Questions tagged [accessible-categories]
For questions about accessible categories, accessible functors, and their properties. Use in conjunction with the tag (category-theory).
12
questions
2
votes
1answer
99 views
$\lambda$-accessible categories, unclear proof
In the context of $\lambda$-accessible categories
consider the proof of the proposition $1.22$ here.
How/where did we use the fact that $\cal D$ in the last but one line has less than $\lambda$ ...
1
vote
1answer
95 views
$\lambda$-pure morphisms in $\lambda$-accessible categories are monos, unclear proof
This is Proposition 2.29 from the book Locally Presentable and Accessible Categories by Jiří Adámek and Jiří Rosický.
Above is a proof that $\lambda$-pure morphisms in $\lambda$-accessible categories ...
7
votes
1answer
178 views
Morita theory for algebras for a monad $T$
There are convincing arguments that support the claim that universal algebra is essentially the theory of $\lambda$-accessible monads $T$ over Set.
Now, given two equivalent categories of algebras ...
0
votes
1answer
69 views
Existence of morphisms in a free completion under directed colimits,$\lambda$-accessible category
Let $\cal K$ be a $\lambda$-accessible category with directed colimits and $\cal C$ be its representative full subcategory consiting of $\lambda$-presentable objects. Let $\cal L$ be free completion ...
2
votes
1answer
104 views
Free cocompletion and preservation of colimits
Let $\mathcal{K}$ be a $\lambda$-accessible category and $\text{Pres}_{\lambda}(\mathcal{K})$ be the small category of its $\lambda$-presentables.
It is well known that $\mathcal{K}$ is the $\...
1
vote
0answers
60 views
Subcategory of accessible category
For a full reflective subcategory $L$ of a locally presentable category $K$ one can state that $L$ is locally presentable.
Sure i do believe that a full reflective subcategory of an accessible one is ...
1
vote
1answer
54 views
How to show the $\kappa$-small functor is $\kappa$-accessible? (coalgebraic logic)
A $\mathtt {Set}$-functor $T:\mathtt {Set} \to \mathtt {Set}$ is defined to be $\kappa$-accessible for a regular cardinal $\kappa$ iff for all sets $X$ and all $x\in TX$ there exists a subset $Y\...
5
votes
1answer
239 views
Is the category of Banach spaces and bounded linear maps accessible?
It's well-known that the category $\mathsf{Ban}_c$ of Banach spaces and linear contractions (i.e. of norm $\leq 1$) is $\omega_1$-accessible. It is also (co)complete, and hence locally $\omega_1$-...
1
vote
1answer
71 views
Category of accessible functors and its closedness
Is the category of $\sf{Set}$ accessible endofunctors right closed w.r.t. composition (as a monoidal structure)? Any hint on how to prove this?
I think that this is true if one works with finitary ...
4
votes
1answer
137 views
Question about accessibility of category of free abelian groups.
I've read, that the accessibility of the category of all free abelian groups is independent on basic set theory (say ZFC). What is the reason for that? And how can I interpret this result? Does it ...
4
votes
1answer
94 views
Why is it crucial that $\kappa$ is a regular cardinal in the definition of $\kappa$-accessible categories?
In the definition of a $\kappa$-accessible (or presentable) category, the cardinal $\kappa$ is always supposed to be regular.
What happens in the irregular case?
1
vote
3answers
364 views
Category of profinite groups
My question is simple:
Is the category of profinite groups an accessible category?
Thank you
Edit: I will add the (hopefully simpler) question:
Is the category of profinite groups complete and ...
