Questions tagged [accessible-categories]

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12 votes
4 answers
1k views

What was Burroni's sketch for topological spaces?

In a 1981 talk, René Guitart cites Albert Burroni as having given "A first interesting example of a mixed sketch...for the category of topological spaces" in 1970. This was apparently done in Burroni'...
Kevin Arlin's user avatar
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32 votes
6 answers
5k views

Reasons to believe Vopenka's principle/huge cardinals are consistent

There are a number of informal heuristic arguments for the consistency of ZFC, enough that I am happy enough to believe that ZFC is consistent. This is true for even some of the more tame large ...
Harry Gindi's user avatar
  • 19.4k
17 votes
2 answers
529 views

Raising the index of accessibility

In the standard reference books Locally presentable and accessible categories (Adamek-Rosicky, Theorem 2.11) and Accessible categories (Makkai-Pare, $\S$2.3), it is shown that for regular cardinals $\...
Mike Shulman's user avatar
15 votes
1 answer
564 views

presentability rank of categories of coalgebras

The following theorem is relatively classical: Theorem: Given an accessible endofunctor, (co)pointed endofunctor or (co)monad $T$ on a locally presentable category $C$, then the category of $T$-(co)...
Simon Henry's user avatar
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11 votes
3 answers
804 views

Relation between Ind-completion and "additive"-ind-completion

Suppose that $\mathcal{C}$ is a skeletally small additive category. To enlarge $\mathcal{C}$ and produce a bigger category whose "small" objects can be identified with those in $\mathcal{C}$,...
3 A's's user avatar
  • 425
11 votes
2 answers
1k views

What are the reflective subcategories of the category of presentable categories?

I am actually interested in the $\infty$-categorical case, but the same question is meaningful in the $1$-categorical situation as well. A nice property of presentable $\infty$-categories is that if ...
KotelKanim's user avatar
  • 2,270
11 votes
1 answer
433 views

Does the homotopy category of spaces admit a weak generating set?

As a follow-up to this question, let $\mathcal C$ be a category and $\mathcal S \subseteq \mathcal C$ a class of objects. Say that $\mathcal S$ is weakly generating if the functors $Hom_{\mathcal C}(S,...
Tim Campion's user avatar
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10 votes
2 answers
373 views

When is the homotopy category of an accessible $\infty$-category accessible?

Let $\mathcal C$ be an accessible $\infty$-category, and let $ho(\mathcal C)$ be its homotopy category. I can think of two "trivial" reasons for $ho(\mathcal C)$ to be accessible: $ho(\mathcal C) = \...
Tim Campion's user avatar
  • 60.6k
8 votes
1 answer
300 views

On the cardinal arithmetic of accessible categories

If $\lambda, \mu$ are regular cardinals, say that $\lambda \trianglelefteq \mu$ if $\lambda \leq \mu$ and $$\forall X, \, |X| < \mu \implies \mathrm{cf} (P_\lambda(X)) < \mu$$ Here $P_\lambda(X)...
Tim Campion's user avatar
  • 60.6k
7 votes
1 answer
299 views

Stability of accessible $\infty$-categories under some operations

I'm reading through Higher Topos Theory, and I can't make sense of a few proofs in the sections about accessible $\infty$-categories. In Proposition 5.4.4.3, Lemma 5.4.4.2 is used, but I don't see ...
Giulio Lo Monaco's user avatar
6 votes
0 answers
256 views

Characterizing the left / right classes of (weak) factorization systems in locally presentable categories

Let $\mathcal M \subseteq Mor(\mathcal C)$ be a class of morphisms in a locally presentable category. It's well-known that $\mathcal M$ is the left half of an accessible orthogonal factorization ...
Tim Campion's user avatar
  • 60.6k
1 vote
1 answer
347 views

How do you prove that the category of weak equivalences of sSet is accessible?

I am trying to prove that the category of simplicial sets is a combinatorial model category by using Proposition A.2.6.15 of Lurie's book, and this requires proving that the weak equivalences are ...
Sebastian H. Martensen's user avatar