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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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 ...
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54 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(...
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50 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 ...
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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 ...
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1answer
107 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 $\lambda$ ...
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101 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 ...
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1answer
246 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 ...
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1answer
83 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 ...
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122 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 $\lambda$...
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79 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 ...
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1answer
60 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\...
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303 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 $\omega_1$-...
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79 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 ...
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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 ...
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1answer
104 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?
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3answers
412 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 ...