Stable Homotopy theory
The tangent (∞,1)-category to a cohesive (∞,1)-topos is itself cohesive: the tangent cohesive (∞,1)-topos.
This is the -topos of parameterized spectra in , hence is the context for cohesive stable homotopy theory.
Stable extension of cohesion
Let be a cohesive (∞,1)-topos.
By the discussion at tangent ∞-category – Examples – Of an ∞-topos the tangent -topos constitutes an extension of by its stabilization :
Stable homotopy types
In a tangent cohesive -topos all the homotopy types in are stable homotopy types.
Cohomology – Twisted bivariant generalized geometric cohomology theory
Where the (∞,1)-categorical hom-space in a general (∞,1)-topos constitute a notion of cohomology, those of a tangent (∞,1)-topos specifically constitute twisted generalized cohomology, in fact twisted bivariant cohomology.
For consider a spectrum object and write for its ∞-group of units. Then the ∞-action of this on is (by the discussion there) exhibited by an object
More generally, for the Picard ∞-groupoid of there is the universal (∞,1)-line bundle
Now for any object we have the trivial sphere spectrum spectrum bundle over
then morphisms in from the latter to the former
are equivalently homotopy commuting diagrams of the form
a choice of twist of E-cohomology , modulating a -principal ∞-bundle;
an element in the -twisted -cohomology of , , hence a section of the associated (∞,1)-line bundle.
If we consider the internal hom instead of the external (∞,1)-categorical hom space then things work even more nicely and we can use just instead of :
For a geometric homotopy type and a spectrum object, then the internal hom/mapping stack
(with respect to the Cartesian closed monoidal (∞,1)-category structure on the (∞,1)-topos is equivalently the mapping spectrum
Notice that as an object of , the object is the constant (∞,1)-presheaf on . By the formula for the internal hom in an (∞,1)-category of (∞,1)-presheaves we have
But since is constant the object is for each object of the presheaf represented by that object. Therefore by the (∞,1)-Yoneda lemma it follows that
This is manifestly the same formula as for the mapping spectrum out of .
Similar kind of arguments give the following more general statement.
In full generality we may formulate the internal hom mapping space in in homotopy type theory notation as follows.
two spectrum bundle dependent types over base homotopy types, , respectively, then the function type between them (regarded as homotopy types in ) is
Let be another spectrum bundle. The cartesian product in is then , with also the coproduct (hence the direct sum), since spectra are stable and hence additive. We compute the mapping space as follows:
In the first line, we curry , apply the induction principle for dependent maps out of , and also apply the universal property of the coproduct . In the second line, we apply the universal property for mapping into Σ-types (the “type-theoretic axiom of choice”) and also that for dependent functions into a product. In the third line we apply the associativity of Σ-types, and also the universal property for mapping into the dependent product of spectra. Finally, in the fourth line, we apply the type-theoretic axiom of choice again in the other direction. The resulting type is the mapping space from to the claimed function type defined above. (See also this discussion.)
We have the following special cases of prop. 3.
If for all , and if , then the function type is
which reproduces the mapping spectrum from prop. 1.
If then then the mapping type is
If for all and for all then the mapping type is
Cohesive and differential refinement
Let be a tangent cohesive -topos and write for the stable (∞,1)-category of spectrum objects inside it. We discuss how every stable homotopy type here canonically sits in the middle of a differential cohomology diagram.
This was observed in (Bunke-Nikolaus-Völkl 13). It is an incarnation of a fracture theorem.
For every the naturality square
(of the flat modality applied to the homotopy fiber of the unit of the shape modality) is an (∞,1)-pullback square.
As before but dually, the diagram extends to a morphism of homotopy cofiber diagrams of the form
and by cohesion the bottom horizontal morphism is an equivalence.
Combining these two statements yields the following (Bunke-Nikolaus-Völkl 13).
For a cohesive (∞,1)-topos every stable homotopy type sits inside a diagram of the form
where the two squares are homotopy pullback squares and the two diagonals are the fiber sequences of the Maurer-Cartan form and its dual.
The idea of forming as a home for nontrivial stable homotopy types was originally suggested by Georg Biedermann and André Joyal, see section 35 of
and see the further references at tangent (infinity,1)-topos.
Discussion of differential cohomology in is in
The above discussion of geometric twisted generalized cohomology as cohomology in the tangent cohesive -topos was presented in
Discussion in a comprehensive context of cohesion is in section 4.2.3 of