17 votes

Kan extensions in the $2$-category of monoidal categories

I believe that this is a particular case of Lurie's "operadic left Kan extension". We may identify a monoidal $\infty$-category $\mathcal{C}$ with a coCartesian fibrations of $\infty$-operads $\...
10 votes

Kan extensions in the $2$-category of monoidal categories

This does not answer the specific questions you ask but is a paper which is focused precisely on computing monoidal Kan extensions: Paul-André Melliès and Nicolas Tabareau. Free models of T-algebraic ...
7 votes
Accepted

How to understand adjoint functors?

Nice question Bumblebee. So, let us start with some "metaphysics of adjointness": THE LEFT AND RIGHT ADJOINTS TO A FUNCTOR $ \mathcal{F}:\mathcal{C}\hookrightarrow\mathcal{D}$ ARE THE FREE (...
6 votes
Accepted

The Kan construction, profunctors, and Kan extensions

I will try to answer the second question. Prop 1. Let ${\bf C} \xleftarrow{i} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $i$ is dense and fully faithful. Moreover $\text{lan}_if$ is ...
6 votes
Accepted

When Kan extensions don't exist

Yes, directions like this have been explored, for all kinds of objects with universal properties (which includes Kan extensions, since as MacLane famously wrote "all concepts are Kan extensions"). ...
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6 votes

Is the singular simplicial functor full

Any non-discrete totally disconnected space gives a counterexample, e.g. the Cantor space $2^{\mathbb{N}}$. If a space $X$ is totally disconnected, then every simplex in it is constant (since the ...
6 votes
Accepted

Is there such a thing as a weighted Kan extension?

Yes. Given $F:C\to D$ and a profunctor $H:E$ ⇸ $C$, i.e. a functor $H : C^{\rm op}\times E\to \rm Set$ (or to the enriching category $V$), the $H$-weighted colimit of $F$ is the functor $L : E \...
  • 60.3k
6 votes

Existence of pointwise Kan extensions in $\infty$-categories

Lurie's approach First let me explain why this is already in HTT 4.3.3. Recall that a (pointwise) left Kan extension of $F: A \to \mathcal{C}$ along an inclusion $i: A \to B$ is a functor $F: B \to \...
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5 votes
Accepted

Kan extensions inside a monoidal category

It is certainly the case that the duals of internal homs have appeared significantly less in the categorical literature. I've included a few more references below, but I am not sure this is a ...
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5 votes
Accepted

Yoneda extension preserving finite products?

(Update: the following answer applies only in the case that $C$ has finite products.) The left Kan extension $\hat{F}$ preserves finite products just when $F$ does. One direction is easy since $F \...
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4 votes

Kan extensions in concrete 2-categories

One situation in which a Kan extension can be "fixed up" is if the category of structure-preserving maps between two structured categories is a reflective or coreflective (full) subcategory of the ...
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4 votes

Kan extensions in the $2$-category of monoidal categories

Here are some miscellaneous remarks/thoughts on these notions. Motivation for Monoidal Co/Limits and Variant Notions. (You've said part of this already, but let me start here for context) If we want ...
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4 votes
Accepted

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

I think you are right that the cube is not a levelwise equivalence - I saw you already put a comment on the Kerodon page so you should get an answer soon. Here's how I would fix the proof: call your ...
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4 votes

How to understand adjoint functors?

If you start with a category and only consider what you can see by looking at functors from groupoids, well you’ll only see the invertible morphisms. So the right adjoint is the core. If you start ...
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4 votes
Accepted

Left and right Kan extensions

That is indeed the right answer: the left adjoint is determined by what it does on (the Yoneda image of) $\mathcal{C}$, since it is a colimit-preserving functor, and there we have $$\mathrm{Hom}_{\...
3 votes

When Kan extensions don't exist

There are several points worth making here. (i) One solution is to use Benabou's theory of 'distributeurs' also called profunctors. A functor F from C to D defines two profunctors basically $D(F-,-):...
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3 votes
Accepted

Faithfulness of Right adjoint to Kan extension

Assuming $C$ is a small category, the right adjoint to $Lan_y(F): PShv(C) = Set^{C^{op}} \to D$ is just the functor $D \to Set^{C^{op}}: d \mapsto \hom_D(F-, d)$; this basic "generalized nerve" ...
  • 50.8k
3 votes

Kan extensions in the $2$-category of monoidal categories

I am not an expert, but I think the best reference for this topic is Algebraic Kan extensions along morphisms of internal algebra classifiers by Mark Weber.
2 votes

Kan extensions in concrete 2-categories

These are the sorts of results that should be in Verity's thesis. The general idea is: suppose we have two "category theories" with some form of 2-functor between them; then what categorical ...
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2 votes

Kan extension of conservative functors

No, take $A$ to be any cocomplete category. For any category $I$ the right Kan extension of $Id:A\to A$ along the embedding $\iota:A\to Fun(I,A)$(object $a$ goes to the constant functor $\underline{a}$...
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2 votes
Accepted

About pointwise Kan extension

The answer is no (I think -- non-pointwise Kan extensions are a pain and I may have messed something up!). I wouldn't lose too much sleep over this, though -- in practice, you never know that some ...
  • 51.1k
2 votes

Are left and right Kan extensions ever isomorphic?

Sure. Consider the left and right Kan extension along the terminal object $t: 1 \to \text{Set}$, applied to a functor $X: 1 \to \text{Sup}$ in the category of sup-lattices. The left Kan extension $\...
  • 50.8k
2 votes

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

Not sure about state of the art, but here are a couple of comments. As is pointed out in your reference, if $F$ has a left adjoint $G$, then $T=FG$. If $F$ happens to be fully faithful, then $G$ is a ...
1 vote
Accepted

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

This is a partial answer. Broadly speaking, representability theorems break down into two types. In both cases, the functor $F$ has to satisfy some exactness condition. For Freyd type theorems, $F$ ...
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1 vote

Kan extensions inside a monoidal category

The common phrase is coexponential object. The internal logic of coclosed categories is cointuitionistic logic. You can broadly think of cointuitionistic logic in terms of pattern matching, ...
1 vote

The Kan construction, profunctors, and Kan extensions

What follows is kind of preliminary and complementary to Ivan's answer. Let $A$ be a small category, $\mathcal B$ and $\mathcal C$ be locally presentable categories and $i\colon A \to \mathcal B$ and ...
1 vote

The Kan construction, profunctors, and Kan extensions

Edit. The construction of $\eta$ below is not correct. See the comments. Ad 2. I think that indeed $\mathrm{Lan}_G(F):\mathbf{C} \to \mathbf{B}$ is left adjoint to $\mathrm{Lan}_F(G):\mathbf{B} \to \...
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