nLab terminal geometric morphism

For the time being, the following speaks in $\infty$-topos theory, for definiteness; but all statements and proofs apply verbatim also to Grothendieck toposes, since they just depend on general abstract category-theoretic properties.

Idea

For $\mathbf{H}$ an $\infty$-topos over the base $Grpd_\infty$, it has an essentially unique geometric morphism to $Grpd_\infty$

(1)$$\mathbf{H} \underoverset {\underset{\Gamma}{\longrightarrow}} {\overset{LConst}{\longleftarrow}} {\;\;\;\;\bot\;\;\;\;} Grpd_\infty$$

where:

• the left adjoint has the interpretation of constructing locally constant $\infty$-stacks,

• the right adjoint has the interpretation of forming global sections.

In other words, $Grpd_\infty$ is the terminal object in the $\infty$-category $Topos_\infty$ of $\infty$-stack $(\infty,1)$-toposes and $\infty$-geometric morphisms between these.

Properties

Proposition

The geometric morphism (1) exists essentially uniquely.

(e.g. Lurie 2009, HTT 6.3.4.1)

Proof

Since every $\infty$-groupoid is an $\infty$-colimit (over itself) of the point (see there):

(2)$$S \,\simeq\, \underset{\underset{S}{\longrightarrow}}{\lim} \,\ast$$

and since the inverse image of a geometric morphism preserves finite limits (by definition), such as the terminal object, and all $\infty$-colimits (since left adjoints preserve colimits), we have that $LConst$ must be given by forming the corresponding $\infty$-colimit of copies of the terminal object $\ast_{\mathbf{H}}$ in $\mathbf{H}$, which does exist:

$$\begin{aligned} LConst(S) & \;\simeq\; LConst \Big( \underset{ \underset{S}{\longrightarrow} }{\lim} \, \ast \Big) \\ & \;\simeq\; \underset{ \underset{S}{\longrightarrow} }{\lim} \, LConst \big( \ast \big) \\ & \;\simeq\; \underset{ \underset{S}{\longrightarrow} }{\lim} \, \ast_{\mathbf{H}} \end{aligned}$$

Proposition

The direct image of the terminal geometric morphism (1) is given by the hom-space out of the terminal object, in that for $X \,\in\, \mathbf{H}$ there is a natural equivalence

$$\Gamma (X) \;\simeq\; \mathbf{H}(\ast_{\mathbf{H}} ,\, X) \,,$$

where $\ast_{\mathbf{H}} \,\in\, \mathbf{H}$ denotes the terminal object.

Proof

For all $S \,\in\, Grpd_\infty$ we have the following sequence of natural equivalences:

$$\begin{array}{lll} Grpd_\infty \big( S ,\, \mathbf{H}(\ast_{\mathbf{H}} ,\, X) \big) & \;\simeq\; Grpd_\infty \Big( \underset{\underset{S}{\longrightarrow}}{\lim} \ast ,\, \mathbf{H}(\ast_{\mathbf{H}} ,\, X) \Big) \\ & \;\simeq\; \underset{\underset{S}{\longleftarrow}}{\lim} \, Grpd_\infty \big( \ast ,\, \mathbf{H}(\ast_{\mathbf{H}} ,\, X) \big) \\ & \;\simeq\; \underset{\underset{S}{\longleftarrow}}{\lim} \, \mathbf{H}(\ast_{\mathbf{H}} ,\, X) \\ & \;\simeq\; \mathbf{H} \Big( \underset{\underset{S}{\longleftarrow}}{\lim} \ast_{\mathbf{H}} ,\, X \Big) \\ & \;\simeq\; \mathbf{H} \Big( \underset{\underset{S}{\longleftarrow}}{\lim} \, LConst(\ast) ,\, X \Big) \\ & \;\simeq\; \mathbf{H} \Big( LConst \big( \underset{\underset{S}{\longleftarrow}}{\lim} \ast \big) ,\, X \Big) \\ & \;\simeq\; \mathbf{H} \big( LConst(S) ,\, X \big) \end{array}$$

(Here we used (2) and that hom-functor preserves limits and that left adjoints preserve colimits and that $LConst$ preserves finite limits such as the terminal object, by definition.)

But this is a hom-equivalence which exhibits (see here) $\mathbf{H}(\ast,-)$ as a right adjoint $\infty$-functor to $LConst$. This implies the claim by essential uniqueness of adjoints.

Literature

For discussion in plain topos theory see any of the references listed there.

Discussion in $\infty$-topos theory includes:

• Jacob Lurie, Section 6.3.4 in: Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press 2009 (pup:8957, pdf)