Editing Exact functor (section)

== Definitions ==
Let '''P''' and '''Q''' be [[abelian categories]], and let {{nowrap|1=''F'': '''P'''→'''Q'''}} be a [[covariant functor|covariant]] [[additive functor]] (so that, in particular, ''F''(0) = 0).  We say that ''F'' is an '''exact functor''' if whenever 

:<math>0 \to A\ \stackrel{f}{\to} \ B\ \stackrel{g}{\to} \ C \to 0</math> 

is a [[short exact sequence]] in '''P''' then 

:<math>0 \to F(A) \ \stackrel{F(f)}{\longrightarrow} \ F(B)\ \stackrel{F(g)}{\longrightarrow} \ F(C) \to 0</math> 

is a short exact sequence in '''Q'''.  (The maps are often omitted and implied, and one says: "if 0→''A''→''B''→''C''→0 is exact, then 0→''F''(''A'')→''F''(''B'')→''F''(''C'')→0 is also exact".) 

Further, we say that ''F'' is   

*'''left-exact''' if whenever 0→''A''→''B''→''C''→0 is exact then 0→''F''(''A'')→''F''(''B'')→''F''(''C'') is exact;
*'''right-exact''' if whenever 0→''A''→''B''→''C''→0 is exact then ''F''(''A'')→''F''(''B'')→''F''(''C'')→0 is exact;
*'''half-exact''' if whenever 0→''A''→''B''→''C''→0 is exact then ''F''(''A'')→''F''(''B'')→''F''(''C'') is exact. This is distinct from the notion of a [[topological half-exact functor]].

If ''G'' is a [[contravariant functor|contravariant]] additive functor from '''P''' to '''Q''', we similarly define ''G'' to be

*'''exact''' if whenever 0→''A''→''B''→''C''→0 is exact then 0→''G''(''C'')→''G''(''B'')→''G''(''A'')→0 is exact;
*'''left-exact''' if whenever 0→''A''→''B''→''C''→0 is exact then 0→''G''(''C'')→''G''(''B'')→''G''(''A'') is exact;
*'''right-exact''' if whenever 0→''A''→''B''→''C''→0 is exact then ''G''(''C'')→''G''(''B'')→''G''(''A'')→0 is exact;
*'''half-exact''' if whenever 0→''A''→''B''→''C''→0 is exact then ''G''(''C'')→''G''(''B'')→''G''(''A'') is exact.

It is not always necessary to start with an entire short exact sequence 0→''A''→''B''→''C''→0 to have some exactness preserved. The following definitions are equivalent to the ones given above:

*''F'' is '''exact''' if and only if ''A''→''B''→''C'' exact implies ''F''(''A'')→''F''(''B'')→''F''(''C'') exact;
*''F'' is '''left-exact''' if and only if 0→''A''→''B''→''C'' exact implies 0→''F''(''A'')→''F''(''B'')→''F''(''C'') exact (i.e. if "''F'' turns kernels into kernels");
*''F'' is '''right-exact''' if and only if ''A''→''B''→''C''→0 exact implies ''F''(''A'')→''F''(''B'')→''F''(''C'')→0 exact (i.e. if "''F'' turns cokernels into cokernels");
*''G'' is '''left-exact''' if and only if ''A''→''B''→''C''→0 exact implies 0→''G''(''C'')→''G''(''B'')→''G''(''A'') exact (i.e. if "''G'' turns cokernels into kernels");
*''G'' is '''right-exact''' if and only if 0→''A''→''B''→''C'' exact implies ''G''(''C'')→''G''(''B'')→''G''(''A'')→0 exact (i.e. if "''G'' turns kernels into cokernels").