Dimension theory | Higher Topos Theory notes

6.5. Dimension theory

The homotopy dimension of a topos measures how far its global sections functor can lower connectivity. For a slice topos $\An_{/X}$ , this recovers a classical finiteness invariant of the anima $$X$$ : it asks whether $$X$$ is a retract of a finite-dimensional CW-complex. Dimension also controls the convergence problem from the preceding section. A local finite-dimensionality hypothesis forces Postnikov towers to converge, and a uniform local bound also ensures that every compatible system of truncations is realized by an object.

6.5.1. Homotopy dimension and examples

Definition 6.67.

Let $\phi_*\colon S \to T$ be a morphism of topoi. We say that $\phi_*$ has dimension $\leq d$ if, for every $n \geq d-1$ , the functor $\phi_*$ sends $$n$$ -connected maps to $$(n-d)$$ -connected maps.

We say that $$T$$ has dimension $\leq d$ if the terminal morphism $\Gamma_* = \Hom_T(*,-)\colon T \to \An$ has dimension $\leq d$ .

Lemma 6.68.

A topos $$T$$ has dimension $\leq d$ if and only if every $$(d-1)$$ -connected object of $$T$$ has a global section.

Proof

For the `only if'-direction, suppose

\(T\)

has dimension

\leq d

, and let

\(X\)

be a

\((d-1)\)

-connected object. Then the unique map

X \to *

\((d-1)\)

-connected. By the dimension condition, the induced map

\Gamma_*(X) \to \Gamma_*(*) = *

\((-1)\)

-connected. A map to the terminal object being

\((-1)\)

-connected means that its domain is nonempty, so

\(X\)

has a global section.For the `if'-direction, suppose every

\((d-1)\)

-connected object of

\(T\)

has a global section. We prove by induction on

n \geq d-1

that

\Gamma_*

sends

\(n\)

-connected maps to

\((n-d)\)

-connected maps.For the initial case, let

f\colon X \to Y

\((d-1)\)

-connected. An effective epimorphism of animae is characterized by having nonempty fibers over every point. Given

y\colon * \to Y

, the fiber of

\Gamma_*(f)

over

\(y\)

\Gamma_*(X) \times_{\Gamma_*(Y)} \{y\} \;\simeq\; \Hom_T(*, X \times_Y *) \;=\; \Gamma_*(X_y),

where

X_y := X \times_Y *

. Since

\(X_y\)

\((d-1)\)

-connected, it has a global section by assumption. Hence the displayed fiber is nonempty, so

\Gamma_*(f)

is an effective epimorphism, i.e. it is

\((-1)\)

-connected.Now let

n \geq d

and let

\(f\)

\(n\)

-connected. It is in particular

\((d-1)\)

-connected, so

\Gamma_*(f)

is an effective epimorphism. Since

\Gamma_*

preserves limits, the diagonal of

\Gamma_*(f)

\Gamma_*(\Delta_f)

. The map

\Delta_f

\((n-1)\)

-connected, so the induction hypothesis shows that its image is

\((n-d-1)\)

-connected. The characterization of connected maps by their diagonals in Theorem 3.22 now shows that

\Gamma_*(f)

\((n-d)\)

-connected.

Definition 6.69.

We say that a topos $$T$$ is locally of dimension $\leq d$ (resp. locally of finite dimension) if it is generated under colimits by objects $$U$$ such that the slice topos $T_{/U}$ has dimension $\leq d$ (resp. has some finite dimension).

Here are some basic sources of dimension bounds.

Example 6.70.

Let $X \in \An$ . If $$X$$ is represented by a $$d$$ -dimensional CW-complex, or by a $$d$$ -dimensional simplicial set, then $\dim(\An_{/X}) \leq d$ .

More precisely, $\dim(\An_{/X}) \leq d$ if and only if $$X$$ is a retract, in the homotopy category of animae, of an anima represented by a $$d$$ -dimensional CW-complex. If $d \neq 2$ , this is further equivalent to $$X$$ itself admitting a $$d$$ -dimensional CW-model. For $$d=2$$ , the retract condition implies that $$X$$ admits a $$3$$ -dimensional CW-model, but it need not admit a $$2$$ -dimensional one. See [Lurie 2009, Example 7.2.1.4].

Example 6.71.

If $$C$$ admits a terminal object $$1$$ , then $\PSh(C)$ has dimension $\leq 0$ . Indeed, global sections are given by evaluation at $$1$$ , and evaluation preserves effective epimorphisms. Equivalently, if $\Ff$ is $$(-1)$$ -connected, then $\Gamma_*(\Ff) \iso \Ff(1)$ is nonempty. See [Lurie 2009, Example 7.2.1.3].

Example 6.72.

