# Questions tagged [accessible-categories]

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41
questions

**9**

votes

**3**answers

411 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 $\mathcal{C}$,...

**5**

votes

**1**answer

166 views

### Why are compactly generated $\infty$-categories closed under limits in $\operatorname{Cat}_{\infty}$?

In Proposition 5.5.7.6 of Lurie's Higher Topos Theory, Lurie states that the $\infty$-category $\operatorname{Pr}^R_{\omega}$ of compactly generated $\infty$-categories and filtered-colimit-preserving ...

**3**

votes

**0**answers

57 views

### Coslices of $\mathbb D$-presentable categories

Let $\mathbb D$ be a sound limit doctrine. When $\mathbb D$ is the doctrine of $\lambda$-small limits, then for any $\mathbb D$-presentable category $\mathscr C$ and $X \in \mathscr C$, the coslice $X/...

**4**

votes

**1**answer

269 views

### Does $\mathsf{Ind}_\lambda^\mu(\mathsf{Ind}_\kappa^\lambda(\mathcal C)) = \mathsf{Ind}_\kappa^\mu(\mathcal C)$?

$\newcommand\Ind{\mathsf{Ind}}\newcommand\Ord{\mathsf{Ord}}\newcommand\Psh{\mathsf{Psh}}$For $\kappa \leq \lambda \leq \Ord$ regular cardinals and $\mathcal C$ an essentially small category, let $\...

**13**

votes

**2**answers

363 views

### In a locally presentable category, is every object (a retract of) the colimit of a chain of smaller objects?

Let $\mathcal C$ be an accessible category. For $C \in \mathcal C$, define the presentability rank $rk(C)$ of $C$ to be the minimal regular $\kappa$ such that $C$ is $\kappa$-presentable. Following ...

**6**

votes

**1**answer

165 views

### If $\mathcal C$ is a $\kappa$-accessible $\infty$-category, then is $Mor \mathcal C$ $\kappa$-accessible?

If $\mathcal C$ is a $\kappa$-accessible 1-category, then the category of morphisms $Mor \mathcal C$ is a $\kappa$-accessible 1-category, with the $\kappa$-presentable objects being those morphisms ...

**4**

votes

**1**answer

171 views

### Almost combinatorial accessible model categories

Theorem: Assume VP. Let $\mathcal{M}$ be an accessible model category
such that there exists a set of generating cofibrations $I$ and such
that all objects are fibrant. Then it is combinatorial.
...

**2**

votes

**0**answers

76 views

### Accessible 2-category of presheaves

Let $A$ be a locally small category and let $\mathbf{Cat}$ be the 2-category of small categories, functors and natural transformations and let $Ps(A)$ be the 2-category of presheaves (the objects are ...

**1**

vote

**1**answer

237 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 ...

**5**

votes

**0**answers

198 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 ...

**9**

votes

**2**answers

285 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) = \...

**15**

votes

**0**answers

220 views

### Is every limit-closed, accessibly-embedded full subcategory of a presentable $\infty$-category reflective?

Let $C$ be a presentable $\infty$-category and let $D\subseteq C$ be a full subcategory closed under limits and sufficiently-filtered colimits. If $D$ is known to be accessible, then by the adjoint ...

**11**

votes

**1**answer

371 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,...

**14**

votes

**1**answer

352 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)...

**8**

votes

**2**answers

374 views

### Can the dual of a finitely-accessible category be accessible?

What is an example of an accessible category $\mathcal C$ which is not essentially small, such that $\mathcal C^{op}$ is finitely-accessible?
More generally, what is an example of an accessible ...

**6**

votes

**3**answers

513 views

### Adjusting the definition of a well-powered category to category theory with universes: size issues

Wikipedia and Borceux (Handbook of Categorical Algebra, Part I) give the following definitions of subobjects and well-powered categories:
A subobject of an object $X$ of a category $\mathsf{C}$ is an ...

**10**

votes

**2**answers

628 views

### Non-small objects in categories

An object $c$ in a category is called small, if there exists some regular cardinal $\kappa$ such that $Hom(c,-)$ preserves $\kappa$-filtered colimits.
Is there an example of a (locally small) ...

**3**

votes

**1**answer

168 views

### Bousfield localization of a left proper accessible model category

What is known about the Bousfield localization of a left proper accessible model category by a set of maps ? (I mean not combinatorial which is already known)

**14**

votes

**2**answers

292 views

### Example of non accessible model categories

By curiosity, I would like to see an example of a model category with the underlying category locally presentable which is not accessible in this sense (and just in case: even by using Vopěnka's ...

**17**

votes

**2**answers

452 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 $\...

**14**

votes

**2**answers

286 views

### Is every accessible category well-powered?

Every locally presentable category is well-powered: since it is a full reflective subcategory of a presheaf topos, its subobject lattices are subsets of those of the latter.
Every accessible category ...

**7**

votes

**1**answer

256 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 ...

