PlanetPhysics/C 1Category2

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A category ${\mathcal {C}}_{1}$ with coproducts is called a $C_{1}$-category  if for every family of

of monomorphisms $\left\{u_{i}:A_{i}\to B_{i}\right\}$ the morphism $\iota :=\oplus _{i}\,u_{i}:\oplus _{i}\,A_{i}\to \oplus _{i}\,B_{i}$ is also a monomorphism ().

With certain additional conditions (as explained in ref. ) ${\mathcal {C}}_{1}$ may satisfy the Grothendieck axiom ${\mathcal {A}}b5$, thus becoming a $C_{3}$-category (Ch. 11 in ).