symmetric monoidal (∞,1)-category of spectra
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
Directly analogous to the concept of completion of a ring is the completion of a module over that ring.
In particular the formal completion or adic completion of a ring $A$ at an ideal $\mathfrak{a}$ has a corresponding analog for modules. Where the adic completion $A_{\mathfrak{a}}^\wedge$ of the ring itself has the geometric interpretation of forming the formal neighbourhood $Spf(A_{\mathfrak{a}}^\wedge)$ of ring spectra $Spec(A/\mathfrak{a}) \hookrightarrow Spec(A)$, so under the interpretation (see here) of $A$-modules as bundles over $Spec(A)$, the $\mathfrak{a}$-adic completion $N_{\mathfrak{a}}^\wedge$ of an $A$-module $N$ has the interpretation of being the restriction of that bundle to that formal neighbourhood.
For $A$ a commutative ring, $\mathfrak{a} \subset A$ an ideal in $A$ and for $N$ an $A$-module, then the $\mathfrak{a}$-adic completion or formal completion at $\mathfrak{a}$ of $N$ is the filtered limit
of quotients of $N$ by the submodules induced by all powers of the ideal.
There is a canonical projection map $N \longrightarrow N^\wedge_{\mathfrak{a}}$. Its kernel is sometimes called the $\mathfrak{a}$-adic residual.
Let $A$ be an E-∞ ring and $\mathfrak{a} \subset \pi_0 A$ a finitely generated ideal of its underlying commutative ring.
An $A$-∞-module $N$ is an $\mathfrak{a}$-torsion module if for all elements $n \in \pi_k N$ and all elements $a \in \mathfrak{a}$ there is $k \in \mathbb{N}$ such that $a^k n = 0$.
(Lurie “Completions”, def. 4.1.3).
is co-reflective and the co-reflector $\Pi_{\mathfrak{a}}$ – the torsion approximation – is smashing.
(Lurie “Completions”, prop. 4.1.12).
For $N \in A Mod_{\leq 0}$ then torsion approximation, prop. , intuced a monomorphism on $\pi_0$
including the $\mathfrak{a}$-nilpotent elements of $\pi_0 N$.
(Lurie “Completions”, prop. 4.1.18).
An $A$-∞-module $N$ is an $\mathfrak{a}$-local module if for every $\mathfrak{a}$-torsion module $T$ (def. ), the derived hom space
is contractible.
(Lurie “Completions”, def. 4.1.9).
For $\mathfrak{a} = (a)$ generated from a single element, then the localization of an (∞,1)-ring-map $A \to A[a^{-1}]$ is given by the (∞,1)-colimit over the sequence of right-multiplication with $a$
(Lurie “Completions”, remark 4.1.11)
of ∞-modules local away from $\mathfrak{a}$ is reflective. The reflector
is called localization.
There is a natural homotopy fiber sequence
relating $\mathfrak{a}$-torsion approximation on the left with $\mathfrak{a}$-localization on the right.
An ∞-module $N$ over $A$ is $\mathfrak{a}$-complete if for all $\mathfrak{a}$-local $\infty$-modules $L$ (def. ) then $Hom_A(L,N) \simeq \ast$.
of the (∞,1)-category of ∞-modules on the $\mathbb{a}$-complete ones is a reflective sub-(∞,1)-category. The reflector
is called $\mathfrak{a}$-completion.
(Lurie “Completions”, def. 4.2.1, lemma 4.2.2).
Definition relates to the traditional definition, def. , as follows
Let $N$ a homotopically discrete ∞-module over the E-∞ ring $A$ which is a Noetherian module in that all its submodules are finitely finitely generated. Then the $\mathfrak{a}$-completion of $N$ in the sense of def. coincides with the traditional definition def. .
(Lurie “Completions”, prop. 4.3.6)
The full sub-(∞,1)-category $A Mod_{\mathfrak{a} comp}$ is a locally presentable (∞,1)-category.
(Lurie “Completions”, prop. 4.1.17)
We discuss how both $\mathfrak{a}$-completion $\flat_{\mathfrak{a}}$ and $\mathfrak{a}$-torsion approximation $\Pi_{\mathfrak{a}}$ on $A Mod$ are monoidal (∞,1)-functors with respect to the smash product of spectra over $A$.
Let $A$ be an E-∞ ring and $\mathfrak{a} \subset \pi_0 A$ a finitely generated ideal of its underlying commutative ring.
The completion reflection $\flat_{\mathfrak{a}}$, def. , is a monoidal (∞,1)-functor.
(Lurie “Completions”, remark 4.2.6).
For the torsion approximation functor $\Pi_{\mathfrak{a}}$ one gets something slightly weaker, it preserves “monoids without unit”:
The full sub-(∞,1)-category of $\mathfrak{a}$-torsion modules, def. , is co-reflective
Moreover, the coreflector $\Pi_{\mathfrak{a}}$ is “smashing”, in that there is $V \in A Mod$ such that $\Pi_{\mathfrak{a}}(-) \simeq V \wedge (-)$ is given by the smash product with $V$. If $\mathfrak{a} = (\{x_i\}_i)$ then $V$ is the tensor product $V =\underset{i}{\otimes} V_i$ over all the homotopy fibers
(Lurie “Completions”, prop. 4.1.12).
From the general properties of smashing localization it follows that
The coreflection $\Pi_{\mathfrak{a}} \colon A Mod \to A Mod$
preserves small (∞,1)-colimits;
is a “monoidal (∞,1)-functor” except possibly for preservation of units.
See also (Lurie “Completions”, cor. 4.1.16).
The homotopy cofiber of $\mathfrak{a}$-completion $\Pi_{\mathfrak{a}}$ is localization away from $\mathfrak{a}$, in that there is a homotopy fiber sequence
with the completion functor of def. on the left and the localization functor of prop. on the right.
(Lurie “Completions”, example 4.1.14, remark 4.1.20)
For suitable ideals $\mathfrak{a}\subset A$ of a commutative ring $A$ or more generally of an E-∞ ring, then the derived functor of $\mathfrak{a}$-adic completion of A-modules forms together with $\mathfrak{a}$-torsion approximation an adjoint modality on the
(∞,1)-category of modules over $A$. See at fracture square for details.
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:
Discussion in the context of higher algebra is in
Discussion of formal completion of (infinity,1)-modules in terms of totalization of Amitsur complexes is in
Last revised on February 2, 2016 at 10:30:39. See the history of this page for a list of all contributions to it.