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If $H$ is the terminal category (=sheaves on the empty space), then $\pi_k^HS^n$ (notation for homotopy groups of "spheres" in $H$) is known! The slice category $H=\mathrm{Spaces}/B$ is an $(\infty,1 …

Jardine has recent a paper called "The Verdier hypercovering theorem", based on an earlier paper called "Cocycle categories", which you should look at if you haven't. He doesn't exactly say it this …

An $n$-monomorphism is a map $A\to B$ for which $Map(X,A)\to Map(X,B)$ has all homotopy fibers $n$-truncated, for all $X$; a space (=$\infty$-groupoid) is $n$-truncated if its homotopy groups all vani …

I don't think so. Take $n=2$, and consider a map $f\colon b\to a$ between objects of $\mathrm{Fun}(I^2, \mathcal{S})$. If $a$ and $b$ are pullback squares, then any map $f$ between them is relativel …

I think the point is that a map $f\colon A\to B$ in a slice $\mathscr{C}_{/Y}$ is $k$-connected if and only if the underlying map in $\mathscr{C}$ is $k$-connected, and this is supposed to be a formal …