nLab tangent cohesive (∞,1)-topos

Idea

The tangent (∞,1)-category $T\mathbf{H}$ to a cohesive (∞,1)-topos is itself cohesive: the tangent cohesive (∞,1)-topos.

This $T \mathbf{H}$ is the $\infty$-topos of parameterized spectra in $\mathbf{H}$, hence is the context for cohesive stable homotopy theory.

Properties

Stable extension of cohesion

Let $\mathbf{H}$ be a cohesive (∞,1)-topos.

By the discussion at tangent ∞-category – Examples – Of an ∞-topos the tangent $\infty$-topos $T \mathbf{H}$ constitutes an extension of $\mathbf{H}$ by its stabilization $Stab(\mathbf{H})$:

$$\array{ && Stab(\mathbf{H}) & \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}} & Stab(\infty Grpd) \simeq Spectra \\ && \simeq && \simeq \\ && T_\ast \mathbf{H} && T_\ast \infty Grpd \\ && \downarrow^{\mathrlap{incl}} && \downarrow^{\mathrlap{incl}} \\ \mathbf{H} &\stackrel{\overset{d}{\longrightarrow}}{\underset{\Omega^\infty \circ tot}{\leftarrow}}& T \mathbf{H} & \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}} & T \infty Grpd \\ && {}^{\mathllap{base}}\downarrow \uparrow^{\mathrlap{0}} && {}^{\mathllap{base}}\downarrow \uparrow^{\mathrlap{0}} \\ && \mathbf{H} & \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} & \infty Grpd } \,.$$

Here

• $\Omega^\infty \circ tot \;\colon\; T \mathbf{H} \longrightarrow \mathbf{H}$ assigns the total space of a spectrum bundle;

  its left adjoint is the tangent complex functor;

• $base \;\colon\; T \mathbf{H} \longrightarrow \mathbf{H}$ assigns the base space of a spectrum bundle;

  its left adjoint produces the 0-bundle;

  together these exhibit $T \mathbf{H}$ as an infinitesimal cohesive (infinity,1)-topos over $\mathbf{H}$.

Stable homotopy types

In a tangent cohesive $\infty$-topos $T \mathbf{H}$ all the homotopy types in $T_\ast \mathbf{H} \hookrightarrow T\mathbf{H}$ 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 $E \in T_\ast \mathbf{H}$ and write $GL_1(E) \in Grp(\mathbf{H})$ for its ∞-group of units. Then the ∞-action of this on $E$ is (by the discussion there) exhibited by an object

$$\left( \array{ E//GL_1(E) \\ \downarrow \\ \mathbf{B}GL_1(E) } \right) \;\;\; \in \;\;\; T_{\mathbf{B}GL_1(E)}\mathbf{H} \hookrightarrow T\mathbf{H} \,.$$

More generally, for $Pic(E) \in \mathbf{H}$ the Picard ∞-groupoid of $E$ there is the universal (∞,1)-line bundle

$$(\widehat{Pic(E)} \to Pic(E)) \in T \mathbf{H} \,.$$

Now for any object $X \in \mathbf{H}$ we have the trivial sphere spectrum spectrum bundle over $X$

$$X \times \mathbb{S} \simeq \left( \array{ X \times \mathbb{S} \\ \downarrow \\ X } \right) \;\;\; \in \;\;\; T_{X}\mathbf{H} \hookrightarrow T\mathbf{H} \,.$$

then morphisms in $T \mathbf{H}$ from the latter to the former

$$\left( \array{ X \times \mathbb{S} \\ \downarrow \\ X } \right) \longrightarrow \left( \array{ E//GL_1(E) \\ \downarrow \\ \mathbf{B}GL_1(E) } \right)$$

are equivalently homotopy commuting diagrams of the form

$$\array{ X \times \mathbb{S} &\stackrel{\sigma}{\longrightarrow}& E//GL_1(E) \\ \downarrow && \downarrow \\ X &\stackrel{\chi}{\longrightarrow}& \mathbf{B}GL_1(E) }$$

and hence

1. a choice of twist of E-cohomology $\chi \;\colon \; X \longrightarrow \mathbf{B}GL_1(E)$, modulating a $GL_1(E)$-principal ∞-bundle;

