nLab tangent (infinity,1)-category

Idea

For $K$ a locally presentable (∞,1)-category whose objects we think of as spaces of sorts, its tangent $(\infty,1)$-category

$$(T_{K^{op}})^{op} \to K$$

is an (∞,1)-category over $K$, whose objects may be thought of as spaces that are infinitesimal thickenings of those of $K$.

More concretely, the tangent $(\infty,1)$-category $T_C \to C$ for $C = K^{op}$ is the fiberwise stabilization of the codomain fibration $Func(\Delta[1], C) \to C$.

This generalizes – as discussed at deformation theory – the classical example of the bifibration Mod $\to$ CRing of the category of all modules over the cateory CRing of all commutative rings:

the fiber of the tangent $(\infty,1)$-category $T_C$ over an object $A \in C$ may be thought of as the $(\infty,1)$-category of square-0-extensions $A \oplus N$ of $A$, for $N$ a module over $A$. Dually, in $K = C^{op}$ we may think of these as being infinitesimal neighbourhoods of 0-sections of vector bundles – or rather of quasicoherent sheaves – over whatever space $A$ is regarded to be the algebra of functions on.

A remarkable amount of information about the geometry of these spaces/objects in $K$ is encoded in the fiber of the tangent $(\infty,1)$-category over them. Notably the left adjoint (∞,1)-functor

$$\Omega : C \to T_C$$

to the domain projection $dom : T_C \to C$ turns out to send each $A$ to its cotangent complex $\Omega(A)$, to be thought of as the module of Kähler differentials on the space that $A$ is functions on.

A 1-categorical approximation to the notion of tangent $(\infty,1)$-category is that of tangent category.

Definition

Let $\mathcal{C}$ be a locally presentable (∞,1)-category.

Definition

(fiberwise stabilization)

For $\mathcal{C}' \to \mathcal{C}$ a categorical fibration, the fiberwise stabilization $Stab(\mathcal{C}' \to \mathcal{C})$ is – roughly – the fibration universal with the property that for each $A \in C$ its fiber over $A$ is the stabilization $Stab(\mathcal{C}'_A)$ of the fiber $\mathcal{C}'_A$ over $A$.

This is (Lurie, section 1.1) formulated in view of (Lurie, remark 1.1.8). There $Stab(\mathcal{C}' \to \mathcal{C})$ is called the stable envelope .

Definition

(tangent $(\infty,1)$-category)

Thetangent $(\infty,1)$-category $T_{\mathcal{C}} \to \mathcal{C}$ is the fiberwise stabilization of the codomain fibration $cod : \mathcal{C}^{\Delta^1} \to \mathcal{C}$:

$$(T_{\mathcal{C}} \stackrel{p}{\to} \mathcal{C}) := Stab(Func(\Delta[1], \mathcal{C}) \stackrel{cod}{\to} \mathcal{C} ) \,.$$

This is DT, def 1.1.12.

For a maybe more explicit definition see below at Tangent ∞-topos – General.

Explicitly, the tangent $\infty$-category is given as follows.

Remark

Given a presentable (∞,1)-category $\mathcal{C}$, the (∞,1)-functor

$$\chi_{cod} \colon \mathcal{C}^{op} \to (\infty,1)Cat$$

which classifies the codomain fibration $cod \colon \mathcal{C}^{\Delta^1} \to \mathcal{C}$ under the (∞,1)-Grothendieck construction factors through the wide non-full inclusion

$$(\infty,1)Cat^R \to (\infty,1)Cat$$

of (∞,1)-functors which are right adjoint (∞,1)-functors. For these the further (now full) inclusion

$$i \colon (\infty,1)StabCat^R \hookrightarrow (\infty,1)Cat^R$$

of the stable (∞,1)-categories has a right adjoint (∞,1)-functor

$$(i \dashv Stab)$$

given by stabilization. (Note that this is not a functor on all of $(\infty,1)Cat$, where instead the obstructions to functoriality are given by Goodwillie calculus.)

So the classifying map of the codomain fibration factors through this and hence we can postcompose with the stabilization functor to obtain

$$i \circ Stab i \chi_{cod} \colon \mathcal{C}^{op} \to (\infty,1)Cat \,.$$

This sends an object $c \in \mathcal{C}$ to the stabilization of the slice (∞,1)-category over $c$:

$$Stab \circ \chi_{cod} \colon c \mapsto Stab(\mathcal{C}_{/c}) \,.$$

Again by the (∞,1)-Grothendieck construction this classifies a Cartesian fibration over $\mathcal{C}$ and this now is the tangent $(\infty,1)$-category projection

$$\array{ T_{\mathcal{C}} \\ \downarrow^{\mathrlap{p}} \\ \mathcal{C} } \,.$$

This is the first part of the proof of DT. prop. 1.1.9.

