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Questions tagged [locally-presentable-categories]

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Presentability rank of tensor product of presentable categories

In this post category means $(\infty, 1)$-category. Let $X, Y$ be two presentable categories. We can then form their tensor product $X \otimes Y \cong \operatorname{ContFun}(X^{\mathrm{op}}, Y)$. Can ...
Brendan Murphy's user avatar
5 votes
1 answer
261 views

Are Euclidean spaces $\Delta$-generated?

From the definition of $\Delta$-generated it seems like $\mathbb R$ should be $\Delta$-generated, as $\mathbb R$ is final with respect to all continuous maps $\mathbb R^n \to \mathbb R$. However, the ...
William B.'s user avatar
0 votes
1 answer
115 views

Is every locally $\kappa$-presentable category, also locally $\tau$-presentable for any $\tau > \kappa$?

Let $\kappa$ be a small regular cardinal and $D$ a locally $\kappa$-presentable category. Is it true that $D$ is also locally $\tau$-presentable for any $\tau > \kappa.$ Adamek und Rosicky show in &...
willie's user avatar
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8 votes
1 answer
218 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 ...
R. van Dobben de Bruyn's user avatar
8 votes
1 answer
212 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 \...
Tim Campion's user avatar
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5 votes
0 answers
145 views

In what algebraic categories do finitely presentable objects form a dense cogenerator?

For each $C$ locally finitely presentable category, the full subcategory of finitely presentable objects $C_{fp}$ is a dense generator, i.e. the natural functor $C \to \mathrm{PSh}(C_{fp})$ is a full ...
Arshak Aivazian's user avatar
6 votes
1 answer
196 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 $...
Tim Campion's user avatar
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1 vote
1 answer
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Proof in Higher Algebra that $\mathcal{C}at(\mathcal{K})$ is presentable

In Higher Algebra Lemma 4.8.4.2, Lurie shows that for $\mathcal{K}$ a small set of simplicial sets, the $\infty$-category $\mathcal{C}at(\mathcal{K})$ of small $\infty$-categories with $K$-shaped ...
Markus Zetto's user avatar
4 votes
1 answer
390 views

Does the rank of a subfunctor not exceed the rank of a functor?

It is known that Vopenka's principle is equivalent to the statement “a subfunctor of a accessible functor is accessible” (Adámek and Rosický, Cor 6.31 in Locally Presentable and Accessible Categories)....
Arshak Aivazian's user avatar
8 votes
1 answer
570 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 ...
Arshak Aivazian's user avatar
8 votes
1 answer
420 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 ...
Arshak Aivazian's user avatar
4 votes
0 answers
271 views

Why is $\rm{Cat}$ a Cartesian-closed category?

I am interested in naturally occurring symmetric closed monoidal structures on locally presentable categories. Two general examples: Grothendieck topos with Cartesian structure. Here, for example, $\...
Arshak Aivazian's user avatar
5 votes
0 answers
237 views

Is there a "relative version" of the theorem that every locally presentable category has all small limits?

Let $\mathcal C$ be a locally presentable category. Then by definition, $\mathcal C$ has all small colimits. Nontrivially, we also have Theorem 1: (Gabriel and Ulmer?) $\mathcal C$ also has all small ...
Tim Campion's user avatar
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Cellular model of a locally presentable category

According to https://ncatlab.org/nlab/show/cellular+model, I call a cellular model of a locally presentable category a set of monomorphisms cofibrantly generating the monomorphisms. I am curious to ...
Philippe Gaucher's user avatar
6 votes
0 answers
163 views

Does $Pr^L_\kappa \in Pr^L$ behave like an "object classifier" or "universe"?

Let $Pr^L$ denote the $\infty$-category of presentable $\infty$-categories and left adjoint functors. Let $Pr^L_\kappa \subset Pr^L$ denote the locally-full subcategory of $\kappa$-compactly-generated ...
Tim Campion's user avatar
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11 votes
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362 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 ...
varkor's user avatar
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2 votes
2 answers
288 views

Is there a "duality involution" on presentable categories?

