# Tag Info

• 60.3k

• 871

### 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 ...
• 60.3k

### 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 ...
• 6,478
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 ...
• 9,690

### 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 ...
• 26.7k
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}_{\...
• 1,807

• 50.8k

### 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, $F$ ...
• 22.7k
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 ...
• 604
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 \...
• 5,314

Only top scored, non community-wiki answers of a minimum length are eligible