Properties

Presentability and limits

Proposition

The tangent $(\infty,1)$-category $T_C$ of the locally presentable (∞,1)-category $C$ is itself a locally presentable $(\infty,1)$-category.

In particular, it admits all (∞,1)-limits and (∞,1)-colimits.

This is (Lurie, prop. 1.1.13).

Moreover:

Proposition

A diagram in the tangent $(\infty,1)$-category $T_{\mathcal{C}}$ is an (∞,1)-(co-)limit precisely if

1. it is a relative (∞,1)-(co-)limit with respect to the projection $p \colon T_{\mathcal{C}} \to \mathcal{C}$;

2. its image under this projection is an (∞,1)-(co-)limit in $\mathcal{C}$.

(Lurie, HigherAlgebra, prop. 7.3.1.12)

Relation to modules

We discuss how the tangent $(\infty,1)$-category construction indeed generalizes the equivalence between the tangent category over CRing and the category Mod of all modules over commutative rings.

Proposition

Let $\mathcal{O}^\otimes$ be a coherent (∞,1)-operad and let $\mathcal{C}^\otimes \to \mathcal{O}^\otimes$ be a stable $\mathcal{O}$-monoidal (∞,1)-category.

Let

$$A \in Alg_\mathcal{O}(\mathcal{C})$$

be an $\mathcal{O}$-algebra in $\mathcal{C}$. Then the stabilization of the over-(∞,1)-category over $A$ is canonically equivalent to $Func_\mathcal{O}(\mathcal{O}, Mod_A^\mathcal{O}(\mathcal{C}))$

$$Stab( Alg_\mathcal{O}(\mathcal{C})/A) \simeq Func_\mathcal{O}(\mathcal{O}, Mod_A^\mathcal{O}(\mathcal{C})) \,.$$

This is (Lurie, theorem 1.5.14).

Proposition

Let $\mathcal{O}^\otimes$ be a coherent (∞,1)-operad and let $\mathcal{C}^\otimes \to \mathcal{O}^\otimes$ be a presentable stable $\mathcal{O}$-monoidal (∞,1)-category. Then there is a canonical equivalence

$$\phi : T_{Alg_\mathcal{O}(\mathcal{C})} \stackrel{\simeq}{\to} Alg_\mathcal{O}(\mathcal{C}) \times_{Func(\mathcal{O}, Alg_\mathcal{O}(\mathcal{C}))} Func_\mathcal{O}(\mathcal{O}, Mod^\mathcal{O}(\mathcal{C}))$$

of presentble fibrations over $Alg_\mathcal{O}(\mathcal{C})$.

This is (Lurie, theorem, 1.5.19).

In words this says that under the given assumptions, objects of $T_{\mathcal{C}}$ may be identified with pairs

$$(A, N)$$

where

• $A$ is an $\mathcal{O}$-algebra in $\mathcal{C}$;

• $N$ is an $A$-module.

Cotangent complex

From its definition as the fiberwise stabilization of the codomain fibration $cod : Func(\Delta[1], C) \to C$ the tangent $(\infty,1)$-category $p : T_C \to C$ inherits a second $(\infty,1)$-functor to $C$, coming from the domain evaluation

$$dom : T_C \to C \,.$$

Definition/Proposition

(cotangent complex)

The domain evaluation $dom : T_C \to C$ admits a left adjoint (∞,1)-functor

$$(\Omega \dashv dom) : T_C \stackrel{\overset{\Omega}{\leftarrow}}{\underset{dom}{\to}} C$$

that is also a section of $p : T_C \to C$ in that

$$(C \stackrel{\Omega}{\to} T_C \stackrel{p}{\to} C) \simeq Id_C$$

and which hence exhibits $C$ as a retract of $T_C$.

This $\Omega$ is the cotangent complex $(\infty,1)$-functor : for $A \in C$ the object $\Omega(A)$ is the cotangent complex of $A$.

This is (Lurie, def. 1.2.2, remark 1.2.3).