$\newcommand\Psh{\mathit{Psh}}\newcommand\Pr{\mathit{Pr}}$Let $\Psh$ be the category of presheaf categories and cocontinuous functors which preserve tiny objects. There is a functor $(-)^\ast : \Psh \...
Tim Campion's user avatar
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5 votes
0 answers
93 views

Left adjoints for functors out of a Deligne-Kelly tensor product

Let $k$ be a field, and let $\mathcal{C}$ be a locally finitely presentable $k$-linear categories. The Deligne-Kelly tensor product $\mathcal{C}\boxtimes\mathcal{C}$ is the 2-universal locally ...
Thibault Décoppet's user avatar
3 votes
0 answers
106 views

Categories in which finite powers commute with filtered colimits

If $\mathcal{C}$ is a category with finite products and filtered colimits, then we say that finite powers commute with filtered colimits in $\mathcal{C}$ if for each natural number $n$, the $n$th ...
User7819's user avatar
  • 203
10 votes
2 answers
379 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 ...
Tim Campion's user avatar
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5 votes
0 answers
240 views

Is the category of topologically free $k[[h]]$-modules locally presentable?

$\newcommand{\colim}{\operatorname{colim}}$ Let $k$ be a field (of characteristic 0, say) and $M$ be a module over $R=k[[h]]$. Recall that the $h$-adic completion of $M$ is $$ \hat M:=\lim M/h^nM, $$ ...
Adrien's user avatar
  • 8,444
11 votes
3 answers
830 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 $\mathcal{C}$,...
3 A's's user avatar
  • 425
7 votes
0 answers
202 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 ...
Tim Campion's user avatar
  • 62k
6 votes
3 answers
497 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 \...
Sam's user avatar
  • 383
9 votes
1 answer
290 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 ...
varkor's user avatar
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6 votes
1 answer
610 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 ...
Bastiaan Cnossen's user avatar
6 votes
0 answers
136 views

Characterisation of essentially algebraic theories with a fixed set of sorts

It is well known (e.g. Palmgren–Vickers's Partial Horn logic and cartesian categories) that many-sorted essentially algebraic theories (equivalently partial Horn theories / quasi-equational theories / ...
varkor's user avatar
  • 8,925
9 votes
1 answer
623 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 ...
Tim Campion's user avatar
  • 62k
2 votes
0 answers
89 views

Coslices of $\mathbb D$-presentable categories

Let $\mathbb D$ be a sound limit doctrine. When $\mathbb D$ is the doctrine of $\lambda$-small limits, then for any $\mathbb D$-presentable category $\mathscr C$ and $X \in \mathscr C$, the coslice $X/...
varkor's user avatar
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5 votes
1 answer
370 views

Does $\mathsf{Ind}_\lambda^\mu(\mathsf{Ind}_\kappa^\lambda(\mathcal C)) = \mathsf{Ind}_\kappa^\mu(\mathcal C)$?

$\newcommand\Ind{\mathsf{Ind}}\newcommand\Ord{\mathsf{Ord}}\newcommand\Psh{\mathsf{Psh}}$For $\kappa \leq \lambda \leq \Ord$ regular cardinals and $\mathcal C$ an essentially small category, let $\...
Tim Campion's user avatar
  • 62k
6 votes
1 answer
398 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? (...
PresentableQCoh's user avatar
9 votes
1 answer
444 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-...
Ivan Di Liberti's user avatar
6 votes
2 answers
236 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 ...
varkor's user avatar
  • 8,925
2 votes
1 answer
205 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)...
varkor's user avatar
  • 8,925
13 votes
2 answers
476 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 ...
Tim Campion's user avatar
  • 62k
6 votes
1 answer
204 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 ...
Tim Campion's user avatar
  • 62k
20 votes
2 answers
718 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{...
Rune Haugseng's user avatar
6 votes
1 answer
276 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 ...
Tim Campion's user avatar
  • 62k
6 votes
0 answers
143 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 ...
Simon Henry's user avatar
  • 40.5k
6 votes
1 answer
183 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 ...
Tim Campion's user avatar
  • 62k
13 votes
0 answers
239 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 $...
Tim Campion's user avatar
  • 62k
6 votes
2 answers
396 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 $Ind_\kappa(F): Ind_\kappa(C) \to Ind_\kappa(D)$ ...
Tim Campion's user avatar
  • 62k
2 votes
3 answers
453 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 \...
Maxime Lucas's user avatar
4 votes
1 answer
317 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\...
Leonid Positselski's user avatar
7 votes
1 answer
469 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 ...
varkor's user avatar
  • 8,925
6 votes
0 answers
261 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 ...
Tim Campion's user avatar
  • 62k
18 votes
1 answer
387 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 ...
Tim Campion's user avatar
  • 62k
8 votes
2 answers
415 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 ...
Tim Campion's user avatar
  • 62k
6 votes
3 answers
738 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 ...
Jxt921's user avatar
  • 1,095
3 votes
1 answer
233 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 ...
Ben MacAdam's user avatar
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