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 ...
crystalline cohomology's user avatar
4 votes
1 answer
178 views

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 ...
crystalline cohomology's user avatar
2 votes
0 answers
55 views

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 ...
julio_es_sui_glace's user avatar
5 votes
0 answers
203 views

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 (...
Jonathan Beardsley's user avatar
4 votes
1 answer
136 views

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 ...
Lorenzo Riva's user avatar
1 vote
1 answer
209 views

Pointwise Kan extensions VS weighted limits

$\newcommand{\Dist}{\operatorname{Dist}} \newcommand{\Ran}{\operatorname{Ran}} \newcommand{\Lim}{\operatorname{Lim}}$ TLDR Given a pointwise kan extension, how can we go from $$ \Dist(B, C)(\phi_c \...
nicolas's user avatar
  • 221
2 votes
1 answer
92 views

Weighted limits and Kan extension in Dist

(noting $\otimes$ for composition in distributors, $\phi_f : A \nrightarrow B = B(-,f=)$ and $\phi^f : B \nrightarrow A = B(f-,=)$ the embeddings of a functor $f:A\to B$ in $Dist$, and $Dist(A,B) = [B^...
nicolas's user avatar
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5 votes
1 answer
283 views

Base change isomorphism for left Kan extensions

Let $\mathscr{C}$, $\mathscr{D}$, and $\mathscr{E}$ be (infinity) categories, and assume we're given a Cartesian diagram: $\require{AMScd}$ \begin{CD} \mathscr{C} \times_{\mathscr{E}} \mathscr{D} @>...
Exit path's user avatar
  • 2,969
6 votes
0 answers
184 views

(Co)cartesian fibrations and left Kan extensions

Let $p: \mathscr{C}\to\mathscr{D}$ be a functor of (small) $\infty$-categories. Let $\mathscr{E}$ be a cocomplete $\infty$-category. Assume that $\mathscr{C}, \mathscr{D}, \mathscr{E}$ admit finite ...
Lao-tzu's user avatar
  • 1,856
12 votes
2 answers
328 views

Conjugacy classes as left Kan extension of forgetful functor

Let $\mathbf{Set}$, $\mathbf{Grp}$, and $\mathbf{Grp}^{\rm conj}$ denote the categories of sets and functions, groups and homomorphisms, and groups and homomorphisms up to conjugation, respectively. (...
David Corwin's user avatar
3 votes
1 answer
243 views

Question about the proof of Kerodon tag 030V (Proposition 7.3.7.1)

$\require{AMScd}$ Related to this, I have a question about the proof given in Kerodon of the following result: Proposition 7.3.7.1: Let $C$ be an $\infty$-category, let $\bar{F} : C^\rhd \to D$ be a ...
daniel gratzer's user avatar
3 votes
0 answers
161 views

Do we really need degeneracies for spectral sequence of homotopy simplicial chain complex?

Let's consider an homotopy simplicial chain complex, that is a functor of $\infty$-categories $X_{\bullet} = \textrm{N}(\Delta^{op}) \to \mathcal{C}_{\ge 0}$, where $\mathcal{C}_{\ge 0}$ is the $\...
Andrea Marino's user avatar
6 votes
0 answers
175 views

Examples of nonpointwise Kan extensions that "play a mathematical role"

Most Kan extensions arising in nature are pointwise, and this observation prompts Kelly to write [1]: Our present choice of nomenclature is based on our failure to find a single instance where a [...
varkor's user avatar
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5 votes
1 answer
129 views

Adjoining extensions in bicategories

Given a bicategory $\mathcal K$, is there a universal construction of a bicategory $\mathcal K'$ and faithful locally fully faithful pseudofunctor $\mathcal K \hookrightarrow \mathcal K'$ such that ...
varkor's user avatar
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2 votes
0 answers
198 views

Why are derived functors triangulated?

I am following Verdier's notion of derived functors as Kan extensions along the localization $K(\mathcal{A}) \to D(\mathcal{A})$ of the homotopy category of complexes to the derived category. In the ...
Ben C's user avatar
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Coherence for closed bicategories

A right-closed bicategory [1] is a bicategory that has all right extensions (i.e. right adjoints to precomposition with a fixed 1-cell). A one-object right-closed bicategory is therefore a right-...
varkor's user avatar
  • 8,635
1 vote
0 answers
257 views

Does QCoh commute with colimits?

If $S$ is a scheme, denote by $QCoh(S)$ the $\infty$-derived category of quasi-coherent sheaves on $S$. If $X$ is a prestack, Gaitsgory and Rozenbluym define in their book $QCoh(X)$ as a right Kan ...
J.P. Gimori's user avatar
3 votes
0 answers
134 views

Reference for "taking adjuncts preserves Kan extensions"

I'm using a result similar to the one below, and I would like to know if there is a reference that I can cite. It's easily proved, by "following your nose". The cell $G\phi.\eta_A$ is often ...
Roald Koudenburg's user avatar
4 votes
1 answer
185 views

Do (co)density (co)monadic constructions stablize?

Under good conditions [1], any functor $F: C \to D$ induces a codensity monad $T: D \to D$ as a right Kan extension of $F$ along itself. It does not say explicitly, but by considering left/right Kan ...
Student's user avatar
  • 4,978
7 votes
0 answers
276 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 (...
gregodom's user avatar
  • 319
6 votes
1 answer
299 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{...
curious math guy's user avatar
11 votes
2 answers
919 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 ...
Bumblebee's user avatar
  • 1,019
7 votes
0 answers
365 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 ...
Eigil Fjeldgren Rischel's user avatar
1 vote
0 answers
102 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 ...
3 A's's user avatar
  • 425
6 votes
1 answer
364 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 $F$ ...
geodude's user avatar
  • 2,129
10 votes
2 answers
485 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 ...
Martin Brandenburg's user avatar
1 vote
1 answer
411 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^{\...
richarddedekind's user avatar
4 votes
0 answers
233 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 ...
FKranhold's user avatar
  • 1,623
3 votes
0 answers
239 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 ...
Arrow's user avatar
  • 10.3k
4 votes
1 answer
882 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 ...
Robin Stoll's user avatar
3 votes
0 answers
204 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 ...
Andrea Marino's user avatar
5 votes
0 answers
365 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 ...
Ender Wiggins's user avatar
13 votes
1 answer
489 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{...
Ivan Di Liberti's user avatar
4 votes
2 answers
701 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 ...
Omer Rosler's user avatar
4 votes
1 answer
183 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 ...
Wolttlow's user avatar
12 votes
1 answer
430 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 ...
L.Guetta's user avatar
  • 175
4 votes
1 answer
613 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 ...
John M.'s user avatar
  • 41
4 votes
1 answer
546 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},\...
user24453's user avatar
  • 333
1 vote
1 answer
314 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 $F$. ...
user84563's user avatar
  • 915
1 vote
0 answers
191 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 ...
nicolas's user avatar
  • 221
2 votes
1 answer
420 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 $\widehat{F}=Lan_{y} F:[C^{op},Set]\rightarrow D$ ...
user84563's user avatar
  • 915
14 votes
3 answers
2k 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 ...
fosco's user avatar
  • 13k
12 votes
4 answers
1k 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 $k$-...
Martin Brandenburg's user avatar
7 votes
2 answers
506 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 ...
pnips's user avatar
  • 71