# ct.category theory – Recovering an abelian category from the Ext of its simple objects

Let $$C$$ be an abelian category, assume for simplicity that $$C$$ is enriched over $$Vect_k$$ (vector spaces over $$k$$) for some fixed field $$k$$.

Suppose also that $$C$$ is both Artinian and Noetherian, so that for any object $$X$$ there is a sequence of objects $$0=X_0 hookrightarrow ldots hookrightarrow X_n = X$$ with $$X_i/X_{i-1}$$ simple. Finally suppose that $$C$$ has enough injective/projective objects so that $$operatorname{Ext}_C$$ can be defined.

Given $$C$$, we build a new category $$S$$, enriched over graded $$k$$-vector spaces, in the following way:

• The objects of $$S$$ are the simple objects of $$C$$
• If $$X,Yin Ob(S)$$ then $$operatorname {Hom}_S(X,Y) = bigoplus_{ngeq0}operatorname{Ext}^n_C(X,Y)$$
• Compositions of morphisms are defined using the natural maps $$operatorname{Ext}^n_C(X,Y)otimes operatorname{Ext}^m_C(Y,Z)tooperatorname{Ext}^{n+m}_C(X,Z)$$

My question is: Can we recover $$C$$ from $$S$$ (say up to equivalence)?

Assuming the answer is “yes”, I guess that there is an analogue for when $$C$$ is only enriched over $$Ab$$, maybe if we redefine $$S$$ so that $$operatorname{Hom}_S(X,Y)=operatorname{Hom}_{D(C)}(X,Y)$$ or something

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