(\mathcal{A})$).

The category $\text{fp}(\mathcal{A})$ is sometimes used in connection with the derived category/triangulated categories (the inclusion $\mathcal{T} \to \text{fp}(\mathcal{T})$ is the universal homological functor on the triangulated category $\mathcal{T}$. It already appears in Verdier's thesis - attributed to Freyd). Closely related are also the hearts of $t$-structures (perverse sheaves).

The most important references are:

1. Peter Freyd, Representations in abelian categories, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), Springer, New York, 1966, pp. 95–120.

2. A. Beligiannis: ''On the Freyd Categories of an Additive Category'', Homology, Homotopy and Applications Vol. 2, No.11 (2000), pp. 147-185.

6 Rollback to Revision 4

Note: I'm only addressing David's question how one one can "add cokernels" to an additive category and how one can use this in order to embed an additive category into an abelian one, even in a universal way.

Concerning the request on minimality in the question, it is unclear to me what exactly it is, Lev wants to achieve. What happens in the mentioned example of sheaves that one identifies an interesting category (vector bundles = finitely generated projective $\mathcal{O}_{X}$-modules) and ends up with the category coherent sheaves. The category constructed below happens to embed into the category of sheaves just because one started out with a class of projective objects in an abelian category already.

The idea is to embed the additive category $\mathcal{A}$ into the category of morphisms $\mathcal{A}^{\to}$. Note that a morphism in $\mathcal{A}^{\to}$ corresponds to a commutative square. More precisely, the idea is that a morphism $(A \to B)$ of $\mathcal{A}$ should represent its cokernel.

Let's pretend this works. Since the cokernel of $0 \to A$ in $\mathcal{A}$ is $A$, we see that we must embed $\mathcal{A}$ via $A \mapsto (0 \to A)$. This embedding of $\mathcal{A} \to \mathcal{A}^{\to}$ is clearly fully faithful. But we're not quite there, yet. There are nonzero morphisms of morphisms $(A^{-1} \to A^{0}) \to (B^{-1} \to B^{0})$ that should induce the zero map on the (putative) cokernels, namely precisely those for which the map $A^{0} \to B^{0}$ factors over $B^{-1}$:

It is easy to see that these morphisms form an ideal $\mathcal{J}$ in $\mathcal{A}^{\to}$ (they are closed under composition and sums), so we may factor this ideal out. In other words, a morphism of $\mathcal{A}^{\to}$ is identified with zero if and only if it lies in $\mathcal{J}$. The resulting category $\text{fp}(\mathcal{A}) = \mathcal{A}^{\to}/\mathcal{J}$ is what we want because we have:

Theorem (Freyd, 1965; Beligiannis, 2000)

1. The category $\text{fp}(\mathcal{A})$ has cokernels. The functor $A \mapsto (0 \to A)$ is fully faithful and universal among functors to additive categories with cokernels: more precisely, every functor $F: \mathcal{A} \to \mathcal{C}$ to a category with cokernels extends uniquely to a cokernel-preserving functor $\text{fp} (\mathcal{A}) \to \mathcal{C}$.
2. Moreover, $\text{fp}{(\mathcal{A})}$ is abelian if and only if $\mathcal{A}$ has weak kernels (a weak kernel has the factorization property of a kernel but uniqueness of the factorization is not required).

So if $\mathcal{A}$ has weak kernels, we're already done. If not, we may play the same game again, using this theorem. We first embed the category with kernels $(\text{fp} \mathcal{A})^{\text{op}}$ into the abelian category $\text{fp}\left( (\text{fp} \mathcal{A})^{\text{op}} \right)$ and then pass to the opposite category $\mathcal{F}{(\mathcal{A})} = \left(\text{fp}\left( (\text{fp} \mathcal{A})^{\text{op}} \right)\right)^{\text{op}}$. The diagrams get a bit unwieldy, but one may check that:

Theorem (Adelman, 1971) The embedding $\mathcal{A} \to \mathcal{F}(\mathcal{A})$ is fully faithful and universal among all functors to abelian categories (every functor to an abelian category extends to an exact functor on $\mathcal{F}(\mathcal{A})$).

The category $\text{fp}(\mathcal{A})$ is sometimes used in connection with the derived category/triangulated categories (the inclusion $\mathcal{T} \to \text{fp}(\mathcal{T})$ is the universal homological functor on the triangulated category $\mathcal{T}$. It already appears in Verdier's thesis - attributed to Freyd). Closely related are also the hearts of $t$-structures (perverse sheaves).

