Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 119 / 6

For a locally small category $\mathcal{C}$, the Yoneda lemma states thatnaturally in both $A\in \mathcal{C}$ and $F:{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$. Taking $F=\mathcal{C}(-,B)$ givesso the Yoneda embedding is full and faithful.

For small $\mathcal{C}$, the presheaf category has pointwise finite limits and colimits, exponentialand a subobject classifier whose elements at $C$ are sieves on $C$. Hence it is a presheaf topos.

If $\mathcal{C}$ has finite limits, the Yoneda embedding preserves them: maps into a limiting object are the corresponding limits of hom-sets. Its essential image is therefore a full subcategory of the presheaf topos closed under finite limits. Since Yoneda is full and faithful, this proves the assertion up to equivalence.

Suppose the small category $\mathcal{C}$ is Cartesian closed. The Yoneda embedding preserves finite products. For $A,B,X\in \mathcal{C}$, the Yoneda lemma and the exponential object adjunction giveThe bijections are natural in $X$, soThus the essential image of $\mathcal{C}$ in its presheaf topos is full and closed under finite products and exponentials.

Assume $\mathcal{C}$ has finite products and every idempotent morphism splits. For a representable presheaf $yA$, the exponential formula givesThus exponentiation by $yA$ is precomposition with $-\times A$. Precomposition has a Right Kan extension as right adjoint, so every representable presheaf is a tiny object.

Conversely, let $P$ be tiny. Then $(-{)}^P$ is a left adjoint and preserves all small colimits. Since $\mathcal{C}$ has a terminal object $1$, the terminal presheaf is $y1$, and evaluation at $1$ preserves colimits. Thereforepreserves all small colimits as a functor of $F$. By the result supplied in the question, splitting idempotents implies that $P$ is representable. Hence the representable presheaves are the tiny objects of an idempotent-complete finite-product category, and Yoneda identifies $\mathcal{C}$ with the full subcategory of tiny objects of its presheaf topos.