Essential morphisms and shape theory | Higher Topos Theory notes

6.6. Essential morphisms and shape theory

Shape theory extracts a pro-object which records how a topos sees constant objects. For an arbitrary morphism of topoi, this construction naturally takes values in a pro-category. It is represented by an ordinary object precisely when the inverse image functor admits a further left adjoint. We first discuss these essential morphisms, then construct relative shape using pro-left adjoints, and finally characterize essentiality by the local representability of relative shape. We closely follow Aizenbud and Carmeli (2021, Section 3.1).

6.6.1. Essential morphisms

Definition 6.77. (Essential morphism)

A morphism of topoi $\phi_* \colon T \to S$ is called essential if the inverse image functor $\phi^*\colon S \to T$ admits a further left adjoint $\phi_\sharp \colon T \to S$ . We thus have an adjoint triple

\phi_\sharp \quad\dashv\quad \phi^* \quad\dashv\quad \phi_*.

We call a topos $$T$$ essential if the terminal morphism $\Gamma_*\colon T \to \An$ is essential.

Example 6.78.

Every étale morphism of topoi is essential: given a topos $$T$$ and an object $U \in T$ , the functor $j^*\colon T \to T_{/U}$ given by $X \mapsto X \times U$ admits a left adjoint $j_\sharp\colon T_{/U} \to T$ given by the forgetful functor.

Example 6.79.

For any small category $$C$$ , the presheaf topos $\PSh(C)$ is essential. Indeed, the terminal morphism $\Gamma_*\colon \PSh(C) \to \An$ is given by taking the limit $\Gamma_*(F) = \lim_{c \in C\catop} F(c)$ , and its left adjoint $\Gamma^*\colon \An \to \PSh(C)$ sends an anima $$A$$ to the constant presheaf $\underline{A}$ . The functor $\Gamma^*$ admits a further left adjoint $\Gamma_\sharp\colon \PSh(C) \to \An$ given by the colimit:

\Gamma_\sharp(F) \quad=\quad \colim_{c \in C\catop} F(c).

6.6.2. Pro-left adjoints and relative shape

For a general morphism of topoi $\phi_*$ , the inverse image functor $\phi^*$ need not admit a left adjoint. It does acquire one after passage to pro-categories. For a presentable category $$C$$ which admits finite limits, we use the large pro-category

\Pro(C) \quad:=\quad \Fun^{\lex,\acc}(C,\An)\catop,

where the superscript indicates the accessible left exact functors. For a small category $$C$$ , this agrees with $\Ind(C\catop)\catop$ . There is a fully faithful embedding $C \hookrightarrow \Pro(C)$ given by $X \mapsto \Hom_C(X,-)$ . An object of $\Pro(C)$ is called pro-constant, or corepresentable, if it lies in the essential image of this embedding.

Proposition 6.80. (Pro-left adjoint)

Let $\phi_*\colon T \to S$ be a morphism of topoi. The induced functor

\Pro(\phi^*)\colon \Pro(S) \longrightarrow \Pro(T)

admits a left adjoint

\phi_\sharp\colon \Pro(T) \longrightarrow \Pro(S).

On corepresentable objects, it is characterized by

\phi_\sharp(X) = \Hom_T(X,\phi^*(-)) \qin \Fun^{\lex,\acc}(S,\An)\catop.

Its restriction $\phi_\sharp\colon T\to\Pro(S)$ preserves colimits. Moreover, $\phi_*$ is essential if and only if this restriction factors through $S\hookrightarrow\Pro(S)$ . In that case, the factorization is a left adjoint to $\phi^*$ .

Proof

Since

\phi^*

is accessible and left exact, the universal property of pro-categories gives the asserted left adjoint. For

X\in T

and

Y\in S

, its adjunction isomorphism reads

\Hom_{\Pro(S)}(\phi_\sharp(X),Y) \;\simeq\; \Hom_{\Pro(T)}(X,\Pro(\phi^*)(Y)) \;\simeq\; \Hom_T(X,\phi^*(Y)),

which gives the displayed formula. To see directly that the restriction to

\(T\)

preserves colimits, let

\((X_i)_i\)

be a diagram in

\(T\)

. For every

Y\in S

, we have \begin{align*} \Hom_{\Pro(S)}\bigl(\phi_\sharp(\colim_iX_i),Y\bigr) &\iso \Hom_T(\colim_iX_i,\phi^*Y) \\ &\iso \lim_i\Hom_T(X_i,\phi^*Y) \\ &\iso \Hom_{\Pro(S)}\bigl(\colim_i\phi_\sharp(X_i),Y\bigr). \end{align*} Since corepresentable objects detect isomorphisms in

\Pro(S)

, the claim follows.Finally, the restriction factors through

\(S\)

if and only if every functor

\Hom_T(X,\phi^*(-))

is corepresentable by an object of

\(S\)

. This says precisely that

\phi^*

admits a left adjoint.

The pro-left adjoint contains more information than we need for shape. Evaluating it on the terminal object records the effect of $\phi^*$ on global sections.

Definition 6.81. (Relative shape)

Let $\phi_*\colon T \to S$ be a morphism of topoi. We define the relative shape of $\phi_*$ as the pro-object

\abs{\phi} \quad:=\quad \phi_\sharp(*) \qquad\in \Pro(S).

Note that this pro-object corepresents the composite functor $\Gamma_{T} \circ \phi^*\colon S \to \An$ , where $\Gamma_{T}\colon T \to \An$ is the global sections functor of $$T$$ .

