Geometric morphism ($f^{\ast }⊣f_{\ast }$)

A geometric morphism $f:\mathcal{E}\to \mathcal{F}$ is an adjunction $f^{\ast }⊣f_{\ast }$ with finite-limit-preserving inverse image $f^{\ast }:\mathcal{F}\to \mathcal{E}$. Inverse image also preserves colimits as a left adjoint. An extra left adjoint $f_{!}⊣f^{\ast }$ makes it essential.

For a geometric morphism $f:\mathcal{E}\to \mathcal{F}$, the displayed internal frame represents $X\mapsto {\mathrm{Sub}}_{\mathcal{E}}(f^{\ast }X)$. The localic part is ${\mathrm{Sh}}_{\mathcal{F}}(L)\to \mathcal{F}$. Its basic open families pull back to the subobjects of $f^{\ast }X$ they represent; gluing these gives a full faithful, subobject-closed inverse image, hence the hyperconnected geometric morphism part. The represented subobject functor recovers the internal frame, proving uniqueness up to equivalence over $\mathcal{F}$.

A geometric morphism $h$ is hyperconnected when its inverse image functor of a geometric morphism is full and faithful and its essential image is closed under subobjects. It then identifies all subobjects of $h^{\ast }X$ with subobjects of $X$, and its image is closed under quotients as well, by descending the kernel-pair relation.

A geometric morphism $p:\mathcal{D}\to \mathcal{F}$ is localic when $\mathcal{D}$ is, over $\mathcal{F}$, the topos of sheaves on an internal locale of $\mathcal{F}$. Equivalently, subobjects of the objects $p^{\ast }X$, with $X\in \mathcal{F}$, form a separating family in $\mathcal{D}$.

Geometric morphisms into a sheaf topos correspond to continuous flat functors from its site into the domain topos, with continuity meaning that covers become jointly epimorphic families. Pulling back sheafified representables gives the functor; the tensor construction gives the inverse image in the other direction. This representation theorem should not be confused with the separately named Diaconescu theorem about choice and excluded middle.

Every Grothendieck topos has a unique geometric morphism to sets, up to isomorphism. Its inverse image is $\Delta S={\coprod }_{s\in S}1$ and its direct image is $\Gamma A=\mathrm{Hom}(1,A)$. A geometric inverse image out of sets must have this form because it preserves coproducts and the terminal object. A local topos is one for which $\Gamma$ itself is an inverse image.

The inverse image is a finite-limit-preserving left adjoint with right adjoint $f_{\ast }$. It preserves arbitrary colimits, and hence images as well as finite limits. These properties preserve the interpretation of geometric formulas. It need not preserve Heyting implication or every universal quantifier.

A geometric morphism is essential when its inverse image has an additional left adjoint. For a functor of small categories, precomposition on presheaves has left Kan extension and Right Kan extension as its two adjoints. Precomposition preserves finite limits pointwise, so this is an essential geometric morphism.

For $f:\mathcal{E}\to \mathcal{F}$, the comonad $G=f^{\ast }{f}_{\ast }$ is Cartesian, and its coalgebra topos $\mathcal{D}$ gives a factorization $f=i\circ p$. Here $p^{\ast }$ is the faithful forgetful functor and $i^{\ast }$ is the comparison functor. Its right adjoint sends $(A,a)$ to $\mathrm{Eq}(f_{\ast }a,{\eta }_{f_{\ast }A})$; applying $f^{\ast }$ identifies the comparison counit with the equalizer $a:A\to GA$ of $Ga,{\delta }_A$, proving that $i_{\ast }$ is full and faithful.

A geometric embedding is a geometric morphism whose direct-image functor is full and faithful. Its inverse image presents a left-exact reflective subcategory of the target topos. It is the second part of the surjection-embedding factorization of a geometric morphism.

A subtopos is a Grothendieck topos embedded by a geometric embedding, considered up to equivalence over the ambient topos. On a site $(\mathcal{C},J)$, subtoposes correspond to Grothendieck topologies $J^{\prime }\supseteq J$. On a classifying topos this is the duality between geometric quotients and subtoposes.

A geometric morphism is surjective when its inverse-image functor is faithful, equivalently conservative for geometric inverse images. Reflection of isomorphisms of subobjects already proves faithfulness by applying the inverse image to equalizers of parallel arrows. The surjection-embedding factorization of a geometric morphism separates this property from a fully faithful direct image.

For a geometric morphism, the characteristic map of $f^{\ast }\top$ induces a comparison $f^{\ast }{\Omega }_{\mathcal{E}}\to {\Omega }_{\mathcal{F}}$ whose adjoint transpose is $\lambda$. Composing with a characteristic map classifies the inverse-image subobject. Thus monicity of $\lambda$ is equivalent to reflection of equality of subobjects under $f^{\ast }$, and hence to faithfulness of the inverse image: apply reflection to equalizers of parallel arrows. Conversely faithfulness reflects when a pulled-back monomorphism is an isomorphism, by comparing its characteristic map with truth; intersections then reflect equality of arbitrary subobjects.

A topos over sets is local when its global sections functor is itself an inverse image functor. For an idempotent-complete small indexing category, its presheaf topos is local exactly when the indexing category has a terminal object: then global sections are evaluation at that object.

Equivalently every open cover of $X$ contains $X$ as a member, so the top element of its open-set lattice is supercompact in the frame sense. Necessity follows from preservation of coproducts and epimorphisms by global sections. Conversely the union of all proper opens misses a point $x$, and the stalk functor there equals global sections. Its right adjoint is a skyscraper sheaf of sets. Empty spaces fail; a T1 space satisfies the condition exactly when it is a singleton.

The constant singleton functor must be a retract in a category of a covariant representable functor. Necessity follows by applying a colimit-preserving global-sections functor to the canonical coproduct of representables onto the terminal functor. Conversely its Hom functor is a retract of evaluation, preserves colimits and has a right adjoint. Such a retract is represented by an initial object in the idempotent completion. For an idempotent-complete category an actual initial object suffices; without this hypothesis an absorbing-zero monoid gives a counterexample to the unqualified claim.

A functor $F:\mathcal{C}\to \mathcal{D}$ induces a geometric morphism from the presheaf topos on $\mathcal{C}$ to that on $\mathcal{D}$. Its inverse image is precomposition with $F^{\mathrm{op}}$, its direct image is Right Kan extension and its extra left adjoint is left Kan extension. The extra left adjoint sends $yC$ to $y(FC)$.

The geometric morphism induced by $T:\mathcal{C}\to \mathcal{D}$ is a geometric embedding exactly when $T$ is full and faithful. Its right adjoint has $(f_{\ast }H)(d)=\mathrm{Nat}(\mathcal{D}(T-,d),H)$. The counit at $c$ is precomposition with $\mathcal{C}(-,c)\to \mathcal{D}(T-,Tc)$. It is invertible for all $H$ precisely when that comparison presheaf is an isomorphism, by the Yoneda lemma, which is precisely full faithfulness of $T$.

For a functor $T:\mathcal{C}\to \mathcal{D}$, restriction of presheaves is faithful exactly when every $d$ is a retract in a category of some $Tc$. Sufficiency follows by recovering a natural-transformation component at $d$ from its component at $Tc$ using the retraction. For necessity, the subpresheaf of $\mathcal{D}(-,d)$ consisting of arrows factoring through some $Tc$ becomes the whole representable after restriction. Faithfulness of a geometric inverse image reflects such a monomorphism being an isomorphism. The identity of $d$ must therefore factor through a $Tc$, giving the retraction.