Let $\mathcal{M}$ be the following category:

• The objects are small abelian categories with chosen zero object, biproducts, kernels, and cokernels.
• The morphisms are functors that preserve the structure strictly.
• Composition and identities are the obvious ones.

If I'm not mistaken, $\mathcal{M}$ is a locally finitely presentable category. At any rate, the forgetful functor $U : \mathcal{M} \to \mathbf{Cat}$ preserves all limits and filtered colimits, so under the assumption that $\mathcal{M}$ is l.f.p., we have a left adjoint $F : \mathbf{Cat} \to \mathcal{M}$.

Question 1. Can the standard model structure on $\mathbf{Cat}$ be transferred to $\mathcal{M}$, i.e. does there exist a model structure on $\mathcal{M}$ where the weak equivalences and fibrations are created by $U : \mathcal{M} \to \mathbf{Cat}$?

Question 2. Assuming the desired model structure on $\mathcal{M}$ exists, is it compatible with the obvious $\mathbf{Cat}$-enrichment, i.e. do we have a model 2-category?

Question 3. Let $\mathfrak{M}$ be the 2-category whose objects and morphisms are as in $\mathcal{M}$ and whose 2-cells are natural transformations; and let $\mathfrak{A}$ be the 2-category whose objects are small abelian categories, whose morphisms are exact functors, and whose 2-cells are natural transformations. We may localise $\mathfrak{M}$ (as a 2-category) with respect to weak equivalences to obtain a (possibly non-locally-small) 2-category $\mathfrak{B}$ with a comparison pseudofunctor $\mathfrak{B} \to \mathfrak{A}$. Is this a biequivalence?

Let $\mathcal{M}$ be the following category:

• The objects are small abelian categories with chosen zero object, biproducts, kernels, and cokernels.
• The morphisms are functors that preserve the structure strictly.
• Composition and identities are the obvious ones.

If I'm not mistaken, $\mathcal{M}$ is a locally finitely presentable category. At any rate, the forgetful functor $U : \mathcal{M} \to \mathbf{Cat}$ preserves all limits and filtered colimits, so under the assumption that $\mathcal{M}$ is l.f.p., we have a left adjoint $F : \mathbf{Cat} \to \mathcal{M}$.

Question 1. Can the standard model structure on $\mathbf{Cat}$ be transferred to $\mathcal{M}$, i.e. does there exist a model structure on $\mathcal{M}$ where the weak equivalences and fibrations are created by $U : \mathcal{M} \to \mathbf{Cat}$?

Question 2. Assuming the desired model structure on $\mathcal{M}$ exists, is it compatible with the obvious $\mathbf{Cat}$-enrichment, i.e. do we have a model 2-category?

Question 3. Let $\mathfrak{M}$ be the 2-category whose objects and morphisms are as in $\mathcal{M}$ and whose 2-cells are natural transformations; and let $\mathfrak{A}$ be the 2-category whose objects are small abelian categories, whose morphisms are exact functors, and whose 2-cells are natural transformations. There is an evident 2-functor $\mathfrak{M} \to \mathfrak{A}$ which is surjective on objects, (locally) injective on morphisms, and (locally) bijective on 2-cells. Does this exhibit $\mathfrak{A}$ as the higher-categorical localisation of $\mathfrak{M}$ with respect to weak equivalences?