## A Characterization of Linear Spaces and Their Affine Maps and a Method of Constructing Categories Related to it |

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affine contractions affine maps affine Pi—structures affine spaces affinely dependent arbitrary arrows C=Set category of affine category Set CHARACTERIZATION OF LINEAR codomain commutative composite conormal pair consider defined definition denotes diagram differentiable functions differentiable manifold domain Dually Eacample embedding theorem empty function epimorphism equivalence classes equivalence relation example F Q(r,m forgetful functor full subcategory func functional f Furthermore hence homomorphisms images inclusion map infinitely differentiable lemma limit cone limits and colimits linear space monics n-set normal pair objects one-point set one-point structure open ball Pi—curve points polynomial of degree projection map proof of theorem proposition prove Q-isomorphic Q-morphism Q-PA Q-structure is isomorphic quotient structures real valued functions restriction satisfying separated Pi—structures separated Q-structure set of affine set of functions set of morphisms Set or SET strong substructure subset of C(r,m substruc tion triple tures unique morphism Yºo