Questions tagged [accessible-categories]

For questions about accessible categories, accessible functors, and their properties. Use in conjunction with the tag (category-theory).

Filter by
Sorted by
Tagged with
4 votes
1 answer
37 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 ...
user avatar
  • 1,479
1 vote
0 answers
58 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 $C, D$ ...
user avatar
1 vote
1 answer
39 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 ...
user avatar
  • 3,578
0 votes
1 answer
47 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 ...
user avatar
  • 3,578
2 votes
1 answer
55 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 ...
user avatar
  • 5,917
0 votes
1 answer
42 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 ${\...
user avatar
0 votes
0 answers
52 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 ...
user avatar
  • 3,578
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(...
user avatar
  • 3,578
2 votes
1 answer
71 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 ...
user avatar
  • 3,578
1 vote
0 answers
34 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 ...
user avatar
  • 1,079
2 votes
1 answer
120 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 $\lambda$ ...
user avatar
  • 3,578
1 vote
1 answer
105 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 ...
user avatar
  • 3,578
9 votes
1 answer
330 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 ...
user avatar
0 votes
1 answer
105 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 ...
user avatar
  • 3,578
2 votes
1 answer
153 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 $\lambda$...
user avatar
2 votes
0 answers
109 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 ...
user avatar
1 vote
1 answer
68 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\...
user avatar
  • 571
5 votes
1 answer
393 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 $\omega_1$-...
user avatar
  • 5,275
1 vote
1 answer
98 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 ...
user avatar
  • 45
4 votes
1 answer
225 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 ...
user avatar
4 votes
1 answer
119 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?
user avatar
  • 6,395
1 vote
3 answers
511 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 ...
user avatar