# nLab tangent (infinity,1)-category

### Context

#### $\left(\infty ,1\right)$-Category theory

(∞,1)-category theory

## Models

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

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

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

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 $\left(\infty ,1\right)$-category ${T}_{C}$ over an object $A\in C$ may be thought of as the $\left(\infty ,1\right)$-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 $\left(\infty ,1\right)$-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 $\Omega \left(A\right)$, 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 $\left(\infty ,1\right)$-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 $\mathrm{Stab}\left(C\prime \to C\right)$ is – roughly – the fibration universal with the property that for each $A\in C$ its fiber over $A$ is the stabilization $\mathrm{Stab}\left(C{\prime }_{A}\right)$ of the fiber $C{\prime }_{A}$ over $A$.

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

###### Definition

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

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

$\left({T}_{C}\stackrel{p}{\to }C\right):=\mathrm{Stab}\left(\mathrm{Func}\left(\Delta \left[1\right],C\right)\stackrel{\mathrm{cod}}{\to }C\right)\phantom{\rule{thinmathspace}{0ex}}.$(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 $\left(\infty ,1\right)$-category ${T}_{C}$ of the locally presentable (∞,1)-category $C$ is itself a locally presentable $\left(\infty ,1\right)$-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 $\left(\infty ,1\right)$-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

$A\in {\mathrm{Alg}}_{𝒪}\left(𝒞\right)$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 ${\mathrm{Func}}_{𝒪}\left(𝒪,{\mathrm{Mod}}_{A}^{𝒪}\left(𝒞\right)\right)$

$\mathrm{Stab}\left({\mathrm{Alg}}_{𝒪}\left(𝒞\right)/A\right)\simeq {\mathrm{Func}}_{𝒪}\left(𝒪,{\mathrm{Mod}}_{A}^{𝒪}\left(𝒞\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$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

$\varphi :{T}_{{\mathrm{Alg}}_{𝒪}\left(𝒞\right)}\stackrel{\simeq }{\to }{\mathrm{Alg}}_{𝒪}\left(𝒞\right){×}_{\mathrm{Func}\left(𝒪,{\mathrm{Alg}}_{𝒪}\left(𝒞\right)\right)}{\mathrm{Func}}_{𝒪}\left(𝒪,{\mathrm{Mod}}^{𝒪}\left(𝒞\right)\right)$\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 ${\mathrm{Alg}}_{𝒪}\left(𝒞\right)$.

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

$\left(A,N\right)$(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 $\mathrm{cod}:\mathrm{Func}\left(\Delta \left[1\right],C\right)\to C$ the tangent $\left(\infty ,1\right)$-category $p:{T}_{C}\to C$ inherits a second $\left(\infty ,1\right)$-functor to $C$, coming from the domain evaluation

$\mathrm{dom}:{T}_{C}\to C\phantom{\rule{thinmathspace}{0ex}}.$dom : T_C \to C \,.
###### Definition/Proposition

(cotangent complex)

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

$\left(\Omega ⊣\mathrm{dom}\right):{T}_{C}\stackrel{\stackrel{\Omega }{←}}{\underset{\mathrm{dom}}{\to }}C$(\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

$\left(C\stackrel{\Omega }{\to }{T}_{C}\stackrel{p}{\to }C\right)\simeq {\mathrm{Id}}_{C}$(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 $\left(\infty ,1\right)$-functor : for $A\in C$ the object $\Omega \left(A\right)$ is the cotangent complex of $A$.

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

## 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 $A$

$\mathrm{Stab}\left({E}_{\infty }/A\right)\simeq A\mathrm{Mod}\left(\mathrm{Spec}\right)$Stab(E_\infty/A) \simeq A Mod(Spec)

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

## References

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

and section 7.3 of

Revised on March 19, 2012 18:35:34 by Urs Schreiber (89.204.153.67)