2. an element in the $\chi$-twisted $E$-cohomology of $X$, $\sigma \in E^{\bullet + \chi}(X,E)$, 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 $X$ instead of $X \times \mathbb{S}$:

Proposition

For $X \in \mathbf{H} \stackrel{0}{\hookrightarrow} T \mathbf{H}$ a geometric homotopy type and $E \in Stab(\mathbf{H}) \simeq T_\ast \mathbf{H} \hookrightarrow T \mathbf{H}$ a spectrum object, then the internal hom/mapping stack

$$[X,E]_{T \mathbf{H}} \in T \mathbf{H}$$

(with respect to the Cartesian closed monoidal (∞,1)-category structure on the (∞,1)-topos is equivalently the mapping spectrum

$$[\Sigma^\infty X, E]_{Stab(\mathbf{H})} \in Stab(\mathbf{H}) \hookrightarrow T \mathbf{H} \,,$$

in that

$$[X,E]_{T \mathbf{H}} \simeq [\Sigma^\infty X,E]_{Stab(\mathbf{H})} \,.$$

Proof

Notice that as an object of $T \mathbf{H} \hookrightarrow \mathbf{H}^{seq}$, the object $X$ is the constant (∞,1)-presheaf on $seq$. By the formula for the internal hom in an (∞,1)-category of (∞,1)-presheaves we have

$$[X,E]_\bullet \simeq \mathbf{H}^{seq}(X \times \bullet, E) \,.$$

But since $X$ is constant the object $X \times \bullet$ is for each object of $seq$ the presheaf represented by that object. Therefore by the (∞,1)-Yoneda lemma it follows that

$$[X,E]_\bullet \simeq [X,E_\bullet] \,.$$

This is manifestly the same formula as for the mapping spectrum out of $\Sigma^\infty X$.

Similar kind of arguments give the following more general statement.

Proposition

For $X \in \mathbf{H} \stackrel{0}{\hookrightarrow} T \mathbf{H}$ a geometric homotopy type, for $E \in E_\infty(\mathbf{H})$ an E-∞ ring with $(\widehat{Pic(E)} \to Pic(E)) \hookrightarrow T \mathbf{H}$ its universal (∞,1)-line bundle over its Picard ∞-groupoid, then the internal hom/mapping stack

$$[X,\widehat{Pic(E)}]_{T \mathbf{H}} \in T \mathbf{H}$$

is the object whose

• base homotopy type is the E-∞ ring $[X, Pic(E)]$ of $E$-twist on $X$;

• whose spectrum bundle is the collection of $\chi$-twisted E-cohomology spectra for all twists $\chi$.

In full generality we may formulate the internal hom mapping space in $T \mathbf{H}$ in homotopy type theory notation as follows.

Proposition

For

$$(a \colon A) \;\vdash\; E_a \colon Spectra(\mathbf{H})$$

and

$$(b \colon B) \;\vdash\; F_b \colon Spectra(\mathbf{H})$$

two spectrum bundle dependent types over base homotopy types, $A,B \colon \mathbf{H}$, respectively, then the function type $(E \to F) \colon T\mathbf{H}$ between them (regarded as homotopy types in $T \mathbf{H}$) is

$$\chi \colon (A \to B); \sigma \colon \underset{a \colon A}{\prod} SpMap(E_a,F_{\chi(a)}) \;\vdash\; \underset{a \colon A}{\prod} F_{\chi(a)} \,.$$

Proof

Let $(x:X)\vdash M_x : Spectra$ be another spectrum bundle. The cartesian product $M\times E$ in $T \mathbf{H}$ is then $(x:X),(a:A) \vdash M_x \oplus E_a$, with $\oplus$ also the coproduct (hence the direct sum), since spectra are stable and hence additive. We compute the mapping space $T\mathbf{H}(M\times E,F)$ as follows:

