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\begin{document}

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\section*{tangent (infinity,1)-category}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{goodwillie_calculus}{}\paragraph*{{Goodwillie calculus}}\label{goodwillie_calculus}

[[!include Goodwillie calculus - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{PresentabilityAndLimits}{Presentability and limits}\dotfill \pageref*{PresentabilityAndLimits} \linebreak
\noindent\hyperlink{RelationToModules}{Relation to modules}\dotfill \pageref*{RelationToModules} \linebreak
\noindent\hyperlink{CotangentComplex}{Cotangent complex}\dotfill \pageref*{CotangentComplex} \linebreak
\noindent\hyperlink{TangentTopos}{Tangent $\infty$-topos of an $\infty$-topos}\dotfill \pageref*{TangentTopos} \linebreak
\noindent\hyperlink{TangentToposGeneral}{General}\dotfill \pageref*{TangentToposGeneral} \linebreak
\noindent\hyperlink{as_the_classifying_topos_for_a_universal_stable_object}{As the classifying $\infty$-topos for a universal stable object}\dotfill \pageref*{as_the_classifying_topos_for_a_universal_stable_object} \linebreak
\noindent\hyperlink{cohesive_tangent_topos_of_a_cohesive_topos}{Cohesive tangent $\infty$-topos of a cohesive $\infty$-topos}\dotfill \pageref*{cohesive_tangent_topos_of_a_cohesive_topos} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{of_rings}{Of $E_\infty$-rings}\dotfill \pageref*{of_rings} \linebreak
\noindent\hyperlink{ExamplesTangentOfAnInfinityTopos}{Of an $\infty$-topos}\dotfill \pageref*{ExamplesTangentOfAnInfinityTopos} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

For $K$ a [[locally presentable (∞,1)-category]] whose objects we think of as [[space]]s of sorts, its \textbf{tangent $(\infty,1)$-category}

\begin{displaymath}
(T_{K^{op}})^{op} \to K
\end{displaymath}
is an [[(∞,1)-category]] over $K$, whose objects may be thought of as spaces that are \textbf{[[infinitesimal space|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 [[module]]s 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 \textbf{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 bundle]]s -- 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|left]] [[adjoint (∞,1)-functor]]

\begin{displaymath}
\Omega : C \to T_C
\end{displaymath}
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 [[category theory|1-categorical]] approximation to the notion of tangent $(\infty,1)$-category is that of [[tangent category]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

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

\begin{defn}
\label{FiberwiseStabilization}\hypertarget{FiberwiseStabilization}{}
\textbf{(fiberwise stabilization)}

For $\mathcal{C}' \to \mathcal{C}$ a [[model structure for quasi-categories|categorical fibration]], [[generalized the|the]] \textbf{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 $A$.

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

\begin{defn}
\label{TangentCategory}\hypertarget{TangentCategory}{}
\textbf{(tangent $(\infty,1)$-category)}

[[generalized the|The]] \textbf{tangent $(\infty,1)$-category} $T_{\mathcal{C}} \to \mathcal{C}$ is [[generalized the|the]] \emph{fiberwise stabilization} of the [[codomain fibration]] $cod : \mathcal{C}^{\Delta^1} \to \mathcal{C}$:

\begin{displaymath}
(T_{\mathcal{C}} \stackrel{p}{\to} \mathcal{C}) := Stab(Func(\Delta[1], \mathcal{C}) \stackrel{cod}{\to} \mathcal{C}  )
  \,.
\end{displaymath}
\end{defn}
This is [[Deformation Theory|DT, def 1.1.12]].

For a maybe more explicit definition see below at \emph{\hyperlink{TangentToposGeneral}{Tangent ∞-topos -- General}}.

