# Questions tagged [locally-presentable-categories]

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34
questions

**9**

votes

**2**answers

493 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) ...

**6**

votes

**1**answer

266 views

### Is the 2-сategory of groupoids locally presentable?

I am wondering if the 2-сategory of groupoids is locally presentable. Locally presentable means the category is accessible and co-complete.
It has been pointed out that the category of groupoids is ...

**4**

votes

**1**answer

128 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)

**4**

votes

**1**answer

104 views

### Why is the category of all small $\mathbf{S}$-enriched categories locally presentable?

In Lurie's Higher Topos Theory Proposition A.3.2.4, the author used Proposition A.2.6.15 to prove that for any combinatorial monoidal model category $\mathbf{S}$
with all objects cofibrant and weak ...

**13**

votes

**2**answers

200 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 ...

**11**

votes

**1**answer

305 views

### Is there any references on the tensor product of presentable (1-)categories?

Is there any references on the tensor product of (locally) presentable categories ?
All I know about this is Lurie's book that deals with the $\infty$-categorical version, and a few references that ...

**11**

votes

**1**answer

250 views

### Accessible functors not preserving lots of presentable objects

Let $F:\cal C\to D$ be an accessible functor between locally presentable categories. By Theorem 2.19 in Adamek-Rosicky Locally presentable and accessible categories, there exist arbitrarily large ...

**14**

votes

**2**answers

343 views

### $\mu$-presentable object as $\mu$-small colimit of $\lambda$-presentable objects

Remark 1.30 of Adámek and Rosicky, Locally Presentable and Accessible Categories claims that in any locally $\lambda$-presentable category, each $\mu$-presentable object (for $\mu\ge\lambda$) can be ...

**5**

votes

**1**answer

124 views

### Rank of presentability of internal Hom of locally presentable categories

Let $C$ and $D$ be locally $\kappa$-presentable categories. It is written on the nLab that the category $\mathrm{Ladj}(C, D)$ of cocontinuous functors from $C$ to $D$ is again locally $\kappa$-...

**9**

votes

**1**answer

233 views

### Closure of presentable objects under finite limits

In a locally presentable category $\cal E$, there are arbitrarily large regular cardinals $\lambda$ such that the $\lambda$-presentable (a.k.a. $\lambda$-compact) objects are closed under pullbacks. ...

**16**

votes

**2**answers

380 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 $\...

**6**

votes

**1**answer

230 views

### Locally presentable categories, universes, and Vopenka's principle

Some aspects of the theory of locally presentable categories depends on set-theoretic assumptions. Namely, a large cardinal axiom known as Vopenka's principle implies several nice properties of such ...

**3**

votes

**1**answer

191 views

### Locally presentable categories

Under category
Let $C$ be a locally presentable category, and let $c$ be an object of $C$. Lets denote by $C^{/c}$ the under category, objects are maps $c\rightarrow x$ and morphisms are the evident ...

**15**

votes

**1**answer

414 views

### When is the category of models of a limit theory a topos?

If $\mathcal{E}$ is a Grothendieck topos on a small base, then it is locally presentable, and hence is equivalent to the category of models of some limit theory.
Is there a characterization of ...

**8**

votes

**0**answers

127 views

### Is every presentable category a limit of locally finitely-presentable categories and finitary left adjoints?

Let $Pr^L$ be the category of presentable categories and left adjoint functors (probably this should be at least a (2,1)-category; anyway ultimately I'm interested in the $(\infty,1)$-setting). For ...