All Questions
Tagged with presentable-categories or locally-presentable-categories
16 questions
15
votes
2
answers
754
views
$\mu$-presentable object as $\mu$-small colimit of $\lambda$-presentable objects
Remark 1.30 of Adámek and Rosický, Locally Presentable and Accessible Categories claims that in any locally $\lambda$-presentable category, each $\mu$-presentable object (for $\mu\ge\lambda$) can be ...
- 25.2k
25
votes
1
answer
2k
views
Locally presentable abelian categories with enough injective objects
I came to the following question when thinking about the (infinitely generated) tilting-cotilting correspondence, where it appears to be relevant.
Does there exist a locally presentable abelian ...
- 15.6k
17
votes
2
answers
557
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 $\...
- 66.8k
15
votes
2
answers
470
views
A locally presentable locally cartesian closed category that is not a quasitopos
This question asks for a locally presentable locally cartesian closed category that is not a topos. All the answers given (at least in the 1-categorical case) are quasitoposes. What is an example of ...
- 66.8k
11
votes
2
answers
1k
views
Is every "nice" abelian category with enough projectives an additive presheaf category?
A "nice" category $\mathcal{C}$ should be (for the purposes of this question) locally presentable at a minimum, and maybe a bit more. One might require $\mathcal{C}$ to be (in roughly order of ...
- 64k
10
votes
2
answers
515
views
Cartesian product of small objects
Let's say we have a locally $\lambda$-presentable category and a pair of $\lambda$-presentable objects $A$ and $B$. Is it true that $A \times B$ is $\lambda$-presentable?
- 4,459
9
votes
1
answer
700
views
Case study: what does it take to formulate and prove Quillen's small object argument in ZFC?
I'm getting a bit lost over at Peter Scholze's interesting question about removing the dependence on universes from theorems in category theory. In particular, I'm being forced to admit that I don't ...
- 64k
7
votes
1
answer
452
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Example of a locally presentable locally cartesian closed category which is not a topos?
The only way I know to get a locally cartesian closed category which is not a topos is to start with a topos and then throw out some objects so that the category is not sufficiently cocomplete to be a ...
- 64k
6
votes
2
answers
659
views
Limit of a sequence of locally presentable categories
Let $\dotsc \to \mathcal{C}_2 \xrightarrow{F_1} \mathcal{C}_1 \xrightarrow{F_0}\mathcal{C}_0$ be a sequence of cocontinuous functors between locally presentable categories. Consider the limit $\...
- 5,482
11
votes
3
answers
927
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 $\mathcal{C}$,...
- 425
9
votes
1
answer
402
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. ...
- 66.8k
9
votes
3
answers
914
views
Enriched locally presentable categories
Is there a standard reference for the theory (if it exists) of $\mathcal{V}$-enriched locally presentable categories? Here $\mathcal{V}$ is a cosmos. Does anything unexpected happens here in contrast ...
- 63.1k
9
votes
2
answers
287
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 $\kappa$-...
- 25.2k
8
votes
2
answers
314
views
When is a locally presentable category (locally) cartesian-closed?
Let $\kappa$ be a regular cardinal. A category $\mathscr C$ is locally $\kappa$-presentable iff it is the free completion of a small $\kappa$-cocomplete category under $\kappa$-filtered colimits. Is ...
- 10.7k
6
votes
0
answers
267
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 ...
- 64k
4
votes
1
answer
339
views
Coreflective subcategories in Grothendieck/locally presentable categories
This question is a reference request for the following result or two results, which I believe are rather easy to prove.
Lemma. Let $\mathcal K$ be a locally presentable category and $\mathcal A\...
- 15.6k