Does mapping cylinder category have enough injectives? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T10:13:27Z http://mathoverflow.net/feeds/question/99908 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99908/does-mapping-cylinder-category-have-enough-injectives Does mapping cylinder category have enough injectives? Hiro 2012-06-18T15:24:26Z 2012-06-18T15:30:12Z <p>Let $A, B$ be two abelian categories, and $\tau : A \to B$ a left exact functor.</p> <p>We define a category $C$ as follows:</p> <p>objects: triples $(M, N, \varphi)$ where $M\in A, N\in B$ and $\varphi: N\to \tau M$ is a morphism in $B$.</p> <p>morphisms: pairs $(f, g): (M, N, \varphi) \to (M^{\prime}, N^{\prime}, \varphi^{\prime})$ where $f:M\to M^{\prime}$ and $g:N\to N^{\prime}$ satisfying $(\tau f) \circ \varphi = \varphi ^{\prime} \circ g$.</p> <p>Then, is the following statement true?</p> <p>If so, then how can one prove it?</p> <p>STATEMENT: If $A, B$ have enough injectives, then so does $C$.</p> <p>For example, let $X$ be a scheme, $Y$ be a closed subscheme of $X$, and $U=X\setminus Y$. If $A$=(etale sheaves on $X$), $B$=(etale sheaves on $U$), then the cagegory $C$ is equivalent to the category of etale sheaves on $Y$. So, $C$ has enough injectives , of course. I wonder whether this kind of situation happens in the general setting above.</p> <p>Please give me any advice.</p>