nLab terminal geometric morphism



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(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

structures in a cohesive (∞,1)-topos


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.


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

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


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



The geometric morphism (1) exists essentially uniquely.

(e.g. Lurie 2009, HTT

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

(2)SlimS* 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 LConstLConst must be given by forming the corresponding \infty-colimit of copies of the terminal object * H\ast_{\mathbf{H}} in H\mathbf{H}, which does exist:

LConst(S) LConst(limS*) limSLConst(*) limS* H \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}


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

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

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


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

Grpd (S,H(* H,X)) Grpd (limS*,H(* H,X)) limSGrpd (*,H(* H,X)) limSH(* H,X) H(limS* H,X) H(limSLConst(*),X) H(LConst(limS*),X) H(LConst(S),X) \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 LConstLConst preserves finite limits such as the terminal object, by definition.)

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


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

Discussion in \infty -topos theory includes:

Last revised on May 26, 2023 at 14:05:00. See the history of this page for a list of all contributions to it.