homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
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prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
A basic fact in number theory is that the natural numbers may be decomposed into the rational numbers and the p-adic integers for all prime numbers $p$. Dually in arithmetic geometry this says that Spec(Z) has a cover by all its formal disks and the complements of finitely many points, a fact that is crucial in the geometric interpretation of the function field analogy and which motivates for instance the geometric Langlands correspondence. (See below.)
Lifting this statement to stable homotopy theory and “higher arithmetic geometry” the arithmetic fracture theorem says that stable homotopy types (and suitably tame plain homotopy types) canonically decompose into their rationalization and their p-completion for all primes $p$, hence into their images in rational homotopy theory and p-adic homotopy theory. Since these images are typically simpler than the original homotopy type itself, this decomposition is a fundamental computational tool in stable homotopy theory, often known under the slogan of “working one prime at a time”. (See below.)
One finds that this arithmetic fracturing in stable homotopy theory is really a statement about the Bousfield localization of spectra with respect to the Moore spectrum for $\mathbb{Q}$ and that of $\mathbb{Q}/\mathbb{Z}$. Viewed this way there is a more general fracture theorem which says that for any suitable pair $E,F$ of spectra/homology theories the Bousfield localization at their coproduct decomposes into the separate Bousfield localizations. This generalized fracture theorem appears for instance in chromatic homotopy theory for localization at Morava K-theory and Morava E-theory. (See below.)
In cohesive homotopy theory every stable homotopy type canonically sits in a fracture square formed from the localizations exhibited by the shape modality and the flat modality. For differential cohesion over infinitesimal cohesion this is a higher geometric analog of the classical artihmetic fracture. (See below.)
The statement in number theory/arithmetic geometry is the following:
The integers $\mathbb{Z}$ are the fiber product of all the p-adic integers $\underset{p\;prime}{\prod} \mathbb{Z}_p$ with the rational numbers $\mathbb{Q}$ over the rationalization of the former, hence there is a pullback diagram in CRing of the form
Equivalently this is the fiber product of the rationals with the integral adeles $\mathbb{A}_{\mathbb{Z}}$ over the ring of adeles $\mathbb{A}_{\mathbb{Q}}$
Since the ring of adeles is the rationalization of the integral adeles $\mathbb{A}_{\mathbb{Q}} = \mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{A}_{\mathbb{Z}}$, this is also (by the discussion here) a pushout diagram in CRing, and in fact in topological commutative rings (for $\mathbb{Q}$ with the discrete topology and $\mathbb{A}_{\mathbb{Z}}$ with its profinite/completion topology).
An original discussion is (Sullivan 05, prop. 1.18). Review includes (Riehl 14, lemma 14.4.2).
Under the function field analogy we may think of
$Spec(\mathbb{Z})$ as an arithmetic curve over F1;
$\mathbb{A}_{\mathbb{Z}}$ as the ring of functions on the formal disks around all the points in this curve;
$\mathbb{Q}$ as the ring of functions on the complement of a finite number of points in the curve;
$\mathbb{A}_{\mathbb{Q}}$ is the ring of functions on punctured formal disks around all points, at most finitely many of which do not extend to the unpunctured disk.
Under this analogy the arithmetic fracture square of prop. 1 says that the curve $Spec(\mathbb{Z})$ has a cover whose patches are the complement of the curve by some points, and the formal disks around these points.
This kind of cover plays a central role in number theory, see for instance the following discussions:
In homotopy theory the corresponding statement is that homotopy types may be decomposed into that of rational homotopy types and p-complete homotopy types of p-local homotopy types.
Let $p$ be a prime number. Let $X$ be a homotopy type/∞-groupoid satisfying at least one of the following sufficient conditions
$X$ is a connected, nilpotent space with finitely generated homotopy groups;
$X$ is p-local homotopy type;
Then $X$ is the homotopy fiber product
of its rationalization $X_{\mathbb{Q}}$ with its p-completion $X_p^\wedge$ over the rationalization $(X_p^\wedge)_{\mathbb{Q}}$of the $p$-completions.
This originates around (Bousfield-Kan 72, VI.8.1). A detailed more modern account is in (May-Ponto, theorem 13.1.4). A quick survey is in (Riehl 14, theorem 14.4.14).
