21
votes
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Are locally presentable categories determined by their objects?
The answer in general is no.
Let $\mathcal C$ be the category of sets, let $\mathcal D$ be the category of pointed sets (with basepoint-preserving maps), and let $f: \mathcal C \to \mathcal D$ be the ...
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17
votes
Accepted
Is the Cartesian product of two finitely presented objects finitely presentable?
No. For counterexamples, see Theorems 3.8, 3.9, and 3.10 of
Finiteness properties of direct products
of algebraic structures
Peter Mayr, Nik Ruškuc
Journal of Algebra 494 (2018) 167-187.
These ...
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16
votes
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Is there any references on the tensor product of presentable (1-)categories?
The canonical reference is Chapter 5 of Greg Bird's thesis.
- 5,539
15
votes
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When is the category of models of a limit theory a topos?
My collaborator Julia Ramos González and I are working on this question precisely in these days.
A part of the answer is already cointained in a paper by Carboni, Pedicchio and Rosický: Syntactic ...
- 9,111
15
votes
Is every "nice" abelian category with enough projectives an additive presheaf category?
The category $[C^{op}, \text{Ab}]$ of $\text{Ab}$-valued presheaves on any (small, for simplicity) $\text{Ab}$-enriched category is about as nice as it gets - locally finitely presentable, ...
- 118k
15
votes
Non-small objects in categories
In the opposite category of the category of sets, and of many algebraic categories, the only small objects are the empty set and the singleton. A conceptual reason for this is Freyd's (or Gabriel and ...
- 3,411
13
votes
Non-small objects in categories
In the category $\mathsf{Top}$ of topological spaces and continuous maps the only $\lambda$-presentable objects are discrete spaces. This appears 1.14(6) in Locally presentable and Accessible ...
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13
votes
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Accessible functors not preserving lots of presentable objects
An example is given in my paper with Tibor Beke,
Abstract elementary classes and accessible categories, Annals Pure Appl. Logic 163 (2012), 2008-2017, doi:10.1016/j.apal.2012.06.003, arXiv:1005....
- 2,160
13
votes
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Tensor product of sites
The category $H$ can be described as the category of $E$-valued sheaves on $D$, or $F$-valued sheaves on $C$.
You get a site by taking the category $C \times D$ and taking the topology generated by ...
- 42.4k
11
votes
Accepted
Example of a locally presentable locally cartesian closed category which is not a topos?
Every Grothendieck quasitopos is presentable and locally cartesian closed. These are categories of separated presheaves on a site. The simplest example of a site whose separated presheaves do not form ...
- 8,972
11
votes
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Cocomplete and finitely complete category with nice pullbacks that is not locally presentable
If you look at the article on quasitoposes, which are locally cartesian closed and therefore satisfy your exactness condition, you'll find a number of examples that are not locally presentable. For ...
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11
votes
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In a locally presentable category, is every object (a retract of) the colimit of a chain of smaller objects?
The last Remark in my joint paper gives a positive answer to the Question.
- 2,160
11
votes
Accepted
Is Qcoh(X) locally presentable?
Zariski descent tells us that
$$\operatorname{QCoh}(X)=\lim_{U\subseteq X} \operatorname{QCoh}(U)$$
where $U$ ranges through all open affines and the limit is taken in the $(2,1)$-categorical sense. ...
- 16.5k
10
votes
$\mu$-presentable object as $\mu$-small colimit of $\lambda$-presentable objects
Here's an argument which I currently believe. As Tim suggested, it does use the fat small object argument. References are to that paper.
Let $\mathcal{K}$ be a locally $\lambda$-presentable category ...
- 25.2k
10
votes
Closure of presentable objects under finite limits
The stronger claim is true at least for locally finitely presentable categories; this follows from Proposition 4.3 in https://arxiv.org/pdf/1005.2910.pdf.
- 2,160
10
votes
Accepted
Can the dual of a finitely-accessible category be accessible?