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

\begin{remark}
\label{TangentCatByExplicitFiberwiseStabilization}\hypertarget{TangentCatByExplicitFiberwiseStabilization}{}
Given a presentable [[(∞,1)-category]] $\mathcal{C}$, the [[(∞,1)-functor]]

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

\begin{displaymath}
(\infty,1)Cat^R \to (\infty,1)Cat
\end{displaymath}
of [[(∞,1)-functors]] which are [[right adjoint|right]] [[adjoint (∞,1)-functors]]. For these the further (now full) inclusion

\begin{displaymath}
i \colon (\infty,1)StabCat^R \hookrightarrow (\infty,1)Cat^R
\end{displaymath}
of the [[stable (∞,1)-categories]] has a [[right adjoint|right]] [[adjoint (∞,1)-functor]]

\begin{displaymath}
(i \dashv Stab)
\end{displaymath}
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

\begin{displaymath}
i \circ Stab i \chi_{cod} \colon \mathcal{C}^{op} \to (\infty,1)Cat
  \,.
\end{displaymath}
This sends an object $c \in \mathcal{C}$ to the [[stabilization]] of the [[slice (∞,1)-category]] over $c$:

\begin{displaymath}
Stab \circ \chi_{cod} \colon c \mapsto Stab(\mathcal{C}_{/c})
  \,.
\end{displaymath}
Again by the [[(∞,1)-Grothendieck construction]] this classifies a [[Cartesian fibration]] over $\mathcal{C}$ and this now is the tangent $(\infty,1)$-category projection

\begin{displaymath}
\itexarray{
    T_{\mathcal{C}}
    \\
    \downarrow^{\mathrlap{p}}
    \\
    \mathcal{C}
  }
  \,.
\end{displaymath}
\end{remark}
This is the first part of the proof of [[Deformation Theory|DT. prop. 1.1.9]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{PresentabilityAndLimits}{}\subsubsection*{{Presentability and limits}}\label{PresentabilityAndLimits}

\begin{prop}
\label{}\hypertarget{}{}
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)-limit]]s and [[(∞,1)-colimit]]s.

\end{prop}
This is (\hyperlink{Lurie}{Lurie, prop. 1.1.13}).

Moreover:

\begin{prop}
\label{}\hypertarget{}{}
A diagram in the tangent $(\infty,1)$-category $T_{\mathcal{C}}$ is an [[(∞,1)-limit|(∞,1)-(co-)limit]] precisely if

\begin{enumerate}%
\item it is a [[relative (infinity,1)-limit|relative (∞,1)-(co-)limit]] with respect to the projection $p \colon T_{\mathcal{C}} \to \mathcal{C}$;


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



\end{enumerate}
\end{prop}
(\hyperlink{LurieHigherAlgebra}{Lurie, HigherAlgebra, prop. 8.3.1.12})

\hypertarget{RelationToModules}{}\subsubsection*{{Relation to modules}}\label{RelationToModules}

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.

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{O}^\otimes$ be a [[coherent (∞,1)-operad]] and let $\mathcal{C}^\otimes \to \mathcal{O}^\otimes$ be a [[stable (∞,1)-category|stable]] $\mathcal{O}$-[[monoidal (∞,1)-category]].

Let

\begin{displaymath}
A \in Alg_\mathcal{O}(\mathcal{C})
\end{displaymath}
be an $\mathcal{O}$-[[algebra over an (infinity,1)-operad|algebra]] in $\mathcal{C}$. Then the [[stabilization]] of the [[over-(∞,1)-category]] over $A$ is canonically equivalent to $Func_\mathcal{O}(\mathcal{O}, Mod_A^\mathcal{O}(\mathcal{C}))$

\begin{displaymath}
Stab( Alg_\mathcal{O}(\mathcal{C})/A)
  \simeq
  Func_\mathcal{O}(\mathcal{O}, Mod_A^\mathcal{O}(\mathcal{C}))
  \,.
\end{displaymath}
\end{prop}
This is (\hyperlink{Lurie}{Lurie, theorem 1.5.14}).

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{O}^\otimes$ be a [[coherent (∞,1)-operad]] and let $\mathcal{C}^\otimes \to \mathcal{O}^\otimes$ be a [[presentable (∞,1)-category|presentable]] [[stable (∞,1)-category|stable]] $\mathcal{O}$-[[monoidal (∞,1)-category]]. Then there is a canonical [[equivalence of (∞,1)-categories|equivalence]]

\begin{displaymath}
\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}))
\end{displaymath}
of presentble [[Cartesian fibration|fibrations]] over $Alg_\mathcal{O}(\mathcal{C})$.

\end{prop}
This is (\hyperlink{Lurie}{Lurie, theorem, 1.5.19}).

