homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
(see also Chern-Weil theory, parameterized homotopy theory)
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Goodwillie calculus – approximation of homotopy theories by stable homotopy theories
A parameterized spectrum is a bundle of spectra (May-Sigurdsson 06), hence a stable homotopy type in parameterized homotopy theory.
Specifically, for $X$ a homotopy type thought of as an ∞-groupoid, then a spectrum parameterized over $X$ is equivalently an (∞,1)-functor $X \longrightarrow Spec$ from $X$ to the stable (∞,1)-category of spectra (Ando-Blumberg-Gepner 11): this assigns to each object of $X$ a spectrum, to each morphism an equivalence of spectra, to each 2-morphism a homotopy between such equivalences, and so forth.
More generally, given an (∞,1)-topos $\mathbf{H}$, then its tangent (∞,1)-topos $T\mathbf{H}$ is the (∞,1)-category of all spectrum objects in $\mathbf{H}$ parameterized over any object of $\mathbf{H}$ (an observation promoted by Joyal).
The intrinsic cohomology of such a tangent (∞,1)-topos of parameterized spectra is twisted generalized cohomology in $\mathbf{H}$, and generally is twisted bivariant cohomology in $\mathbf{H}$.
For more see also at tangent cohesive (∞,1)-topos.
For $X$ a connected homotopy type, then the $X$-parameterized spectra are equivalently the module spectra over the ∞-group ∞-ring $\mathbb{S}[\Omega X]$ of the ∞-group corresponding to the loop space.
To see this, use first that $X$-parameterized spectra are equivalently ∞-functors of (∞,1)-categories of the form
from the delooping ∞-groupoid of $\Omega X$ to the (∞,1)-category of spectra, then use that these are equivalenty Spectrum enriched functors out of the one-object spectrum enriched $\infty$-category with hom-spectrum $\mathbb{S}[\Omega X] \simeq \Sigma^\infty_+ \Omega X$.
Moreover $\mathbb{S}[\Omega X]$-module spectra are equivalent to comodule spectra over the coalgebra $\mathbb{S}[X] = \Sigma^\infty_+ X$ induced from the coalgebra object? structure of $X$ in the Cartesian monoidal (∞,1)-category ∞Grpd via the diagonal (here), and using that $\Sigma^\infty$ is a strong monoidal functor:
(Hess-Shipley 14, theorem 1.2 with prop. 5.18)
See also at A-theory.
For any map $f\colon X\longrightarrow Y$ between ∞-groupoids, the parameterized spectra form a Wirthmüller context version of the yoga of six functors, in that
in that $f^\ast$ is not only a strong monoidal functor but also a strong closed functor, hence that Frobenius reciprocity holds.
Moreover, along (co-)spans of morphisms pull-push $(f_!\dashv f^\ast)$ satisfies the Beck-Chevalley condition (Hopkins-Lurie 14, prop. 4.3.3).
One way to summarize all this is to say that parameterized spectra over ∞Grpd constitute a linear homotopy type theory (Schreiber 14).
Parameterized spectra over a fixed base (in any suitable model category) are discussed in
A comprehensive textbook account on parameterized spectra in ∞Grpd $\simeq$ $L_{whe}$Top is in
A formulation of aspects of this in (∞,1)-category theory is in
See also the further references at (∞,1)-module bundle.
Discussion of the Beck-Chevalley condition is in prop. 4.3.3 of
Discussion of the Kozul duality between $\mathbb{S}[\Omega X]$-module spectra and $\mathbb{S}[X]$-comodule spectra is in
Discussion as a linear homotopy type theory is in