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# Questions tagged [kan-extension]

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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 ...
137 views

### Homotopy Colimit of ?ech 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 ...
167 views

### 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 $F$. ...
84 views

### 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 arrow from ...
208 views

### 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 $\widehat{F}=Lan_{y} F:[C^{op},Set]\rightarrow D$ ...
1k views

### 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 ...
651 views

### 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 $k$-...
295 views

### Kan extensions in concrete 2-categories

Kan extensions make sense in any 2-category. I am interested in Kan extensions in "concrete" 2-categories consisting of actual categories with some sort of structure (e.g., finite products, finite ...

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