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

\begin{displaymath}
(A, N)
\end{displaymath}
where

\begin{itemize}%
\item $A$ is an $\mathcal{O}$-[[∞-algebra over an (∞,1)-operad|algebra]] in $\mathcal{C}$;


\item $N$ is an $A$-[[∞-module over an ∞-algebra over an (∞,1)-operad|module]].



\end{itemize}
\hypertarget{CotangentComplex}{}\subsubsection*{{Cotangent complex}}\label{CotangentComplex}

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 \emph{domain} evaluation

\begin{displaymath}
dom : T_C \to C
  \,.
\end{displaymath}
\begin{prop}
\label{CotangentComplexAdjunction}\hypertarget{CotangentComplexAdjunction}{}
\textbf{(cotangent complex)}

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

\begin{displaymath}
(\Omega \dashv dom) : T_C \stackrel{\overset{\Omega}{\leftarrow}}{\underset{dom}{\to}}
  C
\end{displaymath}
that is also a [[section]] of $p : T_C \to C$ in that

\begin{displaymath}
(C \stackrel{\Omega}{\to} T_C \stackrel{p}{\to} C) \simeq Id_C
\end{displaymath}
and which hence exhibits $C$ as a [[retract]] of $T_C$.

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

\end{prop}
This is (\hyperlink{Lurie}{Lurie, def. 1.2.2, remark 1.2.3}).

In more detail this adjunction is the composite

\begin{displaymath}
(\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
  \,,
\end{displaymath}
where $(\Sigma^\infty_C \dashv \Omega^\infty_C)$ is the fiberwise [[stabilization]] relative adjunction, def. \ref{FiberwiseStabilization}.

\hypertarget{TangentTopos}{}\subsubsection*{{Tangent $\infty$-topos of an $\infty$-topos}}\label{TangentTopos}

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 (\hyperlink{Joyal08}{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 \emph{\href{n-excisive+%28?%2C1%29-functor#nExcisiveApproximation}{n-Excisive functor -- Properties -- n-Excisive reflection}}.

\hypertarget{TangentToposGeneral}{}\paragraph*{{General}}\label{TangentToposGeneral}

\begin{defn}
\label{}\hypertarget{}{}
Let $seq$ be the [[diagram]] category as follows:

\begin{displaymath}
seq 
  \coloneqq
  \left\{
    \itexarray{
    && \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}}
  \,.
\end{displaymath}
\end{defn}
(\hyperlink{Joyal08}{Joyal 08, section 35.5})

\begin{remark}
\label{}\hypertarget{}{}
Given an [[(∞,1)-topos]] $\mathbf{H}$, an [[(∞,1)-functor]]

\begin{displaymath}
X_\bullet \;\colon\; seq \longrightarrow \mathbf{H}
\end{displaymath}
is equivalently

\begin{enumerate}%
\item a choice of [[object]] $B \in \mathbf{H}$ (the image of $\ast \in seq$]);


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


\item 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.



\end{enumerate}
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

\begin{displaymath}
\left\{
  \itexarray{
    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\}
\end{displaymath}
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]]

\begin{displaymath}
\mathbf{H}^{seq}
  \coloneqq
  Func(seq, \mathbf{H})
\end{displaymath}
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 [[equivalence in an (∞,1)-category|equivalences]]. The [[full sub-(∞,1)-category]]

\begin{displaymath}
T \mathbf{H} \hookrightarrow \mathbf{H}^{seq}
\end{displaymath}
on the genuine [[spectrum objects]] is therefore the ``fiberwise [[stabilization]]'' of the self-indexing, hence the tangent $(\infty,1)$-category.

\end{remark}
\begin{lemma}
\label{SpectrificationLemma}\hypertarget{SpectrificationLemma}{}
\textbf{(spectrification is left exact reflective)}

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

\begin{displaymath}
T \mathbf{H}
  \stackrel{\overset{L_{lex}}{\leftarrow}}{\hookrightarrow}
  \mathbf{H}^{seq}
  \,.
\end{displaymath}
\end{lemma}
(\hyperlink{Joyal08}{Joyal 08, section 35.1})

\begin{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

\begin{displaymath}
\theta \;\colon\; id \longrightarrow \Omega
  \,.
\end{displaymath}
This in turn is compatible with $\Omega$ in that

\begin{displaymath}
\theta \circ \Omega \simeq \Omega \circ \theta
  \;\colon\;
  \rho \longrightarrow \rho \circ \rho = \rho^2
  \,.
\end{displaymath}
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 [[reflective sub-(infinity,1)-category|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 [[exact (∞,1)-functor|left exact]] [[reflective sub-(∞,1)-category]].