In more detail this adjunction is the composite

$$(\Omega \dashv dom) \;\colon\; T_C \stackrel{\overset{\Sigma^\infty_C}{\leftarrow}}{\underset{\Omega^\infty_{C}}{\to}} C^{\Delta^1} \stackrel{\overset{const}{\leftarrow}}{\underset{dom}{\to}} C \colon \Omega \,,$$

where $(\Sigma^\infty_C \dashv \Omega^\infty_C)$ is the fiberwise stabilization relative adjunction, def. .

Tangent $\infty$-topos of an $\infty$-topos

We discuss how the tangent $\infty$-category of an (∞,1)-topos is itself an (∞,1)-topos over the tangent $\infty$-category of the original base (∞,1)-topos.

In terms of Omega-spectrum spectrum objects this is due to (Joyal 08) joint with Georg Biedermann. In terms of excisive functors this is due to observations by Georg Biedermann, Charles Rezk and Jacob Lurie, see at $n$-Excisive functor – Properties – $n$-Excisive reflection

General

Definition

Let $seq$ be the diagram category as follows:

$$seq \coloneqq \left\{ \array{ && \vdots && \vdots \\ && \downarrow && \\ \cdots &\to& x_{n-1} &\stackrel{p_{n-1}}{\longrightarrow}& \ast \\ &&{}^{\mathllap{p_{n-1}}}\downarrow &\swArrow& \downarrow^{\mathrlap{i_n}} & \searrow^{\mathrm{id}} \\ &&\ast &\underset{i_n}{\longrightarrow}& x_n &\stackrel{p_n}{\longrightarrow}& \ast \\ && &{}_{\mathllap{id}}\searrow& {}^{\mathllap{p_n}}\downarrow &\swArrow& \downarrow^{\mathrlap{i_{n+1}}} \\ && && \ast &\stackrel{i_{n+1}}{\longrightarrow}& x_{n+1} &\to& \cdots \\ && && && \downarrow \\ && && && \vdots } \right\}_{n \in \mathbb{Z}} \,.$$

(Joyal 08, section 35.5)

Remark

Given an (∞,1)-topos $\mathbf{H}$, an (∞,1)-functor

$$X_\bullet \;\colon\; seq \longrightarrow \mathbf{H}$$

is equivalently

1. a choice of object $B \in \mathbf{H}$ (the image of $\ast \in seq$]);

2. a sequence of objects $\{X_n\} \in \mathbf{H}_{/B}$ in the slice (∞,1)-topos over $B$;

3. a sequence of morphisms $X_n \longrightarrow \Omega_B X_{n+1}$ from $X_n$ into the loop space object of $X_{n+1}$ in the slice.

This is a prespectrum object in the slice (∞,1)-topos $\mathbf{H}_{/B}$.

A natural transformation $f \;\colon \;X_\bullet \to Y_\bullet$ between two such functors with components

$$\left\{ \array{ X_n &\stackrel{f_n}{\longrightarrow}& Y_n \\ \downarrow^{\mathrlap{p_n^X}} && \downarrow^{\mathrlap{p_n^Y}} \\ B_1 &\stackrel{f_b}{\longrightarrow}& B_2 } \right\}$$

is equivalently a morphism of base objects $f_b \;\colon\; B_1 \longrightarrow B_2$ in $\mathbf{H}$ together with morphisms $X_n \longrightarrow f_b^\ast Y_n$ into the (∞,1)-pullback of the components of $Y_\bullet$ along $f_b$.

Therefore the (∞,1)-presheaf (∞,1)-topos

$$\mathbf{H}^{seq} \coloneqq Func(seq, \mathbf{H})$$

is the codomain fibration of $\mathbf{H}$ with “fiberwise pre-stabilization”.

A genuine spectrum object is a prespectrum object for which all the structure maps $X_n \stackrel{\simeq}{\longrightarrow} \Omega_B X_{n+1}$ are equivalences. The full sub-(∞,1)-category

$$T \mathbf{H} \hookrightarrow \mathbf{H}^{seq}$$

on the genuine spectrum objects is therefore the “fiberwise stabilization” of the self-indexing, hence the tangent $(\infty,1)$-category.

