# Questions tagged [accessible-categories]

The accessible-categories tag has no usage guidance.

**12**

votes

**2**answers

187 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 ...

**16**

votes

**2**answers

365 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

234 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

**0**answers

112 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 ...

**2**

votes

**2**answers

121 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

120 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

460 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

147 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

175 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 ...

**8**

votes

**4**answers

675 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

84 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

158 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

242 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

192 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

250 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)...