CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The singular cohomology (also Betti cohomology) of a topological space $X$ is the cohomology in ∞Grpd of its fundamental ∞-groupoid $\Pi(X)$:
for $\mathcal{B}^n \mathbb{Z} \in \infty Grpd$ the Eilenberg-MacLane object with the group $\mathbb{Z}$ in degree $n$, the degree $n$-singular cohomology of $X$ is
With $\infty Grpd$ presented by the category sSet of simplicial sets, the fundamental $\infty$-groupoid $\Pi(X)$ is modeled by the Kan complex
the singular simplicial complex of $X$.
The object $\mathcal{B}^n \mathbb{Z}$ is usefully modeled by the simplicial set
which is
the underlying simplicial set under the forgetful functor
from abelian simplicial groups to simplicial sets;
of the abelian simplicial group $\Xi \mathbb{Z}[n]$ which is the image under the Dold-Kan correspondence
of the chain complex
concentrated in degree $n$.
So in this model we have
A cocycle in this cohomology theory is a cochain on a simplicial set, on the singular complex $Sing X$.
Using the adjunction $(F \dashv U)$ this is isomorphic to
where
is the free abelian simplicial group on the simplicial set $Sing X$: this is the simplicial abelian group of singular chains of $X$. Its elements are formal sums of continuous maps $\Delta^n_{Top} \to X$. In this form
Using next the Dold-Kan adjunction this is
where
is the Moore complex of normalized chains of $\mathbb{Z}[Sing X]$: this is the complex of singular chains, formal sums over $\mathbb{Z}$ of simplices in $X$.
This way singular cohomology is the abelian dual of singular homology.
…
If the topological space $X$ is semi-locally contractible (meaning: any open subset $U\subset X$ has an open cover $W$ by open subsets $W_i\subset U$ that are contractible in $U$), then the sheaf cohomology of $X$ is isomorphic to the singular cohomology of $X$ for any abelian group of coefficients.
This was proved in (Sella 16).
A previous version of this entry led to the following discussion, which later led to extensive discussion by email. Partly as a result of this and similar discussions, there is now more information on how Kan complexes are $\infty$-groupoids at