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The statement is not true (in topological spaces or simplicial sets). The composition $f'' \cdot f'$ will only be multiplication by $d$ "up to higher Atiyah--Hirzebruch filtration". For an explicit …

I would like to understand whether the following construction has been studied. Let me model the pointed sphere as $S^n = I^n / \partial I^n$, and let $\Omega^n (-) := map((I^n, \partial I^n), (-, *)) …

My first remark concerns Segal's (4.1). The technical details are key. Segal begins by choosing a locally finite partition of unity $f_i$ subordinate to the cover $U_i$. The definition of this term on …

If you know the map on k-simplices is (n-k)-connected, you can deduce the map on realisations is n-connected. I don't think you can do better in any sort of generality.

Let $X_\bullet \longrightarrow Y_\bullet \longleftarrow Z_\bullet$ be a diagram of simplicial spaces (=bisimplicial sets, if you like). On p. 14-9 of these notes there is an example which shows that …

In brief: For your first question, no. Let $X_\bullet$ be any semi-simplicial space and $Y_\bullet$ have a point in degree zero and be empty in every other degree. Then $\vert X_\bullet \times Y_\bul …

This is a standard irritation. The issue is that $Top$ is not a category internal to $Top$, because it doesn't have a space of objects (and I don't mean for set-theoretic reasons), so what do you mean …