By Fomin A.A.

Each τ -adic matrix represents either a quotient divisible crew and a torsion-free, finite-rank workforce. those representations are an equivalence and a duality of different types, respectively.

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**Extra resources for A category of matrices representing two categories of Abelian groups**

**Example text**

22 is exact iff K---+ A is monomorphic. B. is exact iff B---+ F is epimorphic. B. is exact iff A ---+ B is an isomorphism. A-B B---+F---+0 0---+ A ---+ B---+ 0 A - B~ B 0 - A ---+ B ---+ C ---+ is exact iff A ---+ B is the zero map. 0 is exact iff A - B is a monomorphism and B ---+ C is a cokernel of A -B. 3. 31 for abelian categories "t The sequence 0 - A ........... A (~) + B-+ . B---+ 0 ts exact. ABELIAN CATEGORIES Proof: . 1c smce A is. To prove that A + B (~) ~ B is a cokernel of uh let <;) A+ B---+ X be a map such that A~ A + B---+ X= 0.

FUNDAMENTALS 25 Sums of the same objects are isomorphic; the notation "t A + B refers to "the" sum of A and B; the maps A---'-+ A +B and B ~ A + Bare "the" associated maps. x > A u B~ X~ A = X~ B and x B. )) X= A~ Gt) "' h X to be t e unique X and 2 A +B--=-+ X= B~ X. 8. COMPLETE CATEGORIES Given an indexed set of objects {A;} 1 in a category, its product is defined to be an object 11 iEIA; together with maps {IIiE/Ai ~ A;}{ 26 ABELIAN CATEGORIES such thatforanyfamily {X~ A;}Jthere is a unique X---+ TI 1 Ai such that X---+ TI 1 A;~ A;= X~ A;.

Every map has a kernel and a cokernel. A 3. A 3*. Every monomorphism is a kernel of a map. Every epimorphism is a cokernel of a map. " Most categories that arise in nature satisfy Axioms A 0 through A 2. Often Axiom A 0 is satisfied by using base points. Many categories satisfy one of A 3 or A 3*. Compact Hausdorfspaces 15 ABELIAN CATEGORIES 36 with base points satisfy A 3; all groups (abelian or not) satisfy A 3*. 1. THEOREMS FOR ABELIAN CATEGORIES Consider an object A. LetS be the family of subobjects of A, Q the family of quotient objects.