# 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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### $\lambda$-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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### $\lambda$-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 ...
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 ...
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### 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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### 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 $\omega_1$-...
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 ...
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 ...
### 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?