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
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).

\end{prop}
(\hyperlink{Joyal08}{Joyal 08, section 35.5})

\begin{proof}
By the the spectrification lemma \ref{SpectrificationLemma} $T \mathbf{H}$ 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

\begin{displaymath}
\itexarray{
    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
  }
  \,.
\end{displaymath}
\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
This statement also follows from the general theory of [[excisive functors]], and in this form it is due to [[Charles Rezk]]. See at \emph{\href{n-excisive+%28?%2C1%29-functor#nExcisiveApproximation}{n-Excisive functor -- Properties -- n-Excisive reflection}} for the above fact and its generalization to ``[[Goodwillie calculus|Goodwillie]] [[jet bundles]]''.

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

\begin{displaymath}
\itexarray{
    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
  }
  \,.
\end{displaymath}
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]].

\end{remark}
\hypertarget{as_the_classifying_topos_for_a_universal_stable_object}{}\paragraph*{{As the classifying $\infty$-topos for a universal stable object}}\label{as_the_classifying_topos_for_a_universal_stable_object}

One may understand $T \mathbf{H}$ as the result of adjoining to $\mathbf{H}$ a universal ``stable object''

\begin{displaymath}
T \mathbf{H}\simeq \mathbf{H}[X_\ast][ (\Sigma \Omega X_\ast^\bullet \to X_\ast^\bullet)^{-1} ]
  \,.
\end{displaymath}
For details see at \emph{[[excisive (∞,1)-functor]]} the section \emph{\href{excisive+%28%E2%88%9E%2C1%29-functor#CharacterizationViaAGenericStableObject}{Characterization via a generic stable object}}.

\hypertarget{cohesive_tangent_topos_of_a_cohesive_topos}{}\paragraph*{{Cohesive tangent $\infty$-topos of a cohesive $\infty$-topos}}\label{cohesive_tangent_topos_of_a_cohesive_topos}

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

\begin{displaymath}
\mathbf{H}
    \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}
  \infty Grpd
\end{displaymath}
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 $\mathbf{H}^{seq}$

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

\begin{displaymath}
T\mathbf{H}
    \stackrel{}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}}
  T\infty Grpd
  \,.
\end{displaymath}
But then we have a further [[left adjoint]] given as the composite

\begin{displaymath}
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
  \,.
\end{displaymath}
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 $Stab(\infty Grpd) \simeq Spec$:

\begin{displaymath}
\itexarray{
    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
  }
  \,.
\end{displaymath}
For more on this see at \emph{[[tangent cohesive (∞,1)-topos]]}.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{of_rings}{}\subsubsection*{{Of $E_\infty$-rings}}\label{of_rings}

\begin{cor}
\label{}\hypertarget{}{}
Let $E_\infty$ be the [[(∞,1)-category]] of [[E-∞ ring]]s and let $A \in E_\infty$. Then the [[stabilization]] of the [[over-(∞,1)-category]] over $A$

\begin{displaymath}
Stab(E_\infty/A) \simeq A Mod(Spec)
\end{displaymath}
is equivalent to the category of $A$-[[module spectra]].

\end{cor}
(\hyperlink{Lurie}{Lurie, cor. 1.5.15}).

\hypertarget{ExamplesTangentOfAnInfinityTopos}{}\subsubsection*{{Of an $\infty$-topos}}\label{ExamplesTangentOfAnInfinityTopos}

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

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

\begin{remark}
\label{SliceOfinfinityGroupoidsIsFunctorCategory}\hypertarget{SliceOfinfinityGroupoidsIsFunctorCategory}{}
For each [[∞-groupoid]]/[[homotopy type]] $X \in \infty Grpd$. there is a [[natural equivalence|natural]] [[equivalence of (∞,1)-categories]]

\begin{displaymath}
\infty Grpd_{/X}
  \simeq
   Func(X, \infty Grpd)
\end{displaymath}
between the [[slice (∞,1)-category]] of [[∞Grpd]] over $X$ and the [[(∞,1)-functor (∞,1)-category]] of maps $X \to \infty Grpd$.

