(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
The notion of Eilenberg–Mac Lane object in an (∞,1)-topos or stable (∞,1)-category generalizes the notion of Eilenberg–Mac Lane space from the (∞,1)-topos Top of topological spaces or the stable (∞,1)-category of spectra:
it is an object $\mathbf{B}^n A$ obtained from an abelian group object $A$ by delooping that $n$ times.
An object that is both $n$-truncated as well as $n$-connected.
Let $\mathbf{H}$ be an (∞,1)-topos.
For $n \in \mathbb{N}$ an Eilenberg-MacLane object $X$ of degree $n$
a pointed object $* \to X \in \mathbf{H}$
which is both $n$-connective as well as $n$-truncated.
This appears as HTT, def. 7.2.2.1
If one drops the condition that $X$ has a global point, then this is the definition of ∞-gerbes.
The next proposition asserts that Eilenberg-MacLane objects defined this way are shifted (∞,1)-categorical group objects:
For $\mathbf{H}$ an (∞,1)-topos, $\mathbf{H}_*$ its (∞,1)-category of pointed objects, $Disc(\mathbf{H})$ the full sub-(∞,1)-category on discrete objects (0-truncated objects) and $n \in \mathbb{N}$, write
for the (∞,1)-functor that assigns the $n$-th categorical homotopy groups.
For $n = 0$ this establishes an equivalence between the full subcategory on degree 0 Eilenberg-MacLane objects and pointed objects of $Disc(\mathbf{H})$; moreover, the restriction $\pi_0:Disc(\mathbf{H}_*)\to Disc(\mathbf{H}_*)$ is equivalent to the identity.
For $n = 1$ this establishes an equivalence between the full subcategory on degree 1 Eilenberg-MacLane objects and the category of group objects in $Disc(\mathbf{H})$.
For $n \geq 2$ this establishes an equivalence between the full subcategory on degree $n$ Eilenberg-MacLane objects and the category of commutative group objects in $Disc(\mathbf{H})$.
This is HTT, prop. 7.2.2.12.
For $\mathbf{H}$ an (∞,1)-topos and $n \in \mathbb{N}$ write $K(-,n)$ for the homotopy inverse to the equivalence induced by $\pi_n$ by the above proposition. For $A \in Disc(\mathbf{H})$ an (abelian) group object we say that
is the degree $n$-Eilenberg-MacLane object of $A$.
We have that
check
The categorical homotopy groups are defined in terms of the canonical powering of $\mathbf{H}$ over ∞Grpd
For fixed ∞-groupoid $K$ this
preserves $(\infty,1)$-limits and hence pullbacks. It follows that the categorical homotopy groups of the loop space object $\Omega K(A,n)$ are those of $K(A,n)$, shifted down by one degree.
By the above proposition on the equivalence between Eilenberg-MacLane objects and group objects, this identifies $\Omega K(A,n) \simeq K(A,n-1)$.
In the archetypical (∞,1)-topos Top$\simeq$ ∞Grpd the notion of Eilenberg-MacLane object reduces to the traditional notion of Eilenberg-MacLane space.
Recall that an (∞,1)-sheaf/∞-stack (∞,1)-topos $\mathbf{H} = Sh_{(\infty,1)}(C)$ may be presented by the model structure on simplicial sheaves on $C$.
In terms of this model the Eilenberg-Mac Lane objects $K(A,n) \in \mathbf{H}$ (for abelian $A$) are the Eilenberg-MacLane sheaves of abelian sheaf cohomology theory.
Under the Dold–Kan correspondence
chain complexes $A[n]$ of abelian groups concentrated in degree $n$ map into simplicial sets
and these to the corresponding constant simplicial sheaves on the site $C$, that we denote by the same symbol, for convenience.
Under the equivalence
of $\mathbf{H}$ with the Kan complex-enriched full subcategory of $sSh(C)$ on fibrant cofibrant objects, this identifies the fibrant reeplacement – the ∞-stackification – of $\Gamma(A[n])$ with the Eilenberg-MacLane object in $\mathbf{H}$.
The notion of cohomology in the (∞,1)-topos $\mathbf{H}$ with coefficients in an object $\mathcal{A} \in \mathbf{H}$ is often taken to be restricted to the case where $\mathcal{A}$ is an Eilenberg-MacLane object.
For $A \in Disc(\mathbf{A})$ an abelian group object, and $n \in \mathbb{N}$, the degree $n$-cohomology of an object $X \in \mathbf{H}$ is the cohomology with coefficients in $K(A,n)$:
The general discussion of Eilenberg-MacLane objects is in section 7.2.2 of
For a discussion of Eilenberg-MacLane objects in the context of the model structure on simplicial presheaves see top of page 4 of