nLab
tangent (infinity,1)-category

Idea

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

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

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 $(\mathrm{\infty },1)$-category $T_C$ over an object $A\in C$ may be thought of as the $(\mathrm{\infty },1)$-category of square-0-extensions $A\oplus N$ of $A$, for $N$ a module over $A$. Dually, in $K=C^{\mathrm{op}}$ 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 $(\mathrm{\infty },1)$-category over them. Notably the left adjoint (∞,1)-functor

$$\Omega :C\to T_C$$\Omega : C \to T_C

to the domain projection $\mathrm{dom}:T_C\to C$ 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 $(\mathrm{\infty },1)$-category is that of tangent category.

Definition

Let $C$ be a locally presentable (∞,1)-category.

Definition

(fiberwise stabilization)

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

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

Definition

(tangent $(\mathrm{\infty },1)$-category)

The tangent $(\mathrm{\infty },1)$-category $T_C\to C$ the fiberwise stabilization of the codomain fibration $\mathrm{cod}:\mathrm{Func}(\Delta [1],C)\to C$:

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

This is DT, def 1.1.12.

Properties

General

Proposition

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

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

This is (Lurie, prop. 1.1.13).

Relation to modules

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

Proposition

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

Let

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

be an $𝒪$-algebra in $𝒞$. Then the stabilization of the over-(∞,1)-category over $A$ is canonically equivalent to ${\mathrm{Func}}_{𝒪}(𝒪,{\mathrm{Mod}}_A^{𝒪}(𝒞))$

$$\mathrm{Stab}({\mathrm{Alg}}_{𝒪}(𝒞)/A)\simeq {\mathrm{Func}}_{𝒪}(𝒪,{\mathrm{Mod}}_A^{𝒪}(𝒞)).$$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 ${𝒪}^{\otimes }$ be a coherent (∞,1)-operad and let ${𝒞}^{\otimes }\to {𝒪}^{\otimes }$ be a presentable stable $𝒪$-monoidal (∞,1)-category. Then there is a canonical equivalence

$$\varphi :T_{{\mathrm{Alg}}_{𝒪}(𝒞)}\stackrel{\simeq }{\to }{\mathrm{Alg}}_{𝒪}(𝒞){\times }_{\mathrm{Func}(𝒪,{\mathrm{Alg}}_{𝒪}(𝒞))}{\mathrm{Func}}_{𝒪}(𝒪,{\mathrm{Mod}}^{𝒪}(𝒞))$$\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 ${\mathrm{Alg}}_{𝒪}(𝒞)$.

This is (Lurie, theorem, 1.5.19).

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

$$(A,N)$$(A, N)

where

• $A$ is an $𝒪$-algebra in $𝒞$;

• $N$ is an $A$-module.

Cotangent complex

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

$$\mathrm{dom}:T_C\to C.$$dom : T_C \to C \,.

Definition/Proposition

(cotangent complex)

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

$$(\Omega ⊣\mathrm{dom}):T_C\stackrel{\stackrel{\Omega }{\leftarrow }}{\underset{\mathrm{dom}}{\to }}C$$(\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

$$(C\stackrel{\Omega }{\to }{T}_C\stackrel{p}{\to }C)\simeq {\mathrm{Id}}_C$$(C \stackrel{\Omega}{\to} T_C \stackrel{p}{\to} C) \simeq Id_C

and hence that $C$ is a retract of $T_C$.

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

This is (Lurie, def. 1.1.2, remark 1.2.3).

Examples

Of $E_{\mathrm{\infty }}$-rings

Corollary

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

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

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

(Lurie, cor. 1.5.15).

References

The definition and study of the notion is tangent $(\mathrm{\infty },1)$-categories is from

• Jacob Lurie, Deformation Theory

and section 7.3 of

• Jacob Lurie, Higher Algebra