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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 ...
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      0answers
      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 ...
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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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      Extending monads along dense functors

      Let $j: \mathsf A \to \mathsf B$ be a fully faithful and dense functor where $\mathsf A$ is a small category and $\mathsf B$ is cocomplete. Let $(T, \eta, \mu)$ be a monad over $\mathsf A$. $\require{...
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      When Kan extensions don't exist

      Kan extension are an incredibly useful concept when they exist. My question is: can we still derive information about a functor when Kan extensions don't exist, and if so, what information and in what ...
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      Kan extension of conservative functors

      Suppose the right Kan extension $\text{Ran}_F G$ of a conservative functor $F$ along a conservative functor $G$ exists (with the category $\text{dom} F=\text{dom} G$ not necessarily small). Is it ...
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      1answer
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      About pointwise Kan extension

      Suppose that you want to look at the left Kan extension of a functor $F : \mathcal{C} \to \mathcal{A}$ along a functor $K : \mathcal{C} \to \mathcal{B}$. It is widely known that if the colimit of the ...
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      1answer
      460 views

      Are left and right Kan extensions ever isomorphic?

      So I wonder if it is possible that a left Kan extension of a fully faithful functor $F$ along some other fully faithful functor $G$ (over a small category) agrees with (is isomorphic to) the right Kan ...
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      1answer
      251 views

      Is the singular simplicial functor full

      A functor $F:\mathcal{C}\to \mathcal{D}$, from an essentially small category to a cocomplete category induces a realisation-nerve adjunction between the categories $\mathbf{Fun}(\mathcal{C}^{op},\...
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      1answer
      196 views

      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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      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 ...
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      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 $\widehat{F}=Lan_{y} F:[C^{op},Set]\rightarrow D$ ...
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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 ...
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      2answers
      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 $k$-...
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      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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