$$\begin{aligned} \sum_{(\phi:X\times A \to B)} \prod_{((x,a):X\times A)} SpMap(M_x\oplus E_a,F_{\phi(x,a)}) &=& \sum_{(\phi:X \to A \to B)} \prod_{(x:X)} \prod_{(a:A)} SpMap(E_a,F_{\phi(x,a)}) \times SpMap(M_x,F_{\phi(x,a)})\\ &=& \prod_{(x:X)} \sum_{(\chi:A \to B)} \left(\prod_{(a:A)} SpMap(E_a,F_{\chi(a)})\right) \times \left( \prod_{(a:A)} SpMap(M_x,F_{\chi(a)}) \right)\\ &=& \prod_{(x:X)} \sum_{\psi : \sum_{(\chi:A \to B)} \prod_{(a:A)} SpMap(E_a,F_{\chi(a)})} SpMap\left(M_x, \prod_{(a:A)} F_{pr_1(\psi)(a)}\right)\\ &=& \sum_{\rho : X \to \sum_{(\chi:A \to B)} \prod_{(a:A)} SpMap(E_a,F_{\chi(a)})} \prod_{(x:A)} SpMap\left(M_x, \prod_{(a:A)} F_{pr_1(\rho(x))(a)}\right) \end{aligned}$$

In the first line, we curry $\phi$, apply the induction principle for dependent maps out of $X\times A$, and also apply the universal property of the coproduct $M_x \oplus E_a$. In the second line, we apply the universal property for mapping into ∞-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 ∞-types?, and also the universal property for mapping into the dependent product $\prod$ 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 $M$ to the claimed function type $(E\to F)$ defined above. (See also this discussion.)

Example

We have the following special cases of prop. .

1. If $E_a = 0$ for all $a \colon A$, and if $B = \ast$, then the function type is

   $$\vdash \; \underset{a \colon A}{\prod} F$$

   which reproduces the mapping spectrum $SpMap(\Sigma^\infty A, F)$ from prop. .

2. If $A = B = \ast$ then the mapping type is

   $$\sigma \colon SpMap(E,F) \;\vdash \; F \colon Spectra$$

3. If $E_a = 0$ for all $a \colon A$ and $F_b = 0$ for all $b \colon B$ then the mapping type is

   $$\chi \colon (A \to B)\;\vdash \; 0 \colon Spectra \,.$$

Cohesive and differential refinement

Let $T\mathbf{H}$ be a tangent cohesive $(\infty,1)$-topos and write $T_\ast \mathbf{H}$ 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.

Proposition

For every $A \in T_\ast \mathbf{H}$ the naturality square

$$\array{ A &\stackrel{}{\longrightarrow}& A/\flat A \\ \downarrow &{}^{(pb)}& \downarrow \\ \Pi(A) &\stackrel{}{\longrightarrow}& \Pi(A/\flat A) }$$

(of the shape modality applied to the homotopy cofiber of the counit of the flat modality) is an (∞,1)-pullback square.

This was observed in (Bunke-Nikolaus-Völkl 13). It is an incarnation of a fracture theorem.

Proof

By cohesion and stability we have the diagram

$$\array{ \flat A &\longrightarrow & A &\stackrel{}{\longrightarrow}& A/\flat A \\ \downarrow^{\mathrlap{\simeq}} && \downarrow && \downarrow \\ \Pi(\flat A) &\longrightarrow& \Pi(A) &\stackrel{}{\longrightarrow}& \Pi(A/\flat A) }$$

where both rows are homotopy fiber sequences. By cohesion the left vertical map is an equivalence. The claim now follows with the homotopy fiber characterization of homotopy pullbacks.

Remark

This means that in stable cohesion every cohesive stable homotopy type is in controled sense a cohesive extension/refinement of its geometric realization geometrically discrete (“bare”) stable homotopy type by the non-discrete part of its cohesive structure;

In particular, $A/\flat A$ may be identified with differential cycle data. Indeed, by stability and cohesion it is the flat de Rham coefficient object

$$A/\flat A = \flat_{dR}\Sigma A$$

of the suspension of $A$, and the map to this quotient is thus the Maurer-Cartan form $\theta_A$. So

$$\array{ A &\stackrel{\theta_A}{\longrightarrow}& \flat_{dR}\Sigma A \\ \downarrow &{}^{(pb)}& \downarrow \\ \Pi(A) &\stackrel{}{\longrightarrow}& \Pi(A/\flat A) }$$

exhibits $A$ as a differential cohomology-coefficient of the generalized cohomology theory $\Pi(A)$.

