On relations:
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Chapter 8: Relations, Question 8.4.8.1.2, on better characterisations of representably full morphisms in $\boldsymbol {\mathsf{Rel}}$. This question also appears as [Emily, What are the 2-categorical mono/epimorphisms in the 2-category of relations?].
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Chapter 8: Relations, Question 8.4.10.1.2, on better characterisations of corepresentably full morphisms in $\boldsymbol {\mathsf{Rel}}$. This question also appears as [Emily, What are the 2-categorical mono/epimorphisms in the 2-category of relations?].
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Chapter 8: Relations, Question 9.2.1.1.2, seeking a characterisation of which left Kan extensions exist in $\boldsymbol {\mathsf{Rel}}$. This question also appears as [Emily, Existence and characterisations of left Kan extensions and liftings in the bicategory of relations II].
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Chapter 8: Relations, Question 9.2.1.1.3, seeking an explicit descriptions of left Kan extensions along relations of the form $f^{-1}$ (which always exist in $\boldsymbol {\mathsf{Rel}}$). This question also appears as [Emily, Existence and characterisations of left Kan extensions and liftings in the bicategory of relations II].
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Chapter 8: Relations, Question 9.2.2.1.2, seeking a characterisation of which left Kan lifts exist in $\boldsymbol {\mathsf{Rel}}$. This question also appears as [Emily, Existence and characterisations of left Kan extensions and liftings in the bicategory of relations II].
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Chapter 8: Relations, Question 9.2.2.1.3, seeking an explicit descriptions of left Kan lifts along relations of the form $\operatorname {\mathrm{Gr}}\webleft (f\webright )$ (which always exist in $\boldsymbol {\mathsf{Rel}}$). This question also appears as [Emily, Existence and characterisations of left Kan extensions and liftings in the bicategory of relations II].
On categories:
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Chapter 11: Categories, Question 11.6.2.1.3, seeking a better characterisation of necessary and sufficient conditions on $F$ for $F^{*}$ to always be full. This question also appears as [Emily, Looking for a nice characterisation of functors $F$ whose precomposition functor $F^*$ is full].
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Chapter 11: Categories, Question 11.6.4.1.3, seeking a characterisation of necessary and sufficient conditions on $F$ for $F^{*}$ or $F_{*}$ to be conservative. This question also appears as [Emily, Characterisations of functors $F$ such that $F^*$ or $F_*$ is [property], e.g. faithful, conservative, etc].
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Chapter 11: Categories, Question 11.6.5.1.2, seeking a characterisation of necessary and sufficient conditions on $F$ for $F^{*}$ or $F_{*}$ to be essentially injective. This question also appears as [Emily, Characterisations of functors $F$ such that $F^*$ or $F_*$ is [property], e.g. faithful, conservative, etc].
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Chapter 11: Categories, Question 11.6.6.1.2, seeking a characterisation of necessary and sufficient conditions on $F$ for $F^{*}$ or $F_{*}$ to be essentially surjective. This question also appears as [Emily, Characterisations of functors $F$ such that $F^*$ or $F_*$ is [property], e.g. faithful, conservative, etc].
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Chapter 11: Categories, Question 11.7.1.1.3, , seeking a characterisation of necessary and sufficient conditions on $F$ for $F^{*}$ or $F_{*}$ to be dominant. This question also appears as [Emily, Characterisations of functors $F$ such that $F^*$ or $F_*$ is [property], e.g. faithful, conservative, etc].
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Chapter 11: Categories, Question 11.7.2.1.3, seeking a characterisation of necessary and sufficient conditions on $F$ for $F^{*}$ or $F_{*}$ to be monic. This question also appears as [Emily, Characterisations of functors $F$ such that $F^*$ or $F_*$ is [property], e.g. faithful, conservative, etc].
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Chapter 11: Categories, Question 11.7.3.1.3, seeking a characterisation of necessary and sufficient conditions on $F$ for $F^{*}$ or $F_{*}$ to be epic. This question also appears as [Emily, Characterisations of functors $F$ such that $F^*$ or $F_*$ is [property], e.g. faithful, conservative, etc].
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Chapter 11: Categories, Question 11.7.5.1.5, seeking a characterisation of necessary and sufficient conditions on $F$ for $F^{*}$ or $F_{*}$ to be pseudoepic. This question also appears as [Emily, Characterisations of functors $F$ such that $F^*$ or $F_*$ is [property], e.g. faithful, conservative, etc].
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Chapter 11: Categories, Question 11.7.5.1.3, seeking a characterisation of pseudoepic functors. This question also appears as [Liberti, Characterization of pseudo monomorphisms and pseudo epimorphisms in Cat].
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Chapter 11: Categories, Question 11.7.5.1.4, which asks whether a pseudomonic and pseudoepic functor must necessarily be an equivalence of categories. This question also appears as [Emily, Is a pseudomonic and pseudoepic functor necessarily an equivalence of categories?].
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Chapter 11: Categories, Question 11.8.4.1.3, seeking a characterisation of functors representably faithful on cores.
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Chapter 11: Categories, Question 11.8.5.1.3, seeking a characterisation of functors representably full on cores.
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Chapter 11: Categories, Question 11.8.6.1.3, seeking a characterisation of functors representably fully faithful on cores.
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Chapter 11: Categories, Question 11.8.7.1.3, seeking a characterisation of functors corepresentably faithful on cores.
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Chapter 11: Categories, Question 11.8.8.1.3, seeking a characterisation of functors corepresentably full on cores.
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Chapter 11: Categories, Question 11.8.9.1.3, seeking a characterisation of functors corepresentably fully faithful on cores.
