# Questions tagged [kan-extension]

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### Kan extensions and restriction

Suppose $A \to B$ be a fully faithful (codense if it helps, i.e. every object in $B$ is the small limit of a diagram in A) embedding of categories. Let $b$ be an object of $B$, $Hom_B(b,-)$ defines (...
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### When do objects in the image of a functor $G$ have a unique action as algebras over the codensity monad of $G$?

Let $G:\mathcal{B}\longrightarrow \mathcal{A}$ be some functor which admits a right Kan extension along itself, $(\operatorname{Ran}_G G, \eta:\operatorname{Ran}_G G \circ G \rightarrow G)$. The ...
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### Is there such a thing as a weighted Kan extension?

The title pretty much sums it up. More in detail. Let $C$, $D$ and $E$ be categories, let $F:C\to D$ and $G:C\to E$ be functors, and let $P:C^{op}\to \mathrm{Set}$ be a presheaf. The colimit of $F$ ...
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### Kan extensions inside a monoidal category

Every monoidal category $(\mathcal{C},\otimes)$ can be seen as a one-object bicategory: the morphisms are the objects of $\mathcal{C}$, and the $2$-morphisms are the morphisms of $\mathcal{C}$. In ...
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### Left and right Kan extensions

Let $F:\mathcal{C}\to\mathcal{D}$ be a functor between small categories. We define the functor \begin{align*} f:\hat{\mathcal{D}}&\longrightarrow\hat{\mathcal{C}} \\ G&\longmapsto G\circ F^{\...
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### Composition of prolongations of $\Gamma$-spaces

Let $S,T:\Gamma^{\text{op}}\to \mathsf{Top}_*$ be two $\Gamma$-spaces ($\Gamma^{\text{op}}$ being the category of finite based sets $r_+=\{*,1,\dotsc,r\}$ with based maps as morphisms. The “op” has ...
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### Pushforward of covering maps

Let $A,B$ be sufficiently connected spaces so as to have the category of coverings equivalent to the category of representations of the fundamental groupoid. Given a covering map of $A$ and a ...
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### Existence of pointwise Kan extensions in $\infty$-categories

This answer by Emily Riehl mentions a to-be-published proof of the fact that $\infty$-categorical (pointwise) Kan extensions exist when the target category admits (co)limits indexed by the relevant ...
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### Homotopy Colimit of Čech Complex

I am studying homotopical cosheaves, and I came up with the following "conjecture". We can see an "additive" precosheaf in chain complexes (such that corestrictions do not commute on the nose) as a ...
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### When do Kan extensions preserve colimits?

Assume that we have a pair of functors $Y:A \to B$ and $F:A \to C$ where $A$ is an essentially small category, $B,C$ are cocomplete categories and $Y,F$ preserve colimits. Assume also that for some ...
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### Faithfulness of Right adjoint to Kan extension

Let $C$ be a category, $D$ be a Grothendieck topos, and suppose we have a fully faithful, left-exact functor $F:C\rightarrow D$. Let $Lan_{y}F:PShv(C)\rightarrow D$ be the Yoneda extension of $F$. ...
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### Limit as a pushout

In Categories for Working Mathematician, Mac Lane describe a cartesian product as a limit for a functor $F$ from a discrete category $|J|$ : Any cone from an object $Z$ to $F$, is a collection of ...
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### Yoneda extension preserving finite products?

Let $C$ be a category and let $F:C\rightarrow D$ be a functor with $D$ locally presentable and cartesian closed. When does the Yoneda extension $\widehat{F}=Lan_{y} F:[C^{op},Set]\rightarrow D$ ...
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### The Kan construction, profunctors, and Kan extensions

It's been a long time since I tried to understand the deep meaning of the "Kan construction", or "nerve-realization" adjunction $$\text{Lan}_y F \dashv N_F = \hom(F,1)$$ that exists among the left ...
### 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 $k$-...