Questions tagged [locally-presentable-categories]
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85 questions
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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 ...
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4
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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 ...
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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 ...
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8
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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
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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)....
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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
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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 ...
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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 &...
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8
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466
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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 ...
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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 ...
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1
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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 ...
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315
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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, $\...
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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 ...
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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 ...
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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}$,...
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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 ...
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407
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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
2
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2
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308
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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 \...
- 64k
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763
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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{...
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1
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414
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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
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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 ...
- 64k
2
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3
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459
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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 \...
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2
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493
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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
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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 ...
- 64k
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50
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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 ...
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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) ...
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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? (...
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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 \...
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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
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3
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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 ...
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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-...
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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}...
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2
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361
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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
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1
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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 $\...
- 64k
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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 ...
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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
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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
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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)
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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 ...
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755
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$\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 ...
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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 $\...
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407
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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
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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 ...
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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,
$$
...
- 8,524
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2
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287
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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
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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
8
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3
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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
6
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1
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187
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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
7
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0
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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
3
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1
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253
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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 ...
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