The most important references are:

1. Peter Freyd, Representations in abelian categories, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), Springer, New York, 1966, pp. 95–120.

2. A. Beligiannis: ''On the Freyd Categories of an Additive Category'', Homology, Homotopy and Applications Vol. 2, No.11 (2000), pp. 147-185.

5 added 2 characters in body

Note: I'm only addressing David's question how one one can "add cokernels" to an additive category and how one can use this in order to embed an additive category into an abelian one, even in a universal way.

Concerning the request on minimality in the question, it is unclear to me what exactly it is, Lev wants to achieve. What happens in the mentioned example of sheaves that one identifies an interesting category (vector bundles = finitely generated projective $\mathcal{O}_{X}$-modules) and ends up with the category coherent sheaves. The category constructed below happens to embed into the category of sheaves just because one started out with a class of projective objects in an abelian category already.

The idea is to embed the additive category $\mathcal{A}$ into the category of morphisms $\mathcal{A}^{\to}$. Note that a morphism in $\mathcal{A}^{\to}$ corresponds to a commutative square. More precisely, the idea is that a morphism $(A \to B)$ of $\mathcal{A}$ should represent its cokernel.

Let's pretend this works. Since the cokernel of $0 \to A$ in $\mathcal{A}$ is $A$, we see that we must embed $\mathcal{A}$ via $A \mapsto (0 \to A)$. This embedding of $\mathcal{A} \to \mathcal{A}^{\to}$ is clearly fully faithful. But we're not quite there, yet. There are nonzero morphisms of morphisms $(A^{-1} \to A^{0}) \to (B^{-1} \to B^{0})$ that should induce the zero map on the (putative) cokernels, namely precisely those for which the map $A^{0} \to B^{0}$ factors over $B^{-1}$:

It is easy to see that these morphisms form an ideal $\mathcal{J}$ in $\mathcal{A}^{\to}$ (they are closed under composition and sums), so we may factor this ideal out. In other words, a morphism of $\mathcal{A}^{\to}$ is identified with zero if and only if it lies in $\mathcal{J}$. The resulting category $\text{fp}(\mathcal{A}) = \mathcal{A}^{\to}/\mathcal{J}$ is what we want because we have:

Theorem (Freyd, 1965; Beligiannis, 2000)

1. The category $\text{fp}(\mathcal{A})$ has cokernels. The functor $A \mapsto (0 \to A)$ is fully faithful and universal among functors to additive categories with cokernels: more precisely, every functor $F: \mathcal{A} \to \mathcal{C}$ to a category with cokernels extends uniquely to a cokernel-preserving functor $\text{fp} (\mathcal{A}) \to \mathcal{C}$.
2. Moreover, $\text{fp}{(\mathcal{A})}$ is abelian if and only if $\mathcal{A}$ has weak kernels (a weak kernel has the factorization property of a kernel but uniqueness of the factorization is not required).

So if $\mathcal{A}$ has weak kernels, we're already done. If not, we may play the same game again, using this theorem. We first embed the category with kernels $(\text{fp} \mathcal{A})^{\text{op}}$ into the abelian category $\text{fp}\left( (\text{fp} \mathcal{A})^{\text{op}} \right)$ and then pass to the opposite category $\mathcal{F}{(\mathcal{A})} = \left(\text{fp}\left( (\text{fp} \mathcal{A})^{\text{op}} \right)\right)^{\text{op}}$. The diagrams get a bit unwieldy, but one may check that:

Theorem (Adelman, 1971) The embedding $\mathcal{A} \to \mathcal{F}(\mathcal{A})$ is fully faithful and universal among all functors to abelian categories (every functor to an abelian category extends to an exact functor on $\mathcal{F}(\mathcal{A})$).

The category $\text{fp}(\mathcal{A})$ is sometimes used in connection with the derived category/triangulated categories (the inclusion $\mathcal{T} \to \text{fp}(\mathcal{T})$ is the universal homological functor on the triangulated category $\mathcal{T}$. It already appears in Verdier's thesis - attributed to Freyd). Closely related are also the hearts of $t$-structures (perverse sheaves).

The most important references are:

1. Peter Freyd, Representations in abelian categories, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), Springer, New York, 1966, pp. 95–120.

2. A. Beligiannis: ''On the Freyd Categories of an Additive Category'', Homology, Homotopy and Applications Vol. 2, No.11 (2000), pp. 147-185.

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