All Questions
Tagged with limits-and-colimits adjoint-functors
11 questions
6
votes
1
answer
281
views
Are adjoints closed under pushouts?
The category $PrL$ of locally presentable categories has all colimits. In particular, if
$A_1 \leftarrow A_0 \rightarrow A_2$ is a diagram of presentable categories, with left adjoint functors between ...
- 521
4
votes
2
answers
314
views
Directed colimit of fully faithful functors
Suppose that for every $n\in\mathbb{N}$ we have a category $\mathcal{C}_n$ and a fully faithful functor $F_n:\mathcal{C}_n\hookrightarrow \mathcal{C}_{n+1}$. My question is whether fully faithful ...
- 119
3
votes
2
answers
987
views
Which functors preserve the number of connected components?
The categories $\mathbf{Top}$ of topological spaces, $\mathbf{sSet}$ of simplicial sets and $\mathbf{Cat}$ of small categories are all equipped with a functor $\pi_0$ into the category $\mathbf{Set}$ ...
11
votes
0
answers
442
views
A right adjoint preserves Phi-colimits if and only if the left adjoint does what?
Let $\Phi$ be a class of categories (e.g. filtered categories), and consider an adjunction $L : \mathbf C \rightleftarrows \mathbf D : R$. A $\Phi$-colimit is a colimit whose diagram is in $\Phi$. We ...
- 10.6k
2
votes
1
answer
706
views
Formula for the left adjoint of the nerve functor?
I recently stumbled upon a formula for the left adjoint of the nerve functor. Let $X$ and $Y$ be simplicial sets, then:
\begin{equation}
\mathbf{sSet}(X,Y)
\cong\mathbf{sSet}(\varinjlim_{\Delta^n\...
9
votes
1
answer
351
views
Algebraically-free monadicity theorem
The monadicity theorem characterises when a functor $u : \mathbf B \to \mathbf E$ is the forgetful functor from the category of algebras for some monad on $\mathbf E$ (up to an equivalence over $\...
- 10.6k
11
votes
2
answers
1k
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 ...
- 1,093
5
votes
1
answer
152
views
Finite well-completeness and the small object argument?
I'm reading a few papers on reflective factorization systems and I've just noticed they're all mentioning a procedure which seems very similar to the small object argument.
First of all, some ...
- 10.5k
3
votes
1
answer
124
views
What do you get when you apply a universal cocone to a colimit functor
Any colimit can be represented as a functor $F$ left adjoint to a particular diagonal functor $\Delta: C \rightarrow C^J$. The unit of this adjunction is the natural transformation $\eta_K: K \...
- 31
7
votes
0
answers
639
views
Constructing pointwise Kan extensions as adjoints to some functor
Background
I'm working on formalizing some category theory in Coq at https://bitbucket.org/JasonGross/catdb. Currently, I'm in the process of formalizing pointwise Kan extensions. Partly because I'...
- 695
12
votes
1
answer
2k
views
Adjoint Functors as Initial Objects of Some Category
Just as universal arrows can be characterized as initial objects of some appropriate comma category, and (co)limits can be characterized as (initial) terminal objects of the appropriate (co)cone ...
- 433