It follows by the discussion at differential cohomology in a cohesive topos that the further differential refinement $\widehat{A}$ of $A$ should be given by a further homotopy pullback

$$\array{ \widehat{A} &\longrightarrow& \Omega^1(-,Lie(A)) \\ \downarrow &{}^{(pb)}& \downarrow \\ A &\stackrel{\theta_A}{\longrightarrow}& \flat_{dR}\Sigma A \\ \downarrow &{}^{(pb)}& \downarrow \\ \Pi(A) &\stackrel{}{\longrightarrow}& \Pi(A/\flat A) } \,.$$

But of course by the generality of the above proposition, such an $\widehat{A}$ sits itself again in its fracture-like pullback diagram.

Dually:

Proposition

For every $A \in T_\ast \mathbf{H}$ the naturality square

$$\array{ \flat(\Pi_{dR} \Sigma^{-1} A) &\longrightarrow & \Pi_{dR}(\Sigma^{-1} A) \\ \downarrow && \downarrow \\ \flat A &\stackrel{}{\longrightarrow}& A }$$

(of the flat modality applied to the homotopy fiber of the unit of the shape modality) is an (∞,1)-pullback square.

Proof

As before but dually, the diagram extends to a morphism of homotopy cofiber diagrams of the form

$$\array{ \flat(\Pi_{dR} \Sigma^{-1} A) &\longrightarrow & \Pi_{dR}(\Omega A) \\ \downarrow && \downarrow \\ \flat A &\stackrel{}{\longrightarrow}& A \\ \downarrow && \downarrow \\ \flat \Pi(A) &\stackrel{\simeq}{\longrightarrow}& \Pi(A) } \,,$$

and by cohesion the bottom horizontal morphism is an equivalence.

Combining these two statements yields the following (Bunke-Nikolaus-Völkl 13).

Corollary

For $\mathbf{H}$ a cohesive (∞,1)-topos every stable homotopy type $A \in Stab(\mathbf{H}) \hookrightarrow T \mathbf{H}$ sits inside a diagram of the form

$$\array{ && \Pi_{dR} \Omega A && \longrightarrow && \flat_{dR}\Sigma A \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{\theta_A}} && \searrow \\ \flat \Pi_{dR} \Omega A && && A && && \Pi \flat_{dR}\Sigma A \\ & \searrow & & \nearrow & & \searrow && \nearrow_{\mathrlap{\Pi \theta_A}} \\ && \flat A && \longrightarrow && \Pi A } \,,$$

where the two squares are homotopy pullback squares and the two diagonals are the fiber sequences of the Maurer-Cartan form $\theta_A$ and its dual.

Remark

The bottom horizontal morphisms in the diagram in prop. are the canonical points-to-pieces transform.

Remark

This kind of diagram under forming $\pi_0$ has been traditionally known from ordinary differential cohomology and from differential K-theory, and had been used in proposals to axiomatize differential cohomology, see for instance (Bunke 12, prop. 4.57) and see at differential cohomology diagram. Here we see that this holds fully generally for every stable cohesive homotopy type. If one still regards this diagram as characteristic of “differential” refinement it hence exhibits every cohesive stable type as a coefficients of some differential cohomology theory. This is a strong version of the synthetic notion “differential cohomology in a cohesive topos” . For more on this see also at smooth spectrum.

Related concepts

• sheaf of spectra

• smooth spectrum

• twisted generalized cohomology

• twisted bivariant cohomology

• twisted differential cohomology

• twisted smooth cohomology in string theory

References

The idea of forming $T_\ast \mathbf{H}$ as a home for nontrivial stable homotopy types was originally suggested by Georg Biedermann and André Joyal, see section 35 of

• André Joyal, Notes on Logoi, 2008 (pdf)

and see the further references at tangent (infinity,1)-topos.

Discussion of differential cohomology in $T_\ast Smooth \infty Grpd \simeq Stab(Smooth \infty Grpd)$ is in

• Ulrich Bunke, Thomas Nikolaus, Michael Völkl, Differential cohomology theories as sheaves of spectra (arXiv:1311.3188)

For details see at differential cohomology hexagon.

The above discussion of geometric twisted generalized cohomology as cohomology in the tangent cohesive $\infty$-topos was presented in

• Urs Schreiber, talk at Twists, generalised cohomology and applications, October 2013 (talk notes)

Discussion in a comprehensive context of cohesion is in section 4.2.3 of

• Urs Schreiber, differential cohomology in a cohesive topos (arXiv:1310.7930)