# Questions tagged [locally-presentable-categories]

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75
questions

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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 ...

5
votes

1
answer

261
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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 ...

0
votes

1
answer

115
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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 &...

8
votes

1
answer

218
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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 ...

8
votes

1
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212
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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 \...

5
votes

0
answers

145
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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 ...

6
votes

1
answer

196
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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 $...

1
vote

1
answer

203
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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 ...

4
votes

1
answer

390
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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)....

8
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1
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570
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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 ...

8
votes

1
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420
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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 ...

4
votes

0
answers

271
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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, $\...

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 ...

1
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0
answers

45
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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 ...

6
votes

0
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163
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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 ...

11
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0
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362
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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 ...

2
votes

2
answers

288
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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 \...

5
votes

0
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93
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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 ...

3
votes

0
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106
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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 ...

10
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2
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379
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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 ...

5
votes

0
answers

240
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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
$$
\hat M:=\lim M/h^nM,
$$
...

11
votes

3
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830
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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 $\mathcal{C}$,...

7
votes

0
answers

202
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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 ...

6
votes

3
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497
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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 \...

9
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1
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290
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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 ...

6
votes

1
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610
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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 ...

6
votes

0
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136
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### 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 / ...

9
votes

1
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623
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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 ...

2
votes

0
answers

89
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### 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/...

5
votes

1
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370
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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 $\...

6
votes

1
answer

398
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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? (...

9
votes

1
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444
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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-...

6
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2
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236
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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 ...

2
votes

1
answer

205
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### 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
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2
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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 ...

6
votes

1
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204
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### 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 ...

20
votes

2
answers

718
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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{...

6
votes

1
answer

276
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### 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
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0
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143
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### 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

1
answer

183
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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 ...

13
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0
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239
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### 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 $...

6
votes

2
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396
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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 $Ind_\kappa(F): Ind_\kappa(C) \to Ind_\kappa(D)$ ...

2
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3
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453
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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 \...

4
votes

1
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317
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### 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

1
answer

469
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### 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 ...

6
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0
answers

261
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### 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 ...

18
votes

1
answer

387
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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 ...

8
votes

2
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415
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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 ...

6
votes

3
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738
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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 ...

3
votes

1
answer

233
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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 ...