nLab
tangent (infinity,1)-category

Context

(,1)(\infty,1)-Category theory

Higher algebra

Stable Homotopy theory

(,2)(\infty,2)-Topos theory

Contents

Idea

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

(T K op) opK (T_{K^{op}})^{op} \to K

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

More concretely, the tangent (,1)(\infty,1)-category T CCT_C \to C for C=K opC = K^{op} is the fiberwise stabilization of the codomain fibration Func(Δ[1],C)CFunc(\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 (,1)(\infty,1)-category T CT_C over an object ACA \in C may be thought of as the (,1)(\infty,1)-category of square-0-extensions ANA \oplus N of AA, for NN a module over AA. Dually, in K=C opK = 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 AA is regarded to be the algebra of functions on.

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

Ω:CT C \Omega : C \to T_C

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

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

Definition

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

Definition

(fiberwise stabilization)

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

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

Definition

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

The tangent (,1)(\infty,1)-category T 𝒞𝒞T_{\mathcal{C}} \to \mathcal{C} is the fiberwise stabilization of the codomain fibration cod:𝒞 Δ 1𝒞cod : \mathcal{C}^{\Delta^1} \to \mathcal{C}:

(T 𝒞p𝒞):=Stab(Func(Δ[1],𝒞)cod𝒞). (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 𝒞\mathcal{C}, the (∞,1)-functor

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

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

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

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

(iStab) (i \dashv Stab)

given by stabilization. (Note that this is not a functor on all of (,1)Cat(\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

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

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

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

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

T 𝒞 p 𝒞. \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 (,1)(\infty,1)-category T CT_C of the locally presentable (∞,1)-category CC is itself a locally presentable (,1)(\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 (,1)(\infty,1)-category T 𝒞T_{\mathcal{C}} is an (∞,1)-(co-)limit precisely if

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

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

(Lurie, HigherAlgebra, prop. 8.3.1.12)

Relation to modules

We discuss how the tangent (,1)(\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 \mathcal{O}^\otimes be a coherent (∞,1)-operad and let 𝒞 𝒪 \mathcal{C}^\otimes \to \mathcal{O}^\otimes be a stable 𝒪\mathcal{O}-monoidal (∞,1)-category.

Let

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

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

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

ϕ:T Alg 𝒪(𝒞)Alg 𝒪(𝒞)× Func(𝒪,Alg 𝒪(𝒞))Func 𝒪(𝒪,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 Alg 𝒪(𝒞)Alg_\mathcal{O}(\mathcal{C}).

This is (Lurie, theorem, 1.5.19).

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

(A,N) (A, N)

where

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

  • NN is an AA-module.

Cotangent complex

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

dom:T CC. dom : T_C \to C \,.
Definition/Proposition

(cotangent complex)

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

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

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

(CΩT CpC)Id C (C \stackrel{\Omega}{\to} T_C \stackrel{p}{\to} C) \simeq Id_C

and which hence exhibits CC as a retract of T CT_C.

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

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

In more detail this adjunction is the composite

(Ωdom):T CΩ C Σ C C Δ 1domconstC:Ω, (\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 (Σ C Ω C )(\Sigma^\infty_C \dashv \Omega^\infty_C) is the fiberwise stabilization relative adjunction, def. 1.

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 seqseq be the diagram category as follows:

seq{ x n1 p n1 * p n1 i n id * i n x n p n * id p n i n+1 * i n+1 x n+1 } n. 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 H\mathbf{H}, an (∞,1)-functor

X :seqH X_\bullet \;\colon\; seq \longrightarrow \mathbf{H}

is equivalently

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

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

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

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

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

{X n f n Y n p n X p n Y B 1 f b B 2} \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:B 1B 2f_b \;\colon\; B_1 \longrightarrow B_2 in H\mathbf{H} together with morphisms X nf b *Y nX_n \longrightarrow f_b^\ast Y_n into the (∞,1)-pullback of the components of Y Y_\bullet along f bf_b.

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

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

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

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

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

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

Lemma

(spectrification is left exact reflective)

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

THL lexH seq. 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 Ω:H seqH seq\Omega \colon \mathbf{H}^{seq} \longrightarrow \mathbf{H}^{seq} and by definition of seqseq this comes with a natural transformation out of the identity

θ:idΩ. \theta \;\colon\; id \longrightarrow \Omega \,.

This in turn is compatible with Ω\Omega in that

θΩΩθ:ρρρ=ρ 2. \theta \circ \Omega \simeq \Omega \circ \theta \;\colon\; \rho \longrightarrow \rho \circ \rho = \rho^2 \,.

Consider then a sufficiently deep transfinite composition ρ tf\rho^{tf}. By the small object argument available in the presentable (∞,1)-category H\mathbf{H} this stabilizes, and hence provides a reflection L:H seqTHL \;\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 H\mathbf{H} an (∞,1)-topos over the base (∞,1)-topos Grpd\infty Grpd, its tangent (∞,1)-category THT \mathbf{H} is an (∞,1)-topos over the base TGrpdT \infty Grpd (and hence in particular also over Grpd\infty Grpd itself).