Lemma

(spectrification is left exact reflective)

The inclusion of spectrum objects into $\mathbf{H}^{seq}$ is left reflective, hence it has a left adjoint (∞,1)-functor $L$ – spectrification – which preserves finite (∞,1)-limits.

$$T \mathbf{H} \stackrel{\overset{L_{lex}}{\leftarrow}}{\hookrightarrow} \mathbf{H}^{seq} \,.$$

(Joyal 08, section 35.1)

Proof

Forming degreewise loop space objects constitutes an (∞,1)-functor $\Omega \colon \mathbf{H}^{seq} \longrightarrow \mathbf{H}^{seq}$ and by definition of $seq$ this comes with a natural transformation out of the identity

$$\theta \;\colon\; id \longrightarrow \Omega \,.$$

This in turn is compatible with $\Omega$ in that

$$\theta \circ \Omega \simeq \Omega \circ \theta \;\colon\; \rho \longrightarrow \rho \circ \rho = \rho^2 \,.$$

Consider then a sufficiently deep transfinite composition $\rho^{tf}$. By the small object argument available in the presentable (∞,1)-category $\mathbf{H}$ this stabilizes, and hence provides a reflection $L \;\colon\; \mathbf{H}^{seq} \longrightarrow T \mathbf{H}$.

Since transfinite composition is a filtered (∞,1)-colimit and since in an (∞,1)-topos these commute with finite (∞,1)-limits, it follows that spectrum objects are an left exact reflective sub-(∞,1)-category.

Proposition

For $\mathbf{H}$ an (∞,1)-topos over the base (∞,1)-topos $\infty Grpd$, its tangent (∞,1)-category $T \mathbf{H}$ is an (∞,1)-topos over the base $T \infty Grpd$ (and hence in particular also over $\infty Grpd$ itself).

(Joyal 08, section 35.5)

Proof

By the the spectrification lemma $T \mathbf{H}$ has a geometric embedding into the (∞,1)-presheaf (∞,1)-topos $\mathbf{H}^{seq}$, and this implies that it is an (∞,1)-topos (by the discussion there).

Moreover, since both adjoint (∞,1)-functor in the global section geometric morphism $\mathbf{H} \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\longrightarrow}} \infty Grpd$ preserve finite (∞,1)-limits they preserve spectrum objects and hence their immediate (∞,1)-presheaf prolongation immediately restricts to the inclusion of spectrum objects

$$\array{ T \mathbf{H} &\stackrel{\overset{T \Delta}{\leftarrow}}{\underset{T \Gamma}{\longrightarrow}}& T \infty Grpd \\ \downarrow^{\mathrlap{incl}} && \downarrow^{\mathrlap{incl}} \\ \mathbf{H} & \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\longrightarrow}} & \infty Grpd } \,.$$

Remark

This statement also follows from the general theory of excisive functors, and in this form it is due to Charles Rezk. See at n-Excisive functor – Properties – n-Excisive reflection for the above fact and its generalization to “Goodwillie jet bundles”.

Remark

We may think of the tangent $\infty$-topos $T \mathbf{H}$ as being an extension of $\mathbf{H}$ by its stabilization $Stab(\mathbf{H}) \simeq T_\ast \mathbf{H}$:

$$\array{ Stab(\mathbf{H}) &\stackrel{\overset{Stab(\Delta)}{\leftarrow}}{\underset{Stab(\Gamma)}{\longrightarrow}}& Sp \\ \downarrow && \downarrow \\ T\mathbf{H} &\stackrel{\overset{T\Delta}{\leftarrow}}{\underset{T\Gamma}{\longrightarrow}}& T \infty Grpd \\ \downarrow^{\mathrlap{base}} && \downarrow^{\mathrlap{base}} \\ \mathbf{H} &\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\longrightarrow}}& \infty Grpd } \,.$$

Crucial for the internal interpretation in homotopy type theory is that the homotopy types in $T_\ast \mathbf{H} \hookrightarrow T \mathbf{H}$ are stable homotopy types.

As the classifying $\infty$-topos for a universal stable object

One may understand $T \mathbf{H}$ as the result of adjoining to $\mathbf{H}$ a universal “stable object”

$$T \mathbf{H}\simeq \mathbf{H}[X_\ast][ (\Sigma \Omega X_\ast^\bullet \to X_\ast^\bullet)^{-1} ] \,.$$

For details see at excisive (∞,1)-functor the section Characterization via a generic stable object.

Cohesive tangent $\infty$-topos of a cohesive $\infty$-topos

Assume that $\mathbf{H}$ is a cohesive (∞,1)-topos over ∞Grpd, in that there is an adjoint quadruple

$$\mathbf{H} \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} \infty Grpd$$

with $Disc, coDisc$ being full and faithful (∞,1)-functors and $\Pi$ preserving finite (∞,1)-products.

