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Stream: learning: reading & references

Topic: ✔ Monadicity of presheaf restriction to right class of OFS

Amar Hadzihasanovic (Jun 14 2025 at 09:31):

Just as a follow-up, I ended up just writing the following coend-y short proof for the explicit description of the monad: the extension of a $C_\mathcal{R}$ presheaf $X$ along $C_\mathcal{R} \to C$ is computed by the coend

$$\int^{c \in C_\mathcal{R}} \mathrm{Hom}_C(-, c) \times X(c)$$

but having an $(\mathcal{L}, \mathcal{R})$ OFS means precisely that

$$\mathrm{Hom}_C(-, c) \simeq \int^{c' \in \mathrm{core}(C)} \mathrm{Hom}_{C_\mathcal{L}}(-, c') \times \mathrm{Hom}_{C_\mathcal{R}}(c', c)$$

so

$$\int^{c \in C_\mathcal{R}} \mathrm{Hom}_C(-, c) \times X(c) \simeq \int^{c \in C_\mathcal{R}, c' \in \mathrm{core}(C)} \mathrm{Hom}_{C_\mathcal{L}}(-, c') \times \mathrm{Hom}_{C_\mathcal{R}}(c', c) \times X(c) \simeq \int^{c \in \mathrm{core}(C)} \mathrm{Hom}_{C_\mathcal{L}}(-, c) \times X(c)$$

Amar Hadzihasanovic (Jun 14 2025 at 09:34):

And the elements of the latter are precisely equivalence classes $[\ell\colon d \to c, x \in X(c)]$ modulo the action of isomorphisms in the middle; when the OFS is strict then there's no quotient

Notification Bot (Jun 14 2025 at 09:34):

Amar Hadzihasanovic has marked this topic as resolved.