\end{remark}
\begin{proof}
By the [[(∞,1)-Grothendieck construction]].

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
For each [[∞-groupoid]]/[[homotopy type]] $X \in \infty Grpd$. there is a [[natural equivalence|natural]] [[equivalence of (∞,1)-categories]]

\begin{displaymath}
T_X (\infty Grpd) \simeq Func(X, Spec)
\end{displaymath}
between the fiber of the tangent (∞,1)-category of [[∞Grpd]] over $X$, def. \ref{TangentCategory}, and the [[(∞,1)-category]] of [[parameterized spectra]] over $X$.

\end{prop}
\begin{proof}
Applying remark \ref{SliceOfinfinityGroupoidsIsFunctorCategory} in remark \ref{TangentCatByExplicitFiberwiseStabilization} yields that

\begin{displaymath}
T_X(\infty Grpd) \simeq Stab(Func(X,\infty Grpd))
  \,.
\end{displaymath}
The statement then follows with the ``\href{stable%20%28infinity,1%29-category#StabGiraud}{stable Giraud theorem}''.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
This means that the tangent $(\infty,1)$-category $T(\infty Grpd)$ is equivalently what in (\hyperlink{Joyal08}{Joyal 08, section 30.34}) is denoted $D(Kan, X)$ in the case that $X = Spec$ is the [[(∞,1)-category of spectra]].

\end{remark}
\begin{prop}
\label{TangentOfInfinityGrpdIsTopos}\hypertarget{TangentOfInfinityGrpdIsTopos}{}
The tangent $(\infty,1)$-category $T (\infty Grpd)$ is itself an [[(∞,1)-topos]].

\end{prop}
\begin{proof}
With the above equivalence this is (\hyperlink{Joyal08}{Joyal 08, section 35.5, 35.6} (with [[Georg Biedermann]])).

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
The [[terminal object in an (∞,1)-category|terminal object]] in $T \infty Grpd$ should be the [[zero spectrum]] regarded as a parameterized spectrum over the point

\begin{displaymath}
0 \colon \ast \to Spec
  \,.
\end{displaymath}
\end{remark}
From this it follows that

\begin{remark}
\label{}\hypertarget{}{}
The [[global elements]]/[[global sections]] functor (which forms the [[derived hom space|(∞,1)-categorical mapping space]] out of the terminal object)

\begin{displaymath}
\Gamma \coloneqq Hom(\ast, -) 
  \;\colon\; T(\infty Grpd) \to \infty Grpd
\end{displaymath}
sends an $X$-[[parameterized spectrum]] to its base homotopy type $X$.

This functor has a [[left adjoint|left]] and [[right adjoint|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]]

\begin{displaymath}
(\cdots base \dashv 0 \dashv base \dashv 0 \dashv \cdots)
  \,.
\end{displaymath}
\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The functor $0$ is a [[full and faithful (∞,1)-functor]]

\begin{displaymath}
0 \;\colon\; \infty Grpd \hookrightarrow T \infty Grpd
\end{displaymath}
and so the tangent $(\infty,1)$-category is [[cohesive (∞,1)-topos|cohesive]] over [[∞Grpd]], hence by prop. \ref{TangentOfInfinityGrpdIsTopos} $T(\infty Grpd)$ is a [[cohesive (∞,1)-topos]]:

\begin{displaymath}
(\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
  \,.
\end{displaymath}
Recalling that here $base = cod \circ \Omega^\infty$, we have one more adjunction, the [[cotangent complex]] adjunction due to prop. \ref{CotangentComplexAdjunction}

