Questions tagged [accessible-categories]
For questions about accessible categories, accessible functors, and their properties. Use in conjunction with the tag (category-theory).
26
questions
2
votes
1
answer
363
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Understanding filtration argument
I'm trying to read the proof of Theorem 3.1 in Cellular Categories and Stable Independence. In the proof ii) $\implies$ iii), the author uses filtrations to proceed with the argument. I'm having some ...
- 454
4
votes
1
answer
92
views
If the domain has only monos, every conservative functor reflects split epis
I'm reading a paper on accessible categories where the authors remark that when the domain has only monos, any conservative, accessible functor reflects split epimorphisms as well. This seemed to be ...
- 454
2
votes
1
answer
94
views
Class of compact cardinals implies every accessible category is co-wellpowered
As the title says, I'm looking for the reference that the existence of a class of compact cardinals implies every accessible category is co-wellpowered. It has been stated in Adamek-Rosicky (Locally ...
- 454
0
votes
2
answers
63
views
any $\lambda-$pure morphism in a $\lambda$-accessible category is a monomorphism
I'm trying to understand the proof of the proposition 2.29 in the book
Locally presentable and accessible categories which is also given in the snippet below.
I've got stuck in the -2nd paragraph ...
- 3,872
4
votes
1
answer
43
views
Compact "non-discrete" objects in non-accessible categories
An object $C$ of a category $\mathcal{C}$ with filtered colimits is compact if its hom-functor $\mathcal{C}(C, -) : \mathcal{C} \to \mathsf{Set}$ preserves $\alpha$-filtered colimits for some regular ...
- 1,601
1
vote
0
answers
78
views
Can we control the "degree" of accessibility of right adjoints between presentable $\infty$-Categories? (HTT 5.4.7.7)
Suppose $g: D \to C$ is a right adjoint between presentable $\infty$-categories. Then by the adjoint functor theorem, $g$ is accessible, i.e. there is a regular cardinal $\kappa$ such that both $C, D$ ...
- 261
1
vote
1
answer
43
views
directed diagrams in accessible categroies
I have a problem with understanding the choice of $i_0$ and $i_1$ in the snippet below.
I believe that they cannot be completely arbitrary and that $\lambda^+$-directedness of $I$ must be used in some ...
- 3,872
0
votes
1
answer
58
views
A paper: Accessible categories, saturation and categoricity
I've been reading a paper on accessible categories, saturation and categoricity by "Jiří Rosický" for quite some time,
but I still cannot understand one detail:
In the snippet below, in the ...
- 3,872
2
votes
1
answer
67
views
Reconciling statements about accessible categories
I have a problem reconciling the following assertions regularly made online about accesssible categories:
The category $\mathrm{Mod}(T)$ of models of some first-order theory $T$ with countable ...
- 9,897
0
votes
1
answer
49
views
A natural transformation
On the page $9$ here I have a very basic question: in the definition $3.1$ $\alpha$ never appears in the condition
"such that if $(U_1f)a\in U_2 HB$ then $a\in U_2 HA$ for each $f:A\to B$ in ${\...
0
votes
0
answers
53
views
Adjointenes of unknown functors
I would like to understand what are here on the page $266$
the types of items in the adjoint euqation $$\mu^{Lim}_{\cal K}\vdash Lim\ \eta_{\cal K}^{Lim}.$$
They should be functors by the definition ...
- 3,872
1
vote
1
answer
55
views
Galois types, factorization 2
I have asked a related question elsewhere but I still have a problem with this proof of the Proposition $6.2$ on page $13$: in the $-3$rd paragraph in the first snippet why
exactly do we enumerate $U(...
- 3,872
2
votes
1
answer
74
views
Galois types, factorization
I do not follow here on the page $13$ in the
proof of proposition $6.2$
what does it mean that $f$ factors over $f_0$ , why such $f_0$ exists and how it relates to definitions $4.1$ and $6.1$ Also why ...
- 3,872
1
vote
0
answers
39
views
Is the 2-Category of Groupoids Locally Presentable?
I am wondering if the 2-Category of groupoids is Locally Presentable? Locally presentable means the category is accessible and co-complete.
Edit:
It has been pointed out that the category of ...
- 1,229
2
votes
1
answer
140
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 $\lambda$ ...
- 3,872
1
vote
1
answer
112
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 ...
- 3,872
13
votes
1
answer
461
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 ...
- 4,831
0
votes
1
answer
119
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 ...
- 3,872
2
votes
1
answer
184
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 $\lambda$...
- 4,831
2
votes
0
answers
130
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 ...
- 4,831
1
vote
1
answer
72
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\...
- 571
5
votes
1
answer
492
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 $\omega_1$-...
- 5,797
1
vote
1
answer
122
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 ...
- 45
4
votes
1
answer
254
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 ...
- 157
4
votes
1
answer
133
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?
- 6,559
1
vote
3
answers
610
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 ...
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