Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 75 / 5

Precomposition definesIt preserves finite limits because presheaf limits are pointwise. Its right adjoint is a Right Kan extension; explicitly, for $H\in \hat{\mathbb{C}}$,Precomposition in $d$ supplies the restriction maps. To verify the adjunction, a map $P\circ T^{\mathrm{op}}\to H$ sends $x\in P(d)$ to the natural family taking $h:Tc\to d$ to the image in $H(c)$ of $P(h)x$. Conversely evaluate such a family at $d=Tc$, $h=1_{Tc}$; naturality makes the two constructions inverse. Thus $f^{\ast }⊣f_{\ast }$ and $f^{\ast }$ is left exact, giving the required geometric morphism

If $d$ is a retract in a category of $Tc$, write $i:d\to Tc$, $r:Tc\to d$ with $ri=1_d$. For a natural transformation $\alpha :P\to Q$ of categorical presheaves, naturality givesConsequently transformations agreeing at all $Tc$ agree at every retract of such an object. If every $d$ is such a retract, restriction $f^{\ast }$ is faithful, and the geometric morphism is surjective.

Conversely suppose $f^{\ast }$ is faithful. Inside the representable presheaf $yd=\mathbb{D}(-,d)$, define $S_d(e)$ to consist of arrows $e\to d$ that factor through some $Tc$. Precomposition preserves this condition, so $S_d$ is a subpresheaf. At $e=Tc$, every arrow factors through that very object, so $f^{\ast }{S}_d=f^{\ast }yd$. A faithful geometric inverse image reflects an inclusion becoming invertible, as proved in Question 4(iii). Hence $S_d=yd$, and $1_d\in S_d(d)$ factors as $d\to Tc\to d$. This is the required retraction. We have proved the retract criterion for surjective presheaf geometric morphisms:

A geometric embedding means $f_{\ast }$ is full and faithful, equivalently that the adjunction counit $f^{\ast }{f}_{\ast }H\to H$ is invertible for every $H$. At $c$, the explicit right-adjoint formula identifies this counit withinduced by ${\tau }_c:\mathbb{C}(-,c)\to \mathbb{D}(T-,Tc)$, sending $h$ to $Th$.

If $T$ is full and faithful, every ${\tau }_c$ is an isomorphism, making all these counit components bijective and hence making $f_{\ast }$ full and faithful. Conversely, if precomposition with ${\tau }_c$ is a bijection for every $H$, then ${\tau }_c$ is invertible by the Yoneda lemma applied in the presheaf category: taking $H=\mathbb{C}(-,c)$ gives a left inverse, and injectivity for $H=\mathbb{D}(T-,Tc)$ makes it a right inverse too. Evaluating ${\tau }_c$ at $c^{\prime }$ therefore says thatis bijective for all $c,c^{\prime }$. This proves the full-faithfulness criterion for presheaf geometric embeddings: