Questions tagged [locally-presentable-categories]

Filter by
Sorted by
Tagged with
9
votes
1answer
249 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-...
5
votes
2answers
103 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 ...
3
votes
1answer
91 views

When is a finitary functor induced by Ind (co)continuous

Let $\mathbf C$ and $\mathbf D$ be small categories. $\mathrm{Ind}(\mathbf C)$ is an accessible category (by definition), and is locally finitely presentable (i.e. cocomplete, or equivalently complete)...
13
votes
2answers
322 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 ...
6
votes
1answer
157 views

If $\mathcal C$ is a $\kappa$-accessible $\infty$-category, then is $Mor \mathcal C$ $\kappa$-accessible?

If $\mathcal C$ is a $\kappa$-accessible 1-category, then the category of morphisms $Mor \mathcal C$ is a $\kappa$-accessible 1-category, with the $\kappa$-presentable objects being those morphisms ...
15
votes
1answer
283 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{...
6
votes
1answer
179 views

Dense generator whose closure under finite colimits takes several steps to form?

Let $\mathcal C$ be a locally finitely presentable category, and let $\mathcal C_0 \subseteq \mathcal C$ be a dense generator of finitely-presentable objects. Then Every object $C \in \mathcal C$ is ...
6
votes
0answers
116 views

Can we make Pres *-autonomous?

The category $\mathbf{Sup}$ of sup-lattices (posets admitting all supremum and supremum preserving map between them) is a well known example of a $*$-autonomous category: The internal Hom is simply ...
6
votes
1answer
142 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 ...
13
votes
0answers
216 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 $...
5
votes
2answers
323 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 $Ind_\kappa(F): Ind_\kappa(C) \to Ind_\kappa(D)$ ...
2
votes
3answers
409 views

Cocomplete and finitely complete category with nice pullbacks that is not locally presentable

I have a result that holds for cocomplete and finitely complete categories such that pullbacks preserve directed colimits, by which I mean $A \times_B (\operatorname{colim}_{i \in I} C_i) = colim_{i \...
4
votes
1answer
184 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\...
7
votes
1answer
248 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 ...
5
votes
0answers
187 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 ...
15
votes
0answers
196 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 ...
8
votes
2answers
364 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 ...
6
votes
3answers
461 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 ...
3
votes
0answers
90 views

The coEilenbeg-Moore category of an Eilenberg-Moore category

Take a category $\mathcal{C}$ with a monad $T$ and construct the the Eilenberg-Moore category $\mathcal{C}^T$, the adjunction that arises is the terminal splitting of the monad $M$. Denote the ...
14
votes
1answer
498 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}...
10
votes
2answers
597 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
1answer
291 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
1answer
157 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
1answer
115 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 ...
14
votes
2answers
280 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
1answer
345 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
1answer
283 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 ...
15
votes
2answers
469 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 ...
5
votes
1answer
151 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
1answer
258 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. ...
17
votes
2answers
433 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
1answer
263 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
1answer
235 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
1answer
483 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
0answers
159 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 ...
2
votes
2answers
190 views

Is a filtered colimit of complete module complete?

This is probably a textbook question but i haven't been able to find a reference. Let $R$ be a complete commutative Noetherian local ring and $I$ its unique maximal ideal (I'm mostly interested in the ...
9
votes
1answer
262 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 ...
14
votes
2answers
346 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 ...
6
votes
1answer
282 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 ...
7
votes
1answer
186 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 ...
11
votes
2answers
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 ...
25
votes
1answer
1k 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 ...
3
votes
1answer
310 views

locally finitely presentable tensor categories

I am looking for examples of locally finitely presentable categories which admit a symmetric monoidal structure, such that the tensor product preserves colimits in each variable, but the unit is not ...
3
votes
0answers
108 views

Colimits of algebras of an endofunctor

I try to understand a proof in Adamek-Rosicky's book "Locally presentable and accessible categories", Cambridge University Press 1994. In Corollary 2.75 (p. 121) it is proven that the category $\...
6
votes
2answers
404 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 $\...
2
votes
0answers
212 views

On the category of $D$-modules

Let $X$ be a smooth variety over an algebraically closed field $k$ of char. $0$. 1) Is the abelian category $M(X)$ of $D$-modules on $X$, which are quasi-coherent as $O$-modules, a Grothendieck ...
1
vote
0answers
225 views

Are finitely presented modules finitely presentable? [closed]

Over a ring $R$ we have a notion of finitely presented module, namely: Definition 1 A module $F$ is finitely presented if there are $m$, $n$ positive integers such that $R^m\to R^n\to F\to 0$ is ...
10
votes
2answers
455 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?
10
votes
0answers
458 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. ...
6
votes
1answer
258 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 ...