All Questions
Tagged with presentable-categories or locally-presentable-categories
85 questions
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
20
votes
2
answers
763
views
Enriched vs ordinary filtered colimits
Filtered categories can be defined as those categories $\mathbf{C}$ such that $\mathbf{C}$-indexed colimits in $\mathrm{Set}$ commute with finite limits.
Similarly, for categories enriched in $\mathbf{...
- 1,947
19
votes
1
answer
414
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 ...
- 64k
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
16
votes
1
answer
584
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 ...
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
15
votes
2
answers
361
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 ...
- 3,901
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
14
votes
1
answer
620
views
Are locally presentable categories determined by their objects?
Let $f: \mathcal{C} \rightarrow \mathcal{D}$ be a colimit-preserving functor between locally presentable categories. Assume that $f$ induces an equivalence between the groupoids underlying $\mathcal{C}...
- 898
13
votes
1
answer
474
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 ...
- 42.4k
13
votes
2
answers
493
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 ...
- 64k
13
votes
0
answers
246
views
Categorification of "Every domain embeds into a field"?
In the category of commutative rings, every domain embeds into a field. Is this true in the category of presentably symmetric monoidal stable $\infty$-categories? Here's what I mean by that.
Let $...
- 64k
12
votes
1
answer
376
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 ...
- 66.8k
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
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
11
votes
0
answers
442
views
A right adjoint preserves Phi-colimits if and only if the left adjoint does what?
Let $\Phi$ be a class of categories (e.g. filtered categories), and consider an adjunction $L : \mathbf C \rightleftarrows \mathbf D : R$. A $\Phi$-colimit is a colimit whose diagram is in $\Phi$. We ...
- 10.7k
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
10
votes
2
answers
407
views
Which abelian groups are $\aleph_1$-filtered colimits of finitely-generated abelian groups?
Observation: Every $\aleph_1$-directed colimit $\varinjlim_i X_i$ of finite sets is finite.
Proof:
Because the $X_i$'s are finite, the Mittag-Leffler condition holds, so by passing to the diagram of ...
- 64k
10
votes
0
answers
510
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. ...
- 1,569
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
739
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) ...
- 511
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
1
answer
478
views
From Topoi to Grothendieck categories
This question is mostly about a reference request. Let $\mathcal{E}$ be a Grothendieck topos. I am looking for a reference of the following two facts. I am aware that $(2) \Rightarrow (1)$ by Gabriel-...
- 9,111
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
9
votes
1
answer
304
views
Internal logic of locally strongly finitely presentable categories
There is a duality between locally strongly finitely presentable (LSFP) categories and (Cauchy complete) cartesian categories, i.e. multisorted algebraic theories [1]. The internal logic of cartesian ...
- 10.7k
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
9
votes
1
answer
415
views
What additional property does the antipode give on the category of all modules over an Hopf algebra?
It is well known that many constructions involving bialgebras extends to monoidal categories, and often becomes more natural in that framework.
If one cares about the category of finite dimensional ...
- 8,524
8
votes
3
answers
2k
views
Is the category of small categories locally presentable?
I was wondering whether the various model structures on the category of small categories are combinatorial. I think that the ones I know are at least cofibrantly generated. In order to be ...
- 15.2k
8
votes
1
answer
584
views
Is the Cartesian product of two finitely presented objects finitely presentable?
Let $C$ be a locally finitely presented category, $A, B$ are two finitely presented (synonym: compact) objects in it. Is it true that $A \times B$ is finitely representable?
At least I have looked at ...
- 6,962
8
votes
1
answer
466
views
Tensor product of sites
Let $C, D$ two Grothendieck sites. Since the corresponding toposes $E, F$ are locally presentable categories, then (by Gabriel-Ulmer duality) they correspond to limit theories $X, Y$ (that is, small ...
- 6,962
8
votes
2
answers
421
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 ...
- 64k
8
votes
1
answer
253
views
Compact objects in slice categories of finitely presentable categories
Given a locally finitely presentable category $\mathscr C$ and an object $X \in \mathscr C$, it is not so hard to show that a morphism $(X \to Y)$ is compact in $\mathscr C_{X/}$ if it can be obtained ...
- 28.7k
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
8
votes
1
answer
216
views
Can finite presentability be tested with respect to sequential colimits?
