(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
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 obtained from an abelian group object by delooping that times.
An object that is both -truncated as well as -connected.
Let be an (∞,1)-topos.
For an Eilenberg-MacLane object of degree
This appears as HTT, def. 22.214.171.124
The next proposition asserts that Eilenberg-MacLane objects defined this way are shifted (∞,1)-categorical group objects:
For an (∞,1)-topos, its (∞,1)-category of pointed objects, the full sub-(∞,1)-category on discrete objects (0-truncated objects) and , write
for the (∞,1)-functor that assigns the -th categorical homotopy groups.
For this establishes an equivalence between the full subcategory on degree 0 Eilenberg-MacLane objects and pointed objects of ; moreover, the restriction is equivalent to the identity.
For this establishes an equivalence between the full subcategory on degree 1 Eilenberg-MacLane objects and the category of group objects in .
For this establishes an equivalence between the full subcategory on degree Eilenberg-MacLane objects and the category of commutative group objects in .
For an (∞,1)-topos and write for the homotopy inverse to the equivalence induced by by the above proposition. For an (abelian) group object we say that
is the degree -Eilenberg-MacLane object of .
We have that
is the -fold delooping of the discrete group object .
The categorical homotopy groups are defined in terms of the canonical powering of over ∞Grpd
For fixed ∞-groupoid this
preserves -limits and hence pullbacks. It follows that the categorical homotopy groups of the loop space object are those of , shifted down by one degree.
By the above proposition on the equivalence between Eilenberg-MacLane objects and group objects, this identifies .
In the archetypical (∞,1)-topos Top ∞Grpd the notion of Eilenberg-MacLane object reduces to the traditional notion of Eilenberg-MacLane space.
In -sheaf -toposes
Recall that an (∞,1)-sheaf/∞-stack (∞,1)-topos may be presented by the model structure on simplicial sheaves on .
In terms of this model the Eilenberg-Mac Lane objects (for abelian ) are the Eilenberg-MacLane sheaves of abelian sheaf cohomology theory.
Under the Dold–Kan correspondence
chain complexes of abelian groups concentrated in degree map into simplicial sets
and these to the corresponding constant simplicial sheaves on the site , that we denote by the same symbol, for convenience.
Under the equivalence
of with the Kan complex-enriched full subcategory of on fibrant cofibrant objects, this identifies the fibrant reeplacement – the ∞-stackification – of with the Eilenberg-MacLane object in .
The notion of cohomology in the (∞,1)-topos with coefficients in an object is often taken to be restricted to the case where is an Eilenberg-MacLane object.
For an abelian group object, and , the degree -cohomology of an object is the cohomology with coefficients in :
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
- Jardine, Fields Lectures: Simplicial Presheaves (pdf)