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D.-C. Cisinski
D.-C. Cisinski
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I give different arguments for this kind of things here. The In short, the main case is the one of the terminal presheaf: one proves by cofinality arguments that the colimit of the Yoneda embedding $S\to\mathcal P(S)$ is isomorphic to the colimit of the identity of $\mathcal P(S)$; since $\mathcal P(S)$ has a terminal object, this colimit has to be the value of the identity at the terminal object, hence the terminal object itself. For a general presheaf $F$, one replaces $S$ by the category of elements $S/F$ and uses the fact that left Kan extending along the projection $S/F\to S$ sends the termnal object of $\mathcal P(S/F)$ to $F$ (and preserves colimits, being a left adjoint).

Here are more precise references. The main observation is that, given any functor $u:A\to B$, the map $S(A)\to (A^{op}\times B)\times_{B^{op}\times B}S(B)$ is limit-cofinal, where $S(A)=Tw(A)^{op}$ is the opposite of the twisted arrow category; this is Prop. 5.6.9 in loc. cit. In the case where $u$ is the Yoneda embedding, this leads to a proof that the left Kan extension along the Yoneda embedding is the identity of the category of presheaves on a small category (this is essentially Lemma 5.8.11 and the proof of Prop. 6.2.13). This leads to a proof of the Yoneda lemma itself (Theorem 5.8.13). This also implies rather directly that the colimit of the Yoneda embedding is the terminal object (Prop. 6.2.13), from which the property you ask for follows right away: any presheaf is a colimit of representable presheaves (Cor. 6.2.16). The fact that presheaves categories are dualizable objects among cocomplete categories is also a direct consequence of the kind of cofinality provided by Prop. 5.6.9 (Theorem 6.7.2).

I give different arguments for this kind of things here. The main observation is that, given any functor $u:A\to B$, the map $S(A)\to (A^{op}\times B)\times_{B^{op}\times B}S(B)$ is limit-cofinal, where $S(A)=Tw(A)^{op}$ is the opposite of the twisted arrow category; this is Prop. 5.6.9 in loc. cit. In the case where $u$ is the Yoneda embedding, this leads to a proof that the left Kan extension along the Yoneda embedding is the identity of the category of presheaves on a small category (this is essentially Lemma 5.8.11 and the proof of Prop. 6.2.13). This leads to a proof of the Yoneda lemma itself (Theorem 5.8.13). This also implies rather directly that the colimit of the Yoneda embedding is the terminal object (Prop. 6.2.13), from which the property you ask for follows right away: any presheaf is a colimit of representable presheaves (Cor. 6.2.16). The fact that presheaves categories are dualizable objects among cocomplete categories is also a direct consequence of the kind of cofinality provided by Prop. 5.6.9 (Theorem 6.7.2).

I give different arguments for this kind of things here. In short, the main case is the one of the terminal presheaf: one proves by cofinality arguments that the colimit of the Yoneda embedding $S\to\mathcal P(S)$ is isomorphic to the colimit of the identity of $\mathcal P(S)$; since $\mathcal P(S)$ has a terminal object, this colimit has to be the value of the identity at the terminal object, hence the terminal object itself. For a general presheaf $F$, one replaces $S$ by the category of elements $S/F$ and uses the fact that left Kan extending along the projection $S/F\to S$ sends the termnal object of $\mathcal P(S/F)$ to $F$ (and preserves colimits, being a left adjoint).

Here are more precise references. The main observation is that, given any functor $u:A\to B$, the map $S(A)\to (A^{op}\times B)\times_{B^{op}\times B}S(B)$ is limit-cofinal, where $S(A)=Tw(A)^{op}$ is the opposite of the twisted arrow category; this is Prop. 5.6.9 in loc. cit. In the case where $u$ is the Yoneda embedding, this leads to a proof that the left Kan extension along the Yoneda embedding is the identity of the category of presheaves on a small category (this is essentially Lemma 5.8.11 and the proof of Prop. 6.2.13). This leads to a proof of the Yoneda lemma itself (Theorem 5.8.13). This also implies rather directly that the colimit of the Yoneda embedding is the terminal object (Prop. 6.2.13), from which the property you ask for follows right away: any presheaf is a colimit of representable presheaves (Cor. 6.2.16). The fact that presheaves categories are dualizable objects among cocomplete categories is also a direct consequence of the kind of cofinality provided by Prop. 5.6.9 (Theorem 6.7.2).

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D.-C. Cisinski
D.-C. Cisinski
  • 13.6k
  • 58
  • 81

I give different arguments for this kind of things here. The main observation is that, given any functor $u:A\to B$, the map $S(A)\to (A^{op}\times B)\times_{B^{op}\times B}S(B)$ is limit-cofinal, where $S(A)=Tw(A)^{op}$ is the opposite of the twisted arrow category; this is Prop. 5.6.9 in loc. cit. In the case where $u$ is the Yoneda embedding, this leads to a proof that the left Kan extension along the Yoneda embedding is the identity of the category of presheaves on a small category (this is essentially Lemma 5.8.11 and the proof of Prop. 6.2.13). This leads to a proof of the Yoneda lemma itself (Theorem 5.8.13). This also implies rather directly that the colimit of the Yoneda embedding is the terminal object (Prop. 6.2.13), from which the property you ask for follows right away: any presheaf is a colimit of representable presheaves (Cor. 6.2.16). The fact that presheaves categories are dualizable objects among cocomplete categories is also a direct consequence of the kind of cofinality provided by Prop. 5.6.9 (Theorem 6.7.2).