**3**

votes

**2**answers

155 views

### Example: Accessible category without colimits

I am looking for intuitive examples of the way(s) that colimits may fail to exist in the category of (Set-valued) models for a limit/colimit sketch.
Bonus points if the sketch and/or the colimit ...

**5**

votes

**0**answers

127 views

### When is $Ind(C)$ small?

Let $C$ be a small category. Then $Ind(C)$ is the free completion of $C$ under filtered colimits. My sense is that typically, $Ind(C)$ is a large category. But sometimes it is small. For example, if $...

**11**

votes

**2**answers

718 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 ...

**5**

votes

**1**answer

162 views

### Can I check the accessibility of a functor on directed colimits of presentable objects?

Let $F: \mathcal{K} \to \mathcal{C}$ be a functor between $\lambda$-accessible categories, you can assume $\mathcal{C}$ to be Set if needed.
Is it true that $F$ is $\lambda$-accessible if and only if ...

**7**

votes

**1**answer

189 views

### Saturated classes, generation by a set and pullbacks of categories

Assume that we have a pullback square
$$
\begin{array}{ccc}
A & \rightarrow & B \\
\downarrow & & \downarrow \\
C & \rightarrow & D \\
\end{array}
$$
with all functors ...

**11**

votes

**4**answers

897 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'...

**5**

votes

**1**answer

93 views

### About small $\omega$-orthogonality classes and Gabriel-Ulmer duality

I am reading the paper http://www.numdam.org/article/CTGDC_2001__42_1_51_0.pdf fixing the implication $(ii)\Rightarrow (i)$ of Theorem 1.39 of Adamek-Rosicky's book. The correct statement is: if $\...

**4**

votes

**1**answer

169 views

### About small-orthogonality classes of a locally presentable category

Let $\mathcal{A} \subset \mathcal{K}$ be two locally presentable
categories. $\mathcal{A}$ reflective and closed under filtered
colimits. Then $\mathcal{A}$ is a small-orthogonality class. Let
$...

**4**

votes

**3**answers

313 views

### About the Yoneda objects of a locally presentable category

This question is a follow-up of Extending functors defined on dense subcategories.
Let $\mathcal{K}$ be a locally presentable category. An object $X$ of
$\mathcal{K}$ is called a Yoneda object if ...

**4**

votes

**1**answer

204 views

### Intuition behind $\lambda$-pure subobjects

While reading about accessible categories in Locally Presentable and Accessible Categories I came accross the notion of $\lambda$-pure subobjects, which seem to be important while characterising ...

**8**

votes

**1**answer

272 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)...

**7**

votes

**1**answer

387 views

### What is known about the large cardinal strength of Shelah's categoricity conjecture?

Shelah's categoricity conjecture states that for every Abstract Elementary Class $\mathcal{K}$ there is a cardinal $\mu$ depending only on $\operatorname{LS}(K)$ (i.e. the Löwenheim–Skolem number of $\...

**7**

votes

**0**answers

180 views

### When are the categories of algebras over props (co)complete?

Suppose P is a (colored) prop in a closed symmetric monoidal locally presentable category C. Is the category Alg_P of algebras over P in C locally presentable?
It seems that one can relatively easy ...

**3**

votes

**0**answers

178 views

### Preimages of accessible full subcategories

My question is ultimately about the model theory of $L_{\infty, \omega}$, but it is more convenient to phrase it in terms of category theory. Suppose I have finitely accessible categories $\mathcal{C},...

**10**

votes

**0**answers

472 views

### Orthogonality relations and accessibility?

Suppose I have a pair of locally presentable categories connected by an accessible functor. Then the preimage of an accessible subcategory of the codomain is an accessible subcategory of the domain. ...

**3**

votes

**1**answer

325 views

### Equivalence of the two definitions of k-compactness/k-presentability

We say that an object $X$ of a category $C$ is $\kappa$-compact (also $\kappa$-presentable and $\kappa$-accessible) for a cardinal $\kappa$ if $h^X(\cdot):=Hom(X,\cdot)$ commutes with all $\kappa$-...

**1**

vote

**1**answer

620 views

### The poset of k-small downward-closed subposets of a poset P is k-filtered when k is a regular cardinal?

Let $\kappa$ be a cardinal, and let $P$ be a poset. Let $\mathcal{P}_\kappa(P)$ denote the poset of $\kappa$-small subposets of $P$ and let $\mathcal{P}_\kappa^\downarrow(P)\subseteq\mathcal{P}_\...

**26**

votes

**6**answers

4k 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 ...

**7**

votes

**1**answer

677 views

### K-good trees and K-compactness of colimits over K-small downwards-closed subposets (500 point bounty if answered by Midnight EST))

Question:
Let $D:A\to (X\downarrow C)$ be a $\kappa$-good $S$-tree rooted at $X$ for a collection of morphisms $S$ in $C$, where $\kappa$ is a fixed uncountable regular cardinal. Then according to ...