# Questions tagged [kan-extension]

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25
questions

**7**

votes

**0**answers

166 views

### 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 (...

**6**

votes

**1**answer

218 views

### (Pro-)representable functors and full subcategories in homotopy theory

$\DeclareMathOperator\Ab{Ab}\DeclareMathOperator\Ho{Ho}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Hotc{Hotc}\DeclareMathOperator\Sm{Sm}$Let $\mathcal{C}\overset{\iota}{\longrightarrow} \mathcal{...

**12**

votes

**2**answers

356 views

### How to understand adjoint functors?

I asked this same question on MathUnderflow two weeks ago but didn't receive any answer. Now that I am thinking more, it feels like the most suitable place for this question is here.
I have a good ...

**5**

votes

**0**answers

126 views

### Left Kan extensions of “strong” monoidal functors

Consider the 2-category $\mathsf{MonCat}$ where objects are monoidal categories,
1-cells are strong monoidal functors, and 2-cells are monoidal natural transformations.
Given arrows $f: \mathsf{C} \to ...

**0**

votes

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50 views

### 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 ...

**5**

votes

**1**answer

188 views

### 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 $F$ ...

**7**

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171 views

### 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 ...

**1**

vote

**1**answer

133 views

### 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^{\...

**4**

votes

**0**answers

222 views

### 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 ...

**2**

votes

**0**answers

119 views

### 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 ...

**2**

votes

**1**answer

335 views

### 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 ...

**2**

votes

**0**answers

160 views

### 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 ...

**4**

votes

**0**answers

207 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 ...

**9**

votes

**0**answers

238 views

### 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{...

**3**

votes

**2**answers

518 views

### 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 ...

**4**

votes

**1**answer

137 views

### 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 ...

**10**

votes

**1**answer

294 views

### 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 ...

**4**

votes

**1**answer

492 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 ...

**4**

votes

**1**answer

292 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},\...

**1**

vote

**1**answer

227 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 $F$. ...

**1**

vote

**0**answers

160 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 ...

**2**

votes

**1**answer

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

**12**

votes

**3**answers

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 ...

**8**

votes

**2**answers

753 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 $k$-...

**7**

votes

**2**answers

330 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 ...