The presheaf category $\PSh(\N)$ on the poset of natural numbers has dimension $\leq 1$ . Indeed, a $$0$$ -connected presheaf is a tower

\cdots \longrightarrow X_2 \longrightarrow X_1 \longrightarrow X_0

of connected animae. Its limit is nonempty: after replacing the tower by a tower of Kan fibrations, one may choose a point of $$X_0$$ and lift it successively through the tower. Thus every $$0$$ -connected object has a global section, and Lemma 6.68 applies.

Example 6.73.

Let $$X$$ be a paracompact Hausdorff space. We say that $$X$$ has covering dimension $\leq d$ if every open covering is refined by one with empty $$(d+2)$$ -fold intersections. Then

\dim \Shv(X) \leq d \quad\Longleftrightarrow\quad \text{$X$ has covering dimension $\leq d$}.

This applies, for example, to every CW-complex of dimension $\leq d$ . See [Lurie 2009, Theorem 7.2.3.6 and Corollary 7.2.3.7].

Example 6.74.

Let $$X$$ be a spectral space of Krull dimension $\leq d$ . Then $\Shv(X)$ has dimension $\leq d$ ; see [Clausen and Mathew 2021, Theorem 3.12].

Example 6.75.

If $$X$$ is a qcqs scheme, then the following bounds hold:

$\dim(X_{\mathrm{Nis}}) \leq \dim_{\mathrm{Krull}}(X)$ ; see [Clausen and Mathew 2021, Theorem 3.18].
$\dim(X_{\mathrm{cdh}}) \leq \dim_{\mathrm{val}}(X)$ ; see [Elmanto et al. 2021, Theorem 2.4.15].

6.5.2. Convergence and Postnikov-completeness

The following result explains why dimension enters the comparison among the completion conditions in Section 6.4. Its first part is local and only concerns Postnikov towers of existing objects. Essential surjectivity of $T \to \widehat T$ requires a uniform local bound.

Proposition 6.76. ([Lurie 2009, Proposition 7.2.1.10 and Corollary 7.2.1.12])

Let $$T$$ be a topos.

If $$T$$ is locally of finite dimension, then for all $X \in T$ , the map $X \iso \lim_n \tau_n X$ is an isomorphism. In particular, $$T$$ is hypercomplete.
If $$T$$ is locally of dimension $\leq d$ for some $$d$$ , then $$T$$ is Postnikov-complete.

Proof

(1) Let

\{U_\alpha\}

be a collection of generators such that every slice

T_{/U_\alpha}

has finite dimension, say at most

d_\alpha

. For every

X \in T

, the dimension bound shows that

\Hom_T(U_\alpha,X) \longrightarrow \Hom_T(U_\alpha,\tau_nX)

(n-d_\alpha)

-connected. The target is

\(n\)

-truncated. The standard convergence criterion for towers of animae therefore identifies

\Hom_T(U_\alpha,X)

with

\lim_n\Hom_T(U_\alpha,\tau_nX)

: in each fixed degree, the homotopy groups and the transition maps stabilize. This is also the mapping-anima argument in the proof of [Lurie 2009, Proposition 7.2.1.10]. Since the

U_\alpha

generate

\(T\)

, it follows that

X \to \lim_n\tau_nX

is an isomorphism.If

f\colon X\to Y

\infty

-connected, then

\tau_nf

is an isomorphism for every

\(n\)

. Applying the convergence statement to

\(X\)

and

\(Y\)

shows that

\(f\)

itself is an isomorphism. Hence

\(T\)

is hypercomplete, in agreement with [Lurie 2009, Corollary 7.2.1.12].(2) By (1), the comparison

T \to \widehat T

is fully faithful. Let

(X_n)_n \in \widehat T

and put

X:=\lim_nX_n

. The transition map

X_{m+1}\to X_m\iso\tau_mX_{m+1}

\(m\)

-connected. The highly connected tower criterion proved in [Lurie 2009, Proposition 7.2.1.10], applied with the uniform bound

\(d\)

, shows that, for every

\(n\)

, the map

X\to X_m

\(n\)

-connected for all sufficiently large

\(m\)

. For

m\geq n

, the composite

X \longrightarrow X_m \longrightarrow X_n

is therefore

\(n\)

-connected. Since

\(X_n\)

\(n\)

-truncated, this identifies

\(X_n\)

with

\tau_nX

. Thus every object of

\widehat T

is the Postnikov tower of an object of

\(T\)

, so

T\to\widehat T

is essentially surjective.

References

Jacob Lurie. Higher topos theory. Ann. Math. Stud. 170, Princeton, NJ: Princeton University Press. 2009.
Dustin Clausen, Akhil Mathew. Hyperdescent and étale $K$-theory. Invent. Math., 225 (3), 981–1076. 2021.
Elden Elmanto, Marc Hoyois, Ryomei Iwasa, Shane Kelly. Cdh descent, cdarc descent, and Milnor excision. Math. Ann., 379 (3--4), 1011–1045. 2021.