All Questions
Tagged with presentable-categories or locally-presentable-categories
19 questions with no upvoted or accepted answers
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
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
10
votes
0
answers
510
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. ...
- 1,569
8
votes
0
answers
226
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 ...
- 64k
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
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
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
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 ...
- 42.4k
6
votes
0
answers
267
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 ...
- 64k
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
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
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 ...
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
$$
\hat M:=\lim M/h^nM,
$$
...
- 8,524
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
3
votes
0
answers
139
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 $\...
- 5,482
3
votes
0
answers
251
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 ...
- 5,562
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
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