Similar statements hold in stable homotopy theory for spectra. There is a stable version of
but more generally there are fracture squares for the coproduct homology theory $E \vee F$ whenever $F$-localization is $E$-acyclic:
For $p$ a prime number write
$L_p$ for Bousfield localization of spectra at the Moore spectrum $S \mathbb{F}_p$, hence for p-completion $(-)_p^\wedge$;
$L_{\mathbb{Q}}$ for the Bousfield localization of spectra at the Moore spectrum/Eilenberg-MacLane spectrum $S \mathbb{Q} \simeq H \mathbb{Q}$, hence for rationalization.
(Sullivan arithmetic square)
For every spectrum $X$ the canonical square
is a homotopy pushout (hence also a homotopy pullback).
Original statements of this include (Bousfield 79, Sullivan 05, prop. 3.20). Review includes (van Koughnett 13, prop. 4.5, Bauer 11, lemma 2.1).
(“one prime at a time”)
The impact of prop. 2 is that it decomposes the study of (stable) homotopy theory into that of
p-adic homotopy theory for each prime $p$.
Both the rationalization $X_{\mathbb{Q}}$ and the p-completion $X_{p}^\wedge$ are typically much easier to analyze than $p$ itself and so the fracture theorem gives a way to decompose the remaining hard part of study of homotopy types into that of $p$-local/$p$-complete spaces. This procedure is known in homotopy theory as working “one prime at a time”.
More generally:
The product of all p-completions is equivalently the Bousfield localization of spectra at the wedge sum $\vee_p S \mathbb{F}_p$ of all Moore spectra
Moreover there is a Bousfield equivalence
and therefore also an equivalence
The first statement originates around (Bousfield 79, prop. 2.6), review includes (van Koughnett 13, prop. 4.4, Bauer 11, below prop. 2.2); the second is highlighted in (Strickland 12, MO comment).
By the discussion at Bousfield localization of spectra and at localization of a space, the rationalization and the p-completion maps on spectra are homotopy cofibers of $E$-acyclifications $G_E(X) \to X$, for $E = S \mathbb{Q} \simeq H \mathbb{Q}$ and $E = S \mathbb{F}_p$ the Moore spectra of $\mathbb{Q}$ and of the cyclic group/finite field $\mathbb{F}_p = \mathbb{Z}/(p)$, respectively (e.g. Lurie 10, lecture 20).
Including this into the statement of prop. 2 says that for spectra $X$ satisfying sufficient conditions as above, then the canonical diagram
is homotopy exact, in that
the square is a homotopy pullback and hence also a homotopy pushout (this is prop. 3);
the diagonals are homotopy cofiber sequences and hence also homotopy fiber sequences (by this proposition at Bousfield localization of spectra);
the top and bottom outer composite sequences are homotopy fiber sequences (and hence homotopy cofiber sequences) (by applying the pasting law to the previous two items).
Together this is like two-thirds of a differential cohomology hexagon – to the extent that $p$-adic completion is adjoint to $p$-torsion approximation. This is indeed the case, as the next proposition asserts
Notice that in view of remark 1 then $X_p^\wedge$ is like the restriction of $X$ from Spec(Z) to all formal disks around the points $(p)$, and hence $G_{S\mathbb{F}_p}$ is like the restriction to the “complement of all formal disks”. Finally $X_{\mathbb{Q}}$ may be understood as the restriction to the Ran space of $Spec(\mathbb{Z})$ (Gaitsgory 11), roughly the colimit of the restriction of $X$ to the complement of finitely many points, as this set of points ranges through all points.
In view of remark 4 we may regard the following fact as an refinement of the traditional arithmetic fracture theorem.
Let $A$ be an E-∞ ring and let $\mathfrak{a} \subset \pi_0 A$ be a finitely generated ideal in its underlying commutative ring.
Then there is an adjoint triple of adjoint (∞,1)-functors
where
$A Mod$ is the stable (∞,1)-category of modules, i.e. of ∞-modules over $A$;
$A Mod_{\mathfrak{a}tors}$ and $A Mod_{\mathfrak{a} comp}$ are the full sub-(∞,1)-categories of $\mathfrak{a}$-torsion and of $\mathfrak{a}$-complete $A$-∞-modules, respectively;
$(-)^{op}$ denotes the opposite (∞,1)-category;
the equivalence of (∞,1)-categories on the left is induced by the restriction of $\Pi_{\mathfrak{a}}$ and $\flat_{\mathfrak{a}}$.
This is effectively the content of (Lurie “Proper morphisms”, section 4):
the existence of $\Pi_{\mathfrak{a}}$ is corollary 4.1.16 and remark 4.1.17
the existence of $\flat_{\mathfrak{a}}$ is lemma 4.2.2 there;
the equivalence of sub-$\infty$-categories is proposition 4.2.5 there.