We say that $\phi_*$ is of constant shape if $\abs{\phi}$ is pro-constant, i.e., if it is corepresentable by an object of $$S$$ .

Definition 6.82. (Shape)

The shape $\abs{T}$ of a topos $$T$$ is the relative shape of the terminal morphism $\Gamma_*\colon T \to \An$ :

\abs{T} \quad:=\quad \abs{\Gamma_*} \qin \Pro(\An).

We say that $$T$$ has constant shape if the pro-anima $\abs{T}$ is pro-constant. We say that an object $X \in T$ has constant shape if the slice topos $T_{/X}$ does.

Example 6.83.

For a presheaf topos $\PSh(C)$ , the terminal morphism is essential by Example 6.79, and its shape is represented by

\abs{\PSh(C)} \quad\simeq\quad \colim_{c \in C\catop} * \;\simeq\; \abs{C},

the classifying anima of $$C$$ . In particular, $\PSh(C)$ has constant shape.

Example 6.84.

Let $$X$$ be a paracompact, locally contractible topological space. Then $\Shv(X)$ is essential and its shape is represented by the singular anima $\Pi_\infty X$ . Without a local contractibility hypothesis, the shape need not be represented by the weak homotopy type of $$X$$ ; the pro-object can retain information which the singular anima loses. See [Lurie 2017, Appendix A].

6.6.3. Local constancy and essentiality

A morphism need not have globally constant relative shape. Essentialness is instead detected after passing to a generating collection of slices.

Definition 6.85. (Locally of constant shape)

Let $\phi_*\colon T \to S$ be a morphism of topoi. Given an object $U \in T$ , we say that $$U$$ has constant shape relative to $\phi_*$ if the composite morphism

T_{/U} \xrightarrow{j_*} T \xrightarrow{\phi_*} S

is of constant shape.

We say that $\phi_*$ is locally of constant shape if there exists a collection of objects $\{U_i\}_{i \in I}$ which generates $$T$$ under colimits such that each $$U_i$$ has constant shape relative to $\phi_*$ .

A topos $$T$$ is locally of constant shape if the terminal morphism $\Gamma_*\colon T \to \An$ is locally of constant shape.

The following proposition provides a key characterization of essential morphisms:

Proposition 6.86. (Aizenbud and Carmeli (2021, Proposition 3.1.5))

Let $\phi_* \colon T \to S$ be a morphism of topoi and let $\{U_i\}_{i \in I}$ be a collection of objects which generates $$T$$ under colimits. Then the following conditions are equivalent:

The morphism $\phi_*$ is essential;
Every object $$U_i$$ has constant shape relative to $\phi_*$ .

Proof

Recall from Example 6.78 that the étale morphism

j_*\colon T_{/U} \to T

is essential. Pro-left adjoints compose, so the relative shape of the composite

\phi_* \circ j_*\colon T_{/U} \to S

is the image of the terminal object of

T_{/U}

under the composite

T_{/U} \xrightarrow{j_\sharp} T \xrightarrow{\phi_\sharp} \Pro(S).

Since

j_\sharp

is the forgetful functor sending

(V \to U)

\(V\)

, we see that the relative shape of

\phi_* \circ j_*

is equivalent to

\phi_\sharp(U) \in \Pro(S)

.If

\phi_*

is essential, then

\phi_\sharp

factors through

\(S\)

, so

\phi_\sharp(U)

is pro-constant for all

U \in T

, and in particular for all

\(U_i\)

.Conversely, assume that

\phi_\sharp(U_i)

is pro-constant for all

i \in I

. The embedding

S\hookrightarrow\Pro(S)

and the functor

\phi_\sharp\colon T\to\Pro(S)

both preserve colimits. Since the collection

\{U_i\}

generates

\(T\)

under colimits, it follows that

\phi_\sharp

factors through

\(S\)

. By Proposition 6.80, this means that

\phi_*

is essential.

Corollary 6.87. ([Lurie 2017, Proposition A.1.8])

A topos $$T$$ is essential if and only if it is locally of constant shape.

Essential morphisms interact well with objects of constant shape:

Lemma 6.88.

Let $\phi_*\colon T_1 \to T_2$ and $\psi_*\colon T_2 \to T_3$ be morphisms of topoi and assume that $\phi_*$ is essential. Let $U \in T_1$ be an object which has constant shape relative to the composite $\psi_* \circ \phi_* \colon T_1 \to T_3$ . Then the object $\phi_\sharp(U) \in T_2$ has constant shape relative to $\psi_*$ .

Proof

The shape of

\phi_\sharp(U) \in T_2

relative to

\psi_*

\psi_\sharp(\phi_\sharp(U)) \in \Pro(T_3)

. Since

\phi_*

is essential, we have

\phi_\sharp(U) \in T_2

, and the shape of

\(U\)

relative to

\psi_* \circ \phi_*

(\psi \circ \phi)_\sharp(U) \;\simeq\; \psi_\sharp(\phi_\sharp(U)).

By assumption, this is pro-constant, so

\phi_\sharp(U)

has constant shape relative to

\psi_*

Corollary 6.89.

Let $\phi_*\colon T_1 \to T_2$ be an essential morphism of topoi. Then $\phi_\sharp\colon T_1 \to T_2$ sends objects of constant shape in $$T_1$$ to objects of constant shape in $$T_2$$ .

References

Avraham Aizenbud, Shachar Carmeli. Relative de Rham theory on Nash manifolds. arXiv preprint arXiv:2105.12417. 2021.
Jacob Lurie. Higher algebra. https://www.math.ias.edu/~lurie/papers/HA.pdf. 2017.