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Questions tagged [accessible-categories]

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

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Confusion about colimits in the category $\mathbf{Set}$

It is well known that $\mathbf{Set}$ is an $\aleph_0$-accessible category, but I'm very inexperienced and I'm not sure how to prove it in detail. In particular, I need to find a set $\Omega$ of ...
Petersu's user avatar
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2 votes
1 answer
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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 ...
interregno's user avatar
4 votes
1 answer
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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 ...
interregno's user avatar
2 votes
1 answer
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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 ...
interregno's user avatar
0 votes
2 answers
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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 ...
user122424's user avatar
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4 votes
1 answer
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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 ...
User7819's user avatar
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1 vote
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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 $C, D$ ...
TheHumanHighway's user avatar
1 vote
1 answer
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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 ...
user122424's user avatar
  • 3,978
0 votes
1 answer
67 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 ...
user122424's user avatar
  • 3,978
2 votes
1 answer
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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 ...
Z. A. K.'s user avatar
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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 ${\...
user113823's user avatar
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0 answers
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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 ...
user122424's user avatar
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1 vote
1 answer
57 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(...
user122424's user avatar
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2 votes
1 answer
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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 ...
user122424's user avatar
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1 vote
0 answers
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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 ...
Ben Sprott's user avatar
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2 votes
1 answer
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$\lambda$-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 $\lambda$ ...
user122424's user avatar
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1 vote
1 answer
113 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 ...
user122424's user avatar
  • 3,978
13 votes
1 answer
496 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 ...
Ivan Di Liberti's user avatar
0 votes
1 answer
128 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 ...
user122424's user avatar
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2 votes
1 answer
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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 $\lambda$...
Ivan Di Liberti's user avatar
2 votes
0 answers
142 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 ...
Ivan Di Liberti's user avatar
1 vote
1 answer
75 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\...
Ak9's user avatar
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6 votes
1 answer
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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 $\omega_1$-...
tcamps's user avatar
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1 vote
1 answer
126 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 ...
carre's user avatar
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6 votes
1 answer
264 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 ...
bbxlmnistvii's user avatar
4 votes
1 answer
138 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?
Damien L's user avatar
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1 vote
3 answers
665 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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