# PlanetPhysics/C 1Category2

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


of monomorphisms ${\displaystyle \left\{u_{i}:A_{i}\to B_{i}\right\}}$ the morphism ${\displaystyle \iota :=\oplus _{i}\,u_{i}:\oplus _{i}\,A_{i}\to \oplus _{i}\,B_{i}}$ is also a monomorphism ([1]).

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

## References

1. See p.81 in ref. ${\displaystyle [266]}$  in the Bibliography for categories and algebraic topology
2. Ref. ${\displaystyle [288]}$  in the Bibliography for categories and algebraic topology