(Joyal 08, section 35.5)

Proof

By the the spectrification lemma 1 THT \mathbf{H} has a geometric embedding into the (∞,1)-presheaf (∞,1)-topos H seq\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 HΓΔGrpd\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

TH TΓTΔ TGrpd incl incl H ΓΔ Grpd. \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 THT \mathbf{H} as being an extension of H\mathbf{H} by its stabilization Stab(H)T *HStab(\mathbf{H}) \simeq T_\ast \mathbf{H}:

Stab(H) Stab(Γ)Stab(Δ) Sp TH TΓTΔ TGrpd base base H ΓΔ Grpd. \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 *HTHT_\ast \mathbf{H} \hookrightarrow T \mathbf{H} are stable homotopy types.

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

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

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

with Disc,coDiscDisc, coDisc being full and faithful (∞,1)-functors and Π\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 H seq\mathbf{H}^{seq}

H seqcoDisc seqΓ seqDisc seqΠ seqGrpd 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 THT \mathbf{H}

THcoDisc seqΓ seqDisc seqTGrpd. 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

THH seqDisc seqΠ seqGrpd seqLTGrpd. 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 LL is a left exact (∞,1)-functor this composite LΠL \Pi preserves finite (∞,1)-products.

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

Stab(H) coDisc seqΓ seqDisc seqLΠ seq Stab(Grpd) incl incl TH coDisc seqΓ seqDisc seqLΠ seq TGrpd base 0 base 0 base 0 base 0 H coDiscΓDiscΠ Grpd. \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 E_\infty-rings

Corollary

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

Stab(E /A)AMod(Spec) Stab(E_\infty/A) \simeq A Mod(Spec)

is equivalent to the category of AA-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 H=\mathbf{H} = ∞Grpd.

Remark

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

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

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

Proposition

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

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

between the fiber of the tangent (∞,1)-category of ∞Grpd over XX, def. 2, and the (∞,1)-category of parameterized spectra over XX.

Proof

Applying remark 5 in remark 1 yields that

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

Proposition

The tangent (,1)(\infty,1)-category T(Grpd)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 TGrpdT \infty Grpd should be the zero spectrum regarded as a parameterized spectrum over the point

0:*Spec. 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)

ΓHom(*,):T(Grpd)Grpd \Gamma \coloneqq Hom(\ast, -) \;\colon\; T(\infty Grpd) \to \infty Grpd

sends an XX-parameterized spectrum to its base homotopy type XX.

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

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

(base0base0). (\cdots base \dashv 0 \dashv base \dashv 0 \dashv \cdots) \,.
Remark

The functor 00 is a full and faithful (∞,1)-functor

0:GrpdTGrpd 0 \;\colon\; \infty Grpd \hookrightarrow T \infty Grpd

and so the tangent (,1)(\infty,1)-category is cohesive over ∞Grpd, hence by prop. 8 T(Grpd)T(\infty Grpd) is a cohesive (∞,1)-topos:

(ΠDiscΓcoDisc):T(Grpd)0base0baseGrpd. (\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=codΩ base = cod \circ \Omega^\infty, we have one more adjunction, the cotangent complex adjunction due to prop. 5

GrpddomΩ ΩT(Grpd)0base0baseGrpd. \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 H\mathbf{H} a general (∞,1)-topos the above discussion goes through essentially verbatim. If H\mathbf{H} is itself cohesive, then we end up with

HdomΩTH0base0baseHcoDiscΓDiscΠGrpd. \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 H\mathbf{H} a locally ∞-connected (∞,1)-topos (hence in particular for a cohesive (∞,1)-topos), there are canonical (∞,1)-functors

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

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

TH TΓ TGrpd 0 base 0 base H ΓDiscΠ Grpd \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, THT \mathbf{H} is the (∞,1)-Grothendieck construction of

Xlim(ΩH /X */ΩH /X */Ω). 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 Ω\Omega is an (∞,1)-limit construction and since the right adjoint global section functor Γ\Gamma preserves all (∞,1)-limits, there is a homotopy-commuting diagram

Ω H /X */ Ω H /X */ Ω Γ Γ Ω Grpd /Γ(X) */ Ω Grpd /Γ(X) */ Ω \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(H /X)Stab(Grpd /Γ(X)) Stab(\mathbf{H}_{/X}) \longrightarrow Stab(\infty Grpd_{/\Gamma(X)})

and hence a morphism

TH XHStab(H /X) XHStab(Grpd Γ(X)). 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 H\mathbf{H}) Disc:GrpdHDisc \;\colon\; \infty Grpd \longrightarrow \mathbf{H}

TH XHStab(H /X) XHStab(Grpd /Γ(X)) SGrpdStab(Grpd /S)TGrpd, 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 ΓDiscid\Gamma \circ Disc \simeq id.

Remark

When THT \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….

References

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

and section 7.3 of

The (infinity,1)-topos structure on tangent (,1)(\infty,1)-categories is discussed in 35.5 of

Revised on October 20, 2013 08:59:54 by Urs Schreiber (89.204.130.236)