Since (∞,1)-limits and (∞,1)-colimits in an (∞,1)-presheaf (∞,1)-topos are computed objectwise, this adjoint quadruple immediately prolongs to $\mathbf{H}^{seq}$

$$\mathbf{H}^{seq} \stackrel{\overset{\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}} \infty Grpd^{seq} \,.$$

Moreover, all three right adjoints preserves the (∞,1)-pullbacks involved in the characterization of spectrum objects and hence restrict to $T \mathbf{H}$

$$T\mathbf{H} \stackrel{}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}} T\infty Grpd \,.$$

But then we have a further left adjoint given as the composite

$$T\mathbf{H} \hookrightarrow \mathbf{H}^{seq} \stackrel{\overset{\Pi^{seq}}{\longrightarrow}}{\underset{Disc^{seq}}{\leftarrow}} \infty Grpd^{seq} \stackrel{\overset{L}{\longrightarrow}}{\underset{}{\leftarrow}} T \infty Grpd \,.$$

Again since $L$ is a left exact (∞,1)-functor this composite $L \Pi$ preserves finite (∞,1)-products.

So it follows in conclusion that if $\mathbf{H}$ is a cohesive (∞,1)-topos then its tangent $(\infty,1)$-category $T \mathbf{H}$ is itself a cohesive (∞,1)-topos over the tangent $(\infty,1)$-category $T \infty Grpd$ of the base (∞,1)-topos, which is an extension of the cohesion of the $\infty$-topos $\mathbf{H}$ over $\infty Grpd$ by the cohesion of the stable $\infty$-category $Stab(\mathbf{H})$ over $Stab(\infty Grpd) \simeq Spec$:

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

For more on this see at tangent cohesive (∞,1)-topos.

Examples

Of $E_\infty$-rings

Corollary

Let $E_\infty$ be the (∞,1)-category of E-∞ rings and let $A \in E_\infty$. Then the stabilization of the over-(∞,1)-category over $A$

$$Stab(E_\infty/A) \simeq A Mod(Spec)$$

is equivalent to the category of $A$-module spectra.

(Lurie, cor. 1.5.15).

Of an $\infty$-topos

We discuss here aspects of the tangent $\infty$-categories of (∞,1)-toposes.

First consider the base (∞,1)-topos $\mathbf{H} =$ ∞Grpd.

Remark

For each ∞-groupoid/homotopy type $X \in \infty Grpd$. there is a natural equivalence of (∞,1)-categories

$$\infty Grpd_{/X} \simeq Func(X, \infty Grpd)$$

between the slice (∞,1)-category of ∞Grpd over $X$ and the (∞,1)-functor (∞,1)-category of maps $X \to \infty Grpd$.

Proof

By the (∞,1)-Grothendieck construction.

Proposition

For each ∞-groupoid/homotopy type $X \in \infty Grpd$. there is a natural equivalence of (∞,1)-categories

$$T_X (\infty Grpd) \simeq Func(X, Spec)$$

between the fiber of the tangent (∞,1)-category of ∞Grpd over $X$, def. , and the (∞,1)-category of parameterized spectra over $X$.

Proof

Applying remark in remark yields that

$$T_X(\infty Grpd) \simeq Stab(Func(X,\infty Grpd)) \,.$$

The statement then follows with the “stable Giraud theorem”.

Remark

This means that the tangent $(\infty,1)$-category $T(\infty Grpd)$ is equivalently what in (Joyal 08, section 30.34) is denoted $D(Kan, X)$ in the case that $X = Spec$ is the (∞,1)-category of spectra.

Proposition

The tangent $(\infty,1)$-category $T (\infty Grpd)$ is itself an (∞,1)-topos.

Proof

With the above equivalence this is (Joyal 08, section 35.5, 35.6 (with Georg Biedermann)).

Remark

The terminal object in $T \infty Grpd$ should be the zero spectrum regarded as a parameterized spectrum over the point

$$0 \colon \ast \to Spec \,.$$

From this it follows that

Remark

The global elements/global sections functor (which forms the (∞,1)-categorical mapping space out of the terminal object)

$$\Gamma \coloneqq Hom(\ast, -) \;\colon\; T(\infty Grpd) \to \infty Grpd$$

sends an $X$-parameterized spectrum to its base homotopy type $X$.