The traditional arithmetic fracture square of prop. 2, regarded as in remark 4, is the left part of the “differential cohomology diagram” induced by the adjoint modality $(\Pi_{\mathfrak{a}} \dashv \flat_{\mathfrak{a}} )$ of prop. 8, for the special case that $X = \mathbb{S}$ is the sphere spectrum and $\mathfrak{a} = (p)$ a prime ideal
cohesion in E-∞ arithmetic geometry:
cohesion modality | symbol | interpretation |
---|---|---|
flat modality | $\flat$ | formal completion at |
shape modality | $ʃ$ | torsion approximation |
dR-shape modality | $ʃ_{dR}$ | localization away |
dR-flat modality | $\flat_{dR}$ | adic residual |
the differential cohomology hexagon/arithmetic fracture squares:
The special case of prop. 8 where $A$ is the Eilenberg-MacLane spectrum of a plain commutative ring, and hence – by the stable Dold-Kan correspondence – the case where $A$-∞-modules are equivalently chain complexes, has a longer tradition in the existing literature. This we highlight separately below.
By the discussion at completion of modules in the section Monoidalness, the adjoint modality of prop. 8 is a monoidal (∞,1)-functor without, possibly, respect the tensor unit in $A Mod$. This means that $(\Pi_{\mathfrak{a}}\dashv \flat_{\mathfrak{a}})$ passes to “commutative ∞-monoids-without unit” in $A Mod$, hence to (formal duals of) nonunital E-∞ algebras. By this proposition (Lurie “Algebra”, prop. 5.2.3.15) nonunital E-∞ rings are equivalent to augmented E-∞ rings over the sphere spectrum, hence this is E-∞ arithmetic geometry under $Spec(\mathbb{S})$.
Notice that in addition $\Pi_{\mathfrak{a}}$ here should preserve finite products (because by the discussion at completion of a module – monoidalness the underlying $\Pi_{\mathfrak{a}} \colon A Mod \to A Mod$ preserves all small (∞,1)-colimits and because by this proposition finite coproducts in $CRng(A Mod)$ are computed in the underlying $A Mod$.
Therefore we may think of $\Pi_{\mathfrak{a}}$ as a shape modality and of $\flat_{\mathfrak{a}}$ as a sharp modality on affine E-∞-arithmetic geometry under $Spec(\mathbb{S})$ – namely on formal duals of nonunital E-∞ rings .
(It may be entertaining to note that on the level of ∞-groups of units then E-∞ arithmetic geometry under $Spec(\mathbb{S})$ translates to abelian ∞-groups of twists over the sphere spectrum – which has been argued to be the homotopy-theoretic incarnation of superalgebra, see at superalgebra – abstract idea for more on this.)
For more discussion of this see also differential cohesion and idelic structure.
We discuss here arithmetic fracturing on chain complexes of modules. Under the stable Dold-Kan correspondence this is a special case of prop. 8, but this special case has a longer tradition in the literature – going back to Grothendieck local duality and sometimes called Greenlees-May duality due to (Greenlees-May 92) – and we point to these original proofs.
Let $A$ be a commutative ring, let $\mathfrak{a} \subset A$ be be an ideal generated by a single regular element (i.e. not a zero divisor). Write $A Mod_{\infty}^{op}$ for the opposite (∞,1)-category of the (∞,1)-category of modules over $A$.
Write
$\flat_{\mathfrak{a}}\colon A Mod_\infty^{op} \to A Mod_{\infty}^{op}$ for the derived functor of formal completion (adic completion) of modules at $\mathfrak{a}$;
with canonical natural transformation
$\epsilon_{\mathfrak{a}} \colon \flat_{\mathfrak{a}} \longrightarrow id$
$\Pi_{\mathfrak{a}} \colon A Mod_\infty^{op} \to A Mod_\infty^{op}$ for the total derived functor of the $\mathfrak{a}$-torsion approximation-functor;
with canonical natural transformation
$\eta_{\mathfrak{a}}\colon id \longrightarrow \Pi_{\mathfrak{a}}$
Finally write
for the full (∞,1)-subcategories of objects $X$ for which, $\epsilon_{\mathfrak{a}}(X)$ or $\eta_{\mathfrak{a}}(X)$ is an equivalence in an (∞,1)-category, respectively.
