1
$\begingroup$

Let $F$ be a contravariant functor from some category of spaces (e.g. smooth manifolds or (compact?) topological Hausdorff spaces), to Abelian groups. Assume that for any open sets $U, V \subseteq X$ such that $U\cup V = X$, we have that $$ F(X) \stackrel{j_U^* \oplus j_V^*}{\longrightarrow} F(U) \oplus F(V) \stackrel{i_U - i_V}{\longrightarrow} F(U\cap V)$$ is exact in the middle (here, $j_U, j_V$ are the inclusions of $U$ and $V$ into $X$ and $i_U, i_V$ are the inclusions of $U\cap V$ into $X$.

Under what circumstances can we construct functors $F^{i}$ together with boundary maps that continue the sequence to the left and/or to the right? Is there some general construction? What if additionally, we assume $F$ to be a homotopy functor?

Of course, this is something like a "left derived functor", but $\mathrm{Man}$ or $\mathrm{Top}$ are not abelian categories, so this doesn't make sense.

\Edit: Think about $F = K^0$, complex K-theory.

\Edit2: I deleted the zero to the right of the sequence.

$\endgroup$
7
  • $\begingroup$ Presumably $F$ should satisfy some sort of excision axiom? $\endgroup$ May 15, 2014 at 14:05
  • 2
    $\begingroup$ $F$ is a presheaf on any manifold and the mayer-vietoris axiom is a special case of the gluing axiom. If one has an additional continuity assumption (I'm thinking of something along the lines of $F(U) = \lim F(U_i)$ for any directed system of open subsets), it should be possible to derive the full gluing axiom. Then you have a sheaf of abelian groups and can do the usual construction. $\endgroup$ May 15, 2014 at 14:16
  • 4
    $\begingroup$ Or maybe you do not need to bother with direct limits (or with requiring $F(\emptyset)=0$). It appears that $F$ is a sheaf with respect to the topology generated by ordinary two-set open covers, so that sheaf cohomology for that topology should do the job. $\endgroup$ May 15, 2014 at 15:03
  • 2
    $\begingroup$ Derived functors make sense in a great deal more generality than abelian categories! See, for example, math.harvard.edu/~eriehl/cathtpy.pdf. Anyway, I think a keyword you might want to look up is "homotopy sheafification." $\endgroup$ May 15, 2014 at 17:21
  • 1
    $\begingroup$ Oh, the left!? I take it all back. But what kind of $K$-theory gives you a zero on the right like that? $\endgroup$ May 15, 2014 at 21:28

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.