This functor has a left and right adjoint (∞,1)-functor both given by sending $X$ to the zero spectrum bundle over $X$.

So we have an infinite chain of adjoint (∞,1)-functors

$$(\cdots base \dashv 0 \dashv base \dashv 0 \dashv \cdots) \,.$$

Remark

The functor $0$ is a full and faithful (∞,1)-functor

$$0 \;\colon\; \infty Grpd \hookrightarrow T \infty Grpd$$

and so the tangent $(\infty,1)$-category is cohesive over ∞Grpd, hence by prop. $T(\infty Grpd)$ is a cohesive (∞,1)-topos:

$$(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) \; \colon \; T(\infty Grpd) \stackrel{\overset{base}{\longrightarrow}}{\stackrel{\overset{0}{\leftarrow}}{\stackrel{\overset{base}{\longrightarrow}}{\underset{0}{\leftarrow}}}} \infty Grpd \,.$$

Recalling that here $base = cod \circ \Omega^\infty$, we have one more adjunction, the cotangent complex adjunction due to prop.

$$\infty Grpd \stackrel{ \overset{\Omega}{\longrightarrow} } {\underset{dom\circ \Omega^{\infty}}{\leftarrow}} T(\infty Grpd) \stackrel{\overset{base}{\longrightarrow}}{\stackrel{\overset{0}{\leftarrow}}{\stackrel{\overset{base}{\longrightarrow}}{\underset{0}{\leftarrow}}}} \infty Grpd \,.$$

Remark

For $\mathbf{H}$ a general (∞,1)-topos the above discussion goes through essentially verbatim. If $\mathbf{H}$ is itself cohesive, then we end up with

$$\mathbf{H} \stackrel{\overset{\Omega}{\longrightarrow}}{\underset{dom}{\leftarrow}} T\mathbf{H} \stackrel{\overset{base}{\longrightarrow}}{\stackrel{\overset{0}{\leftarrow}}{\stackrel{\overset{base}{\longrightarrow}}{\underset{0}{\leftarrow}}}} \mathbf{H} \stackrel{}{\stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}} \infty Grpd \,.$$

Proposition

For $\mathbf{H}$ a locally ∞-connected (∞,1)-topos (hence in particular for a cohesive (∞,1)-topos), there are canonical (∞,1)-functors

$$T \mathbf{H} \stackrel{\overset{T Disc}{\leftarrow}}{\underset{T \Gamma}{\longrightarrow}} T \infty Grpd$$

and such that $T \Gamma$ covers the global section geometric morphism $\Gamma \;\colon\; \mathbf{H} \longrightarrow \infty Grpd$ in that it fits into a square

$$\array{ T \mathbf{H} &\stackrel{}{\stackrel{}{\stackrel{\overset{T\Gamma}{\longrightarrow}}{}}}& T \infty Grpd \\ {}^{\mathllap{0}}\uparrow\downarrow^{base} && {}^{\mathllap{0}}\uparrow\downarrow^{base} \\ \mathbf{H} &\stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{}}}& \infty Grpd }$$

Proof

By definition of stabilization, $T \mathbf{H}$ is the (∞,1)-Grothendieck construction of

$$X \mapsto \underset{\leftarrow}{\lim} \left( \cdots \stackrel{\Omega}{\to} \mathbf{H}^{\ast/}_{/X} \stackrel{\Omega}{\to} \mathbf{H}^{\ast/}_{/X} \stackrel{\Omega}{\to} \cdots \right) \,.$$

Since the loop space object (∞,1)-functor $\Omega$ is an (∞,1)-limit construction and since the right adjoint global section functor $\Gamma$ preserves all (∞,1)-limits, there is a homotopy-commuting diagram

$$\array{ \cdots &\stackrel{\Omega}{\to}& \mathbf{H}^{\ast/}_{/X} &\stackrel{\Omega}{\to}& \mathbf{H}^{\ast/}_{/X} &\stackrel{\Omega}{\to}& \cdots \\ && \downarrow^{\mathrlap{\Gamma}} && \downarrow^{\mathrlap{\Gamma}} \\ \cdots &\stackrel{\Omega}{\to}& \infty Grpd^{\ast/}_{/\Gamma(X)} &\stackrel{\Omega}{\to}& \infty Grpd^{\ast/}_{/\Gamma(X)} &\stackrel{\Omega}{\to}& \cdots }$$

in (∞,1)Cat. This induces a natural morphism

$$Stab(\mathbf{H}_{/X}) \longrightarrow Stab(\infty Grpd_{/\Gamma(X)})$$

and hence a morphism

$$T \mathbf{H} \simeq \int_{X \in \mathbf{H}} Stab(\mathbf{H}_{/X}) \longrightarrow \int_{X \in \mathbf{H}} Stab(\infty Grpd_{\Gamma(X)}) \,.$$

