under construction

Contents

Idea

There are several different notions of what what a relative notion of cohomology is.

Push-forward not to the point, but to another space

In ordinary cohomology – as described there – for a given ambient (∞,1)-topos $H$ the cohomology of an object $X\in H$ with coefficients in an object $A\in H$ is the set/group of connected components of the derived hom space

More generally, we may consider a kind of internal hom instead, whose components are themselves (∞,1)-presheaves.

Fix $p:X\to Y$ a morphism in $H$. Let $\mathrm{Op}(Y)\subset H_{/Y}$ be some subcategory of the over quasi-category of $H$ over $Y$, to be regarded as the category of open subsets of $Y$ in $H$.

Then the functor

with $X{\mid }_U:=X{\times }_{Y}U$ the fiber product, produces an $(\mathrm{\infty }\mathrm{,1})$-presheaf on $\mathrm{Op}(Y)$. This can be regarded as being a cocycle in the relative cohomology of $X$ with coefficients in $A$ .

As the formula shows, this is the direct image of $A{\mid }_X$ along $p:X\to Y$.

Notice that in the case the $Y=*$ the terminal object in $H$ and taking the only “open subset” of $*$ to be the point itself, this produces an $(\mathrm{\infty }\mathrm{,1})$-presheaf on the point, which canonically identifies with an ∞-groupoid that coincides with the one used in the non-relative version of cohomology.

Quotient cocycles

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