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This is a crosspost of this MSE question.

The direct image sheaf functor $f_\ast$ and inverse image sheaf functor $f^\ast$ (here I mean the usual inverse image sheaf functor often denoted by $f^{-1}$) form a well-known adjunction for $\mathsf{Set}$-valued sheaves (I think also for sheaves valued in algebraic categories).

Is there anything interesting to be said about $f^\ast,g^\ast$ and $f_\ast,g_\ast$ when $f\simeq g$? What about $f_!,g_!$ and $f^!,g^!$?

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    $\begingroup$ I'm making this a comment since I'm not sure of the answer, but I believe that a homotopy between $f$ and $g$ gives rise to a natural isomorphism between $f_\ast f^\ast$ and $g_\ast g^\ast$, making a commutative triangle with the two unit maps for the adjunctions. I'm not sure what $f_!$ means for sheaves of sets but for abelian sheaves I think the analogous claim holds for $f_!f^!$ and $g_!g^!$ if $f$ and $g$ are proper homotopy equivalent. $\endgroup$ – Dan Petersen Sep 12 '15 at 16:16
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    $\begingroup$ @DanPetersen this sounds really interesting. Why, intuitively, should homotopy relate the functors you mentioned? If you could post an answer providing "just" intuition that would be great. $\endgroup$ – Arrow Sep 12 '15 at 19:19
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    $\begingroup$ I'm not sure the relation goes in that direction. What is true is that there is a continuous endpoint-preserving map from the standard interval to the Sierpiński space, and this ultimately implies that natural transformations of geometric morphisms give rise to homotopies of geometric morphisms – not the other way around. $\endgroup$ – Zhen Lin Sep 13 '15 at 0:19
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    $\begingroup$ @ZhenLin Right, that was stupid of me. I was thinking of the following topological fact: if $p_X$ and $p_Y$ are projections to the point, and if $f$ and $g$ are homotopic maps $X \to Y$, then the maps $Rp_{Y\ast} p_Y^\ast \to Rp_{X\ast} p_X^\ast$ induced by the two unit maps for the adjunctions corresponding to $f$ and $g$ coincide. $\endgroup$ – Dan Petersen Sep 13 '15 at 6:46
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    $\begingroup$ The only interesting relation you might find are when you restrict yourself to locally contant sheaves. Otherwise sheaves are not rigid at all and hav no reasons to have nay sort of good properties with respect to homotopy. You can look to what happen when $f$ and $g$ are the inclusions of the two endpoints of the interval... $\endgroup$ – Simon Henry Oct 7 '15 at 13:09
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By naturality, it suffices to consider the projection map $p : X \times I \to X$. If $X$ and $I$ are locally conctractible Hausdorff spaces and $\mathcal{F}$ is a sheaf of abelian groups on $X$, then the Vietoris-Begle theorem (Bredon, Sheaf theory, II.13) states that the unit $\mathcal{F} \to p_* p^* \mathcal{F}$ is an isomorphism. There are also cohomological and Ext-functor versions of this result. See here in section 2.d.

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