The transformation $\epsilon_{\mathfrak{a}}$ exhibits $(A Mod_\infty^{op})^{\mathfrak{a}com}\hookrightarrow A Mod_\infty$ as a reflective (∞,1)-subcategory, hence $\flat_{\mathfrak{a}}$ as an idempotent (∞,1)-monad.
The transformation $\eta_{\mathfrak{a}}$ exhibits $(A Mod_\infty^{op})^{\mathfrak{a}tor}\hookrightarrow A Mod_\infty$ as a co-reflective $(\infty,1)$-category, hence $\Pi_{\mathfrak{a}}$ as an idempotent $(\infty,1)$-comonad.
Restricted to these sub-$(\infty,1)$-categories both $\flat_{\mathfrak{a}}$ as well as $\Pi_{\mathfrak{a}}$ become equivalences of (∞,1)-categories, hence exhibiting $(\Pi_{\mathfrak{a}} \dashv \flat_{\mathfrak{a}})$ as a higher adjoint modality.
This is a paraphrase of the results in (Dwyer-Greenlees 99) and (Porta-Shaul-Yekutieli 10) from the language of derived categories to (∞,1)-category theory.
First of all, by our simplifying assumption that $\mathfrak{a}$ is generated by a single regular element, the running assumption of “weak proregularity” in (Porta-Shaul-Yekutieli 10, def.3.21) is satisfied.
Then in view of (Porta-Shaul-Yekutieli 10, corollary 3.31) the statement of (Porta-Shaul-Yekutieli 10, theorem 6.12) is the characterization of reflectors-category#CharacterizationOfReflectors) as discussed at reflective sub-(∞,1)-category, and formally dually so for the coreflection. With the fully faithfulness that goes with this the equivalence of the two inclusions on the level of homotopy categories given by (Hovey-PalieriS-trickland 97, 3.3.5, Dwyer-Greenlees 99, theorem 2.1 Porta-Shaul-Yekutieli 10, theorem 6.11) implies the canonical equivalence of the two sub-(∞,1)-categories and this means that $\Pi_{\mathfrak{a}}$ and $\flat_{\mathfrak{a}}$ are the adjoint pair induced from the reflection/coreflection adjoint triple.
This adjoint triple is stated more explicitly in (Dwyer-Greenlees 99, section 4), see also (Porta-Shaul-Yekutieli 10, end of remark 6.14).
Therefore arithmetic fracture squares in the homotopy theory of chain complexes are induced by this as in corollary 1 above.
By prop. 3 the arithmetic fracture square of prop. 2 is equivalently of the form
In this form the statement holds much more generally:
Let $E, F, X$ be spectra such that the $F$-localization of $X$ is $E$-acyclic, i.e. $E_\bullet(L_F X) \simeq 0$, then the canonical square diagram
is a homotopy pullback (and hence by stability also a homotopy pushout).
e.g. (Bauer 11, prop. 2.2)
The general version of the fracture statement in prop. 6 is used frequently in chromatic homotopy theory for decomposition in Morava K-theory and Morava E-theory-localizations. For example there is a chromatic fracture square:
In particular it is used for instance in the construction of tmf, see example 2 below.
From another perspective:
Suppose that $L \colon Spectra \to Spectra$ is a smashing localization given by smash product with some spectrum $T$. Write $F$ for the homotopy fiber
Then there is a fracture diagram of operations
where $[F,-]$ and $T \wedge (-) \colon Spectra \to Spectra$ are idempotent (∞,1)-monads and $[T,-]$, $[F,-]$ are idempotent $\infty$-comonad, the diagonals are homotopy fiber sequences.
(Charles Rezk, MO comment,August 2014)
For $T = S \mathbb{Z}[p^{-1}]$ the Moore spectrum of the integers localized away from $p$, then
and hence
$\Sigma^{-1} S (\mathbb{Z}[p^{-1}]/\mathbb{Z}) \wedge (-)$ is $p$-torsion approximation;
$[\Sigma^{-1} S (\mathbb{Z}[p^{-1}]/\mathbb{Z}),-]$ is $p$-completion;
$S \mathbb{Z}[p^{-1}] \wedge (-)$ is $p$-rationalization;
$[T,-]$ is forming $p$-adic residual.
With (Bousfield 79, prop.2.5)
Let $A$ be an E-∞ ring and let $\mathfrak{a} \subset \pi_0 A$ be a finitely generated ideal in its underlying commutative ring.
