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Abconditions $Ab3$ and $Ab5$ conditions
1 Abconditions: $Ab3$ and $Ab5$ conditions
1. (Ab3). Let us recall that an Abelian category $\mathcal{A}b$ is cocomplete (or an $\mathcal{A}b3$category) if it has arbitrary direct sums.
2. (Ab5). A cocomplete Abelian category $\mathcal{A}b$ is said to be an $\mathcal{A}b5$category if for any directed family $\left\{A_{i}\right\}_{{i\in I}}$ of subobjects of $\mathcal{A}$, and for any subobject $B$ of $\mathcal{A}$, the following equation holds
$(\sum_{{i\in I}}A_{i})\bigcap B=\sum_{{i\in I}}(A_{i}\bigcap B).$
1.0.1 Remarks

One notes that the condition Ab3 is equivalent to the existence of arbitrary direct limits.

Furthermore, Ab5 is equivalent to the following proposition: there exist inductive limits and the inductive limits over directed families of indices are exact, that is, if $I$ is a directed set and $0\to A_{i}\to B_{i}\to C_{i}\to 0$ is an exact sequence for any $i\in I$, then
$0\to\limdir{(A_{i})}\to\limdir{(B_{i})}\to\limdir{(C_{i})}\to 0$ is also an exact sequence.
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