The morphism in question is the postcomposition of this with pullback/restriction of the (∞,1)-Grothendieck construction along the reflective inclusion (by assumption on $\mathbf{H}$) $Disc \;\colon\; \infty Grpd \longrightarrow \mathbf{H}$

$$T \mathbf{H} \simeq \int_{X \in \mathbf{H}} Stab(\mathbf{H}_{/X}) \longrightarrow \int_{X \in \mathbf{H}} Stab(\infty Grpd_{/\Gamma(X)}) \longrightarrow \int_{S \in \infty Grpd} Stab(\infty Grpd_{/S}) \simeq T \infty Grpd \,,$$

where we used that by reflectivity $\Gamma \circ Disc \simeq id$.

Remark

When $T \mathbf{H}$ is an $\infty$-topos it should carry another structure $\otimes$ of a symmetric monoidal (∞,1)-category, induced by fiberwise smash product of spectrum objects….

Related concepts

• parameterized spectra

• model structure for parameterized spectra

• Joyal locus

• doubly closed monoidal (infinity,1)-category

• differentiable (∞,1)-category

• Goodwillie calculus

• jet (∞,1)-category

• tangent cohesion

• deformation theory

References

Discussion of model category models is in

• Stefan Schwede, section 3 of Spectra in model categories and applications to the algebraic cotangent complex, Journal of Pure and Applied Algebra 120 (1997) 104

The $\infty$-category theoretic definition and study of the notion of tangent $(\infty,1)$-categories is from

• Jacob Lurie, Deformation Theory

and

• Jacob Lurie, section 7.3 of Higher Algebra

The (infinity,1)-topos structure on tangent $(\infty,1)$-categories (cf. Joyal loci):

• André Joyal, §35.5 of Notes on Logoi (2008) [pdf, pdf]

with expository emphasis in:

• Mathieu Anel, André Joyal, §4.2.3 in: Topo-logie, in New Spaces for Mathematics and Physics, Cambridge University Press (2021) 155-257 [doi:10.1017/9781108854429.007, pdf]

Presentation by model categories is discussed in

• Yonatan Harpaz, Joost Nuiten, Matan Prasma, Tangent categories of algebras over operads (arXiv:1612.02607)

• Yonatan Harpaz, Joost Nuiten, Matan Prasma, The abstract cotangent complex and Quillen cohomology of enriched categories (arXiv:1612.02608)

• Yonatan Harpaz, Joost Nuiten, Matan Prasma, The tangent bundle of a model category, Theory and Applications of Categories Vol. 34, 2019, No. 33, pp 1039-1072, journal web site

• Vincent Braunack-Mayer, Combinatorial parametrised spectra (based on the PhD thesis),

  Algebr. Geom. Topol. 21 (2021) 801-891 [arXiv:1907.08496, doi:10.2140/agt.2021.21.801]

Generalization to parameterized objects in any stable (∞,1)-category is discussed in:

• Marc Hoyois, Topoi of parametrized objects, Theory and Applications of Categories, Vol. 34, 2019, No. 9, pp 243-248. (arXiv:1611.02267, tac:34-09)

A more general notion of a tangent structure on an $(\infty, 1)$-category was attempted to reformulate the Goodwillie calculus in:

• Kristine Bauer, Matthew Burke, Michael Ching, Tangent $\infty$-categories and Goodwillie calculus (arXiv:2101.07819).

Note that this is an extension to $(\infty, 1)$-categories of the concept of a tangent bundle category. In the following slides it is claimed that the tangent $(\infty, 1)$-category construction of this page is a particular case of their more general construction (see slide 3):

• Michael Ching, Tangent ∞-categories and Goodwillie calculus (slides)

Two further examples of tangent structures on an $(\infty,1)$-category, on (∞,1)-Topos and its opposite, are given in:

• Michael Ching, Dual tangent structures for infinity-toposes, (arXiv:2101.08805)

Exposition:

• Eric Finster: A Tour of Parameterized Spectra, talk at Running HoTT 2024, CQTS@NYUAD (April 2024) [video:kt]