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

\begin{displaymath}
\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
  \,.
\end{displaymath}
\end{remark}
\begin{prop}
\label{DifferentialOfCohesiveGlobalSections}\hypertarget{DifferentialOfCohesiveGlobalSections}{}
For $\mathbf{H}$ a [[locally ∞-connected (∞,1)-topos]] (hence in particular for a [[cohesive (∞,1)-topos]]), there are canonical [[(∞,1)-functors]]

\begin{displaymath}
T \mathbf{H} \stackrel{\overset{T Disc}{\leftarrow}}{\underset{T \Gamma}{\longrightarrow}} T \infty Grpd
\end{displaymath}
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

\begin{displaymath}
\itexarray{
    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
  }
\end{displaymath}
\end{prop}
\begin{proof}
By definition of [[stabilization]], $T \mathbf{H}$ is the [[(∞,1)-Grothendieck construction]] of

\begin{displaymath}
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)
  \,.
\end{displaymath}
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]]

\begin{displaymath}
\itexarray{
    \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
  }
\end{displaymath}
in [[(∞,1)Cat]]. This induces a natural morphism

\begin{displaymath}
Stab(\mathbf{H}_{/X})
  \longrightarrow
  Stab(\infty Grpd_{/\Gamma(X)})
\end{displaymath}
and hence a morphism

\begin{displaymath}
T \mathbf{H} 
  \simeq 
  \int_{X \in \mathbf{H}} Stab(\mathbf{H}_{/X}) \longrightarrow \int_{X \in \mathbf{H}} Stab(\infty Grpd_{\Gamma(X)})
  \,.
\end{displaymath}
The morphism in question is the postcomposition of this with pullback/restriction of the [[(∞,1)-Grothendieck construction]] along the [[reflective sub-(∞,1)-category|reflective inclusion]] (by assumption on $\mathbf{H}$) $Disc \;\colon\; \infty Grpd \longrightarrow \mathbf{H}$

\begin{displaymath}
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
  \,,
\end{displaymath}
where we used that by reflectivity $\Gamma \circ Disc \simeq id$.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
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]]\ldots{}.

\end{remark}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[parameterized spectra]]


\item [[differentiable (∞,1)-category]]


\item [[Goodwillie calculus]]


\item [[jet (∞,1)-category]]


\item [[tangent cohesion]]


\item [[deformation theory]]



\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

Discussion of [[model category]] models is in

\begin{itemize}%
\item [[Stefan Schwede]], section 3 of \emph{Spectra in model categories and applications to the algebraic cotangent complex}, Journal of Pure and Applied Algebra 120 (1997) 104

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

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Deformation Theory]]}

\end{itemize}
and

\begin{itemize}%
\item [[Jacob Lurie]], section 7.3 of \emph{[[Higher Algebra]]}

\end{itemize}
The [[(infinity,1)-topos]] structure on tangent $(\infty,1)$-categories is discussed in 35.5 of

\begin{itemize}%
\item [[André Joyal]], \emph{Notes on Logoi}, 2008 (\href{http://www.math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf}{pdf})

\end{itemize}
Presentation by [[model categories]] is discussed in

\begin{itemize}%
\item [[Yonatan Harpaz]], [[Joost Nuiten]], [[Matan Prasma]], \emph{Tangent categories of algebras over operads} (\href{https://arxiv.org/abs/1612.02607}{arXiv:1612.02607})


\item [[Yonatan Harpaz]], [[Joost Nuiten]], [[Matan Prasma]], \emph{The abstract cotangent complex and Quillen cohomology of enriched categories} (\href{https://arxiv.org/abs/1612.02608}{arXiv:1612.02608})


\item [[Vincent Braunack-Mayer]], \emph{Combinatorial parametrised spectra}, based on the [[schreiber:thesis Braunack-Mayer|PhD thesis]] (\href{https://arxiv.org/abs/1907.08496}{arXiv:1907.08496})



\end{itemize}
Generalization to parameterized objects in any [[stable (∞,1)-category]] is discussed in:

\begin{itemize}%
\item [[Marc Hoyois]], \emph{Topoi of parametrized objects}, Theory and Applications of Categories, Vol. 34, 2019, No. 9, pp 243-248. (\href{https://arxiv.org/abs/1611.02267}{arXiv:1611.02267}, \href{http://www.tac.mta.ca/tac/volumes/34/9/34-09abs.html}{tac:34-09})

\end{itemize}
[[!redirects tangent (infinity,1)-categories]]

[[!redirects tangent (∞,1)-category]] [[!redirects tangent (∞,1)-categories]]

[[!redirects tangent (∞,1)-topos]] [[!redirects tangent (∞,1)-toposes]] [[!redirects tangent (infinity,1)-topos]] [[!redirects tangent (infinity,1)-toposes]]



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