In Accessible Categories: The Foundations of Categorical Model Theory by Makkai and Paré, there is the example of a finitely accessible self-dual category. Apparently the example is due to Isbell. ...
- 9,111
9
votes
A locally presentable locally cartesian closed category that is not a quasitopos
The 1-categorical version of motivic spaces is locally cartesian closed but not a quasitopos. The idea is that there exists an $\mathbb A^1$-contractible scheme that is covered by $\mathbb A^1$-rigid ...
- 8,972
9
votes
Accepted
Raising the index of accessibility
Under GCH, if $\lambda < \mu$ are regular cardinals, then $\lambda \lhd \mu$ implies $\lambda \ll\mu$. The proof uses the following standard fact:
Lemma. Suppose $\lambda \leq \gamma$ are ...
- 8,099
9
votes
Accepted
Locally presentable categories
Over and under categories of a presentable category are presentable. This is Proposition 1.57 in Adámek, Rosický, Locally Presentable and Accessible Categories.
If $T : \mathcal{C} \to \mathcal{C}$ ...
- 4,459
9
votes
Accepted
Example of non accessible model categories
I don't know whether set-theoretic hypotheses are necessary to answer this question. But if we assume the negation of Vopěnka's principle, here is an example: by Example 6.12 of Adamek-Rosicky ...
- 66.7k
9
votes
From Topoi to Grothendieck categories
Johnstone's "Topos theory" has a chapter on cohomology; your (1) is the first result the author proves. I don't remember about (2), but maybe it's also there?
- 13.6k
9
votes
Accepted
Why are compactly generated $\infty$-categories closed under limits in $\operatorname{Cat}_{\infty}$?
By lemma 5.4.5.5., the forgetful functor $Pr^R_\omega\to Cat_\infty$ preserves pullbacks : the projection functors in the pullback preserve filtered colimits.
It's easy to prove that the same thing ...
- 15.8k
8
votes
Accepted
Is the 2-сategory of groupoids locally presentable?
This is true.
Since the $2$-category of groupoids is equivalent to the $2$-category of $1$-truncated spaces, the statement follows immediately from 5.5.1.8 and 5.5.6.21 in Lurie's Higher Topos ...
- 2,197
8
votes
A locally presentable locally cartesian closed category that is not a quasitopos
Thomas Holder points out to me that this question is answered in Borceux and Pedicchio, A characterization of quasi-toposes, with an example that is reproduced in C4.2.4 of Sketches of an Elephant: ...
8
votes
Accepted
Locally presentable categories, universes, and Vopenka's principle
In the Grothendieck universe approach to category theory, as you say, we replace all small sets with $\mathcal{U}$-small sets. Let's look at the definition of a locally presentable $\mathcal{U}$-...
- 25.2k
8
votes
Adjusting the definition of a well-powered category to category theory with universes: size issues
Your dilemma can be resolved by Scott's trick, if your universes are cumulative von Neumann universes.
Briefly, given an equivalence relation $E \subseteq C \times C$ on a class $C$ and $x \in C$, ...
- 48.8k
8
votes
Cocomplete and finitely complete category with nice pullbacks that is not locally presentable
Filtered colimits commute with finite limits in any Grothendieck topos. A Grothendieck topos does not need to be locally finitely presentable; the presentability rank of a topos is tightly related to ...
- 9,111
8
votes
Cocomplete and finitely complete category with nice pullbacks that is not locally presentable
Every localization (= full reflective subcategory such that the reflector preserves finite limits) of a locally finitely presentable category satisfies this property. More can be found in Localisation ...
- 2,160
8
votes
Accepted
Can a locally presentable category have a proper class of accessible localizations?
A limit closure of a set of objects of a locally presentable category $\mathcal K$ is reflective. In this way one gets an increasing chain of reflective full subcategories of $\mathcal K$. If this ...
- 2,160
8
votes
Enriched vs ordinary filtered colimits
For Q1: something related is dealt with in a context more general than the classical one by Adamek, Borceux, Lack and Rosicky in their paper "A classification of accessible categories". They ...
- 373
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