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Cameron
Cameron
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Given a geometric morphismgeometric morphism $$f:\mathscr{F}\to\mathscr{E}$$ where $\mathscr{F},\mathscr{E}$ are toposes, we know that $f^*$ does not preserve internal homs, i.e. $f^*[X,Y]\ncong[f^*X,f^*Y]$. We do have a morphism $\Phi:f^*[X,Y]\to[f^*X,f^*Y]$, though no extra conditions on this morphism (SGA4, Tome 1, Exposé iv, 10.5). Similarly, if $(\mathscr{E},A)$ is a ringed topos, $f^*$ does not preserve internal Ext, i.e. $$f^*\underline{\text{Ext}}^n_A(M,N)\ncong\underline{\text{Ext}}^n_{f^*A}(f^*M,f^*N),$$ where $\underline{\text{Ext}}^n_A(M,N)$ is the right derived functor of the internal $A$-hom $[M,N]_A$. Does there exist a correction to this, e.g. a spectral sequence that relates the two?

Given a geometric morphism $$f:\mathscr{F}\to\mathscr{E}$$ we know that $f^*$ does not preserve internal homs, i.e. $f^*[X,Y]\ncong[f^*X,f^*Y]$. We do have a morphism $\Phi:f^*[X,Y]\to[f^*X,f^*Y]$, though no extra conditions on this morphism (SGA4, Tome 1, Exposé iv, 10.5). Similarly, if $(\mathscr{E},A)$ is a ringed topos, $f^*$ does not preserve internal Ext, i.e. $$f^*\underline{\text{Ext}}^n_A(M,N)\ncong\underline{\text{Ext}}^n_{f^*A}(f^*M,f^*N),$$ where $\underline{\text{Ext}}^n_A(M,N)$ is the right derived functor of the internal $A$-hom $[M,N]_A$. Does there exist a correction to this, e.g. a spectral sequence that relates the two?

Given a geometric morphism $$f:\mathscr{F}\to\mathscr{E}$$ where $\mathscr{F},\mathscr{E}$ are toposes, we know that $f^*$ does not preserve internal homs, i.e. $f^*[X,Y]\ncong[f^*X,f^*Y]$. We do have a morphism $\Phi:f^*[X,Y]\to[f^*X,f^*Y]$, though no extra conditions on this morphism (SGA4, Tome 1, Exposé iv, 10.5). Similarly, if $(\mathscr{E},A)$ is a ringed topos, $f^*$ does not preserve internal Ext, i.e. $$f^*\underline{\text{Ext}}^n_A(M,N)\ncong\underline{\text{Ext}}^n_{f^*A}(f^*M,f^*N),$$ where $\underline{\text{Ext}}^n_A(M,N)$ is the right derived functor of the internal $A$-hom $[M,N]_A$. Does there exist a correction to this, e.g. a spectral sequence that relates the two?

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Cameron
Cameron
  • 171
  • 4

Is there a correction to the failure of geometric morphisms to preserve internal homs?

Given a geometric morphism $$f:\mathscr{F}\to\mathscr{E}$$ we know that $f^*$ does not preserve internal homs, i.e. $f^*[X,Y]\ncong[f^*X,f^*Y]$. We do have a morphism $\Phi:f^*[X,Y]\to[f^*X,f^*Y]$, though no extra conditions on this morphism (SGA4, Tome 1, Exposé iv, 10.5). Similarly, if $(\mathscr{E},A)$ is a ringed topos, $f^*$ does not preserve internal Ext, i.e. $$f^*\underline{\text{Ext}}^n_A(M,N)\ncong\underline{\text{Ext}}^n_{f^*A}(f^*M,f^*N),$$ where $\underline{\text{Ext}}^n_A(M,N)$ is the right derived functor of the internal $A$-hom $[M,N]_A$. Does there exist a correction to this, e.g. a spectral sequence that relates the two?