Then there is an adjoint triple of adjoint (∞,1)-functors
where
$A Mod$ is the stable (∞,1)-category of modules, i.e. of ∞-modules over $A$;
$A Mod_{\mathfrak{a}tor}$ and $A Mod_{\mathfrak{a} comp}$ are the full sub-(∞,1)-categories of $\mathfrak{a}$-torsion and of $\mathfrak{a}$-complete $A$-∞-modules, respectively;
$(-)^{op}$ denotes the opposite (∞,1)-category;
the equivalence of (∞,1)-categories on the left is induced by the restriction of $\Pi_{\mathfrak{a}}$ and $\flat_{\mathfrak{a}}$.
This is effectively the content of (Lurie “Completions”, section 4):
In cohesive homotopy theory every stable homotopy type $X$ sits in a fracture square of the form
where $\flat$ is the flat modality and $\Pi_{dR}$ the homotopy fiber of the unit $X\to \Pi X$ of the shape modality. This is the left part of the differential cohomology hexagon for $X$, see there for details.
Here $\Pi_{dR} X$ is such that for any other stable cohesive homotopy type $\hat E$ then functions $\Pi_{dR} X \to \hat E$ are equivalent to functions $X \to \flat_{dR} \hat E$, where $\hat E \to \flat_{dR} \hat E$ is a generalized form of rationalization in the sense discussed at differential cohomology hexagon. In particular if $\hat E$ is a Hopkins-Singer-type differential cohomology refinement of a plain spectrum $E$, then $E\to \flat_{dR} E$ is its ordinary rationalization given by the Chern character and $\hat E \to \flat_{dR} \hat E$ is the corresponding map on Chern curvature forms.
Moreover, if the ambient cohesion is differential cohesion over a base of infinitesimal cohesion, then the flat modality $\flat$ takes any space $X$ to the union of all its formal disks. (See at differential cohesion and idelic structure.) Accordingly the collection of functions $\flat X \to \hat E$ in this case behave like the product of all formal power series of $\hat E$-valued functions around all global points of $X$, analogous to remark 1.
An example of this are synthetic differential ∞-groupoids regarded as cohesive over their formal moduli problems, as its its complex analytic incarnation by synthetic differential complex analytic ∞-groupoids. In this context if $X = \Sigma$ is a complex curve then $\flat \Sigma$ is precisely the analog of the integral adeles as it is predicted by the function field analogy.
The construction of the tmf-spectrum – the spectrum of global sections of the derived Deligne-Mumford stack of derived elliptic curves – as described in (Behrens 13) proceeds by first applying the arithmetic fracture square of prop. 1, prop. 2 to decompose the moduli stack of elliptic curves into rational and $p$-adic curves, and then in a second step in applying in turn the general fracture square of prop. 6 for Morava K-theory to the remaining $p$-adic pieces.
See at tmf – Decomposition via arithmetic fracture squares for more on this.
Aldridge Bousfield, Daniel Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol 304, Springer 1972
Aldridge Bousfield, The localization of spectra with respect to homology , Topology vol 18 (1979) (pdf)
Dennis Sullivan, Geometric topology: localization, periodicity and Galois symmetry, volume 8 of K- Monographs in Mathematics. Springer, Dordrecht, 2005. The 1970 MIT notes, Edited and with a preface by Andrew Ranicki (pdf)
Tilman Bauer, Bousfield localization and the Hasse square, 2011 (pdf)
Paul VanKoughnett, Spectra and localization, 2013 (pdf)
Emily Riehl, Categorical homotopy theory, new mathematical monographs 24, Cambridge University Press 2014 (published version)
Peter May, Kate Ponto, chapters 7 and 8 of More concise algebraic topology: Localization, completion, and model categories (pdf)
Michael Shulman, The Propositional Fracture Theorem, (blog post)
Related MO-discussion:
Discussion of rational functions as functions on the Ran space is in
Discussion of $\mathfrak{a}$-adic completion and $\mathfrak{a}$-torsion approximation as derived idempotent (co-)monads on a derived category of chain complexes of modules – Greenlees-May duality – is in
building on
John Greenlees, Peter May, Derived functors of I-adic completion and local homology, J. Algebra 149 (1992), 438–453 (pdf)
Leovigildo Alonso, Ana Jeremías, Joseph Lipman, Local Homology and Cohomology on Schemes (arXiv:alg-geom/9503025)
Mark Hovey, John Palmieri, Neil Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114.
William Dwyer, John Greenlees, Complete modules and torsion modules, Amer. J. Math. 124, No. 1, (1999) (pdf)
Discussion of this in stable homotopy theory and the full generality of higher algebra is in
And in the context of commutative DG-rings in
This and further generalization is in
See also