# Questions tagged [kan-extension]

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### Existence and characterisations of left Kan extensions and liftings in the bicategory of relations II

This is the second part to a previous question regarding left Kan extensions/lifts in the bicategory $\mathsf{Rel}$ of sets, relations, and inclusions of relations, which has now been split into two ...
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### Existence and characterisations of left Kan extensions and liftings in the bicategory of relations I

The bicategory $\mathsf{Rel}$ of sets, relations, and inclusions of relations has right Kan extensions and right Kan lifts¹, however I believe it does not have all left Kan extensions/lifts. Is it ...
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### Example of a factorisation of functors $F = HK$ for which the Kan extension of $F$ along $K$ is not $H$

I was reading Emily Riehl's book: Categorical Homotopy theory, and I encountered exercise 1.1.3: Exercise 1.1.3: Construct a toy example to illustrate that if $F$ factors through $K$ along some ...
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### Lax monoidal structure on the right Kan extension of a partially monoidal Γ-set

First some preliminaries. Let me write $Fin_\ast$ for the skeleton of the category of finite pointed sets and pointed maps between them on the objects $n_+=\{0,1,...,n\}$, where $0$ is the base point (...
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### Relationship between Kan extensions and internal hom

Let $\mathcal{C}$ be a (sufficiently complete and cocomplete) closed monoidal category with internal hom $[-,-]$. Let $F : \mathcal{A} \to \mathcal{C}$ be a functor obtained as the left Kan extension ...
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### Kan extensions in the $2$-category of monoidal categories

Kan extensions make sense in any $2$-category. But so far I have only really seen them in the case of the $2$-category of categories, functors, natural transformations and the $2$-category of $k$-...
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