Let k be a field. We show that locally presentable, k-linear categories \({\mathcal {C}}\) dualizable in the sense that the identity functor can be recovered as \(\coprod _i x_i\otimes f_i\) for objects \(x_i\in {\mathcal {C}}\) and left adjoints \(f_i\) from \({\mathcal {C}}\) to \(\mathrm {Vect}_k\) are products of copies of \(\mathrm {Vect}_k\). This partially confirms a conjecture by Brandenburg, the author and T. Johnson-Freyd. Motivated by this, we also characterize the Grothendieck categories containing an object x with the property that every object is a copower of x: they are precisely the categories of non-singular injective right modules over simple, regular, right self-injective rings of type I or III.

令k为一个字段。我们表明，在本地呈现的，ķ -线性类别\（{\ mathcal {C}} \） dualizable在这个意义上的身份算符可以回收作为\（\ COPROD _i X_I \ otimes f_i \）为对象\（X_I \ {\ mathcal {C}} \）中的左伴点\（f_i \）从\（{\ mathcal {C}} \}到\（\ mathrm {Vect} _k \）是\（\ mathrm {Vect} _k \）。这在一定程度上证实了勃兰登堡，作者和约翰逊·弗雷德（T. Johnson-Freyd）的猜想。因此，我们还对包含对象x的Grothendieck类别进行了刻画，其属性为每个对象都是x：它们恰好是I或III型简单，规则，正确的自我注射环上的非奇异注射权利模块的类别。