Let $\mathcal C$ be a locally finitely-presentable category, and let $C \in \mathcal C$ be an object such that for all sequential colimits, the map $$\varinjlim \operatorname{Hom}(C, X_i) \to \...
- 64k
8
votes
0
answers
226
views
Is every presentable category a limit of locally finitely-presentable categories and finitary left adjoints?
Let $Pr^L$ be the category of presentable categories and left adjoint functors (probably this should be at least a (2,1)-category; anyway ultimately I'm interested in the $(\infty,1)$-setting). For ...
- 64k
7
votes
1
answer
452
views
Example of a locally presentable locally cartesian closed category which is not a topos?
The only way I know to get a locally cartesian closed category which is not a topos is to start with a topos and then throw out some objects so that the category is not sufficiently cocomplete to be a ...
- 64k
7
votes
1
answer
396
views
Locally presentable categories, universes, and Vopenka's principle
Some aspects of the theory of locally presentable categories depends on set-theoretic assumptions. Namely, a large cardinal axiom known as Vopenka's principle implies several nice properties of such ...
- 4,459
7
votes
1
answer
563
views
Characterisation of essentially algebraic theories as monads
The following correspondence between algebraic theories and monads on $\mathbf{Set}$ is well-known (see, for example, Algebraic Theories: A Categorical Introduction to General Algebra).
The ...
- 10.7k
7
votes
1
answer
216
views
Saturated classes, generation by a set and pullbacks of categories
Assume that we have a pullback square
$$
\begin{array}{ccc}
A & \rightarrow & B \\
\downarrow & & \downarrow \\
C & \rightarrow & D \\
\end{array}
$$
with all functors ...
- 660
7
votes
0
answers
207
views
Higher categorical / operadic approach to homotopy associative, homotopy commutative, $H_\infty$ ring spectra?
Let $\mathcal C$ be a symmetric monoidal $\infty$-category, and let $O$ be an operad (for example, $O$ could be an $A_m$ or $E_n$ operad or a tensor product thereof, and $\mathcal C$ could be spaces ...
- 64k
6
votes
3
answers
784
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 ...
- 1,115
6
votes
1
answer
439
views
Is Qcoh(X) locally presentable?
Let $X$ be a scheme. Is the category $QCoh(X)$ of quasi-coherent sheaves on $X$ locally presentable? If so, can we say anything about the $\kappa$ for which $QCoh(X)$ is locally $\kappa$-presentable? (...
6
votes
1
answer
771
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 ...
- 556
6
votes
1
answer
187
views
Can a locally presentable category have a proper class of accessible localizations?
Question: What is an example of a locally presentable category $\mathcal C$ such that there exists a proper class of accessible localizations $(\mathcal C \to \mathcal D_i)_{i < ORD}$?
In other ...
- 64k
6
votes
1
answer
355
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 ...
- 1,313
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
6
votes
2
answers
407
views
Can conservativity depend on the universe?
Question 1: Let $F: C \to D$ be a conservative, $\kappa$-cocontinuous functor between small, $\kappa$-cocomplete categories. Is the induced functor $Ind_\kappa(F): Ind_\kappa(C) \to Ind_\kappa(D)$ ...
- 64k
6
votes
1
answer
275
views
Do cocontinuous SET-valued functors separate points?
Let $C$ be a category. For the purposes of this question, I would like to avoid cases where the answer might be "no" simply because $C$ is "too large", and so I will ask that $C$ has a set of ...
- 54.6k
6
votes
3
answers
523
views
Contramodule as direct limit of its finitely generated subcontramodules
$\DeclareMathOperator\Hom{Hom}$Let $K$ be a field. Let $C$ be a $K$-coalgebra. A contramodule $M$ over $C$ is a $K$-space with a $K$-linear map $\pi_M:\Hom_K(C,M)\longrightarrow M$ such that $\pi_M \...
- 383
6
votes
1
answer
201
views
In a weak factorization system, the left class is left cancellative iff the right class is what?
Let $\mathcal K$ be a locally presentable category, and let $(\mathcal L, \mathcal R)$ be a cofibrantly-generated weak factorization system thereon. We say that $\mathcal L$ is left cancellative if $...
- 64k