All Questions
Tagged with presentable-categories or locally-presentable-categories
85 questions
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
4
votes
1
answer
217
views
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 ...
- 15.9k
5
votes
1
answer
270
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 ...
0
votes
1
answer
133
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 &...
- 15.6k
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
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 ...
- 42.4k
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 \...
- 15.9k
5
votes
0
answers
149
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 ...
- 6,962
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
1
vote
1
answer
214
views
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 ...
- 898
4
votes
1
answer
394
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)....
- 64k
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
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
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 ...
- 53.3k
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 ...
- 42.4k
5
votes
0
answers
315
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, $\...
- 6,962
5
votes
0
answers
255
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 ...
- 64k
1
vote
0
answers
50
views
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 ...
- 3,901
3
votes
2
answers
226
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)
- 42.4k
6
votes
0
answers
179
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 ...
- 64k
3
votes
1
answer
253
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 ...
- 10.7k
2
votes
2
answers
308
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 \...
- 15.9k
5
votes
0
answers
105
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 ...
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 ...
- 9,111
3
votes
0
answers
115
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 ...
- 203
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$-...
- 64k
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{...
- 499
5
votes
0
answers
255
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,
$$
...
- 35.5k
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 $\mathcal{C}$,...
- 499
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
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 \...
- 15.6k
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 ...
- 3,065
6
votes
0
answers
139
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 / ...
- 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
5
votes
1
answer
384
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 $\...
- 64k
2
votes
0
answers
92
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/...
- 10.7k
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? (...
- 16.5k
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-...
- 13.6k
2
votes
1
answer
215
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)...
- 15.9k
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
6
votes
1
answer
213
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 ...
- 64k
6
votes
1
answer
283
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 ...
- 64k
6
votes
0
answers
145
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 ...
- 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 ...
CommunityBot
- 1
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 ...
- 37.8k
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 $Ind_\kappa(F): Ind_\kappa(C) \to Ind_\kappa(D)$ ...
- 42.4k
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 ...
- 2,160
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
2
votes
3
answers
459
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 \...
- 21.3k