Questions tagged [accessible-categories]
The accessible-categories tag has no usage guidance.
35
questions
4
votes
1answer
162 views
Almost combinatorial accessible model categories
Theorem: Assume VP. Let $\mathcal{M}$ be an accessible model category
such that there exists a set of generating cofibrations $I$ and such
that all objects are fibrant. Then it is combinatorial.
...
2
votes
0answers
66 views
Accessible 2-category of presheaves
Let $A$ be a locally small category and let $\mathbf{Cat}$ be the 2-category of small categories, functors and natural transformations and let $Ps(A)$ be the 2-category of presheaves (the objects are ...
1
vote
1answer
205 views
How do you prove that the category of weak equivalences of sSet is accessible?
I am trying to prove that the category of simplicial sets is a combinatorial model category by using Proposition A.2.6.15 of Lurie's book, and this requires proving that the weak equivalences are ...
6
votes
0answers
179 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 ...
9
votes
2answers
272 views
When is the homotopy category of an accessible $\infty$-category accessible?
Let $\mathcal C$ be an accessible $\infty$-category, and let $ho(\mathcal C)$ be its homotopy category. I can think of two "trivial" reasons for $ho(\mathcal C)$ to be accessible:
$ho(\mathcal C) = \...
15
votes
0answers
188 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 ...
11
votes
1answer
353 views
Does the homotopy category of spaces admit a weak generating set?
As a follow-up to this question, let $\mathcal C$ be a category and $\mathcal S \subseteq \mathcal C$ a class of objects. Say that $\mathcal S$ is weakly generating if the functors $Hom_{\mathcal C}(S,...
13
votes
1answer
304 views
presentability rank of categories of coalgebras
The following theorem is relatively classical:
Theorem: Given an accessible endofunctor, (co)pointed endofunctor or (co)monad $T$ on a locally presentable category $C$, then the category of $T$-(co)...
8
votes
2answers
359 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 ...
6
votes
3answers
442 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 ...
9
votes
2answers
582 views
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) ...
4
votes
1answer
155 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)
14
votes
2answers
277 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 ...
16
votes
2answers
421 views
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 $\...
14
votes
2answers
261 views
Is every accessible category well-powered?
Every locally presentable category is well-powered: since it is a full reflective subcategory of a presheaf topos, its subobject lattices are subsets of those of the latter.
Every accessible category ...
7
votes
0answers
133 views
Stability of accessible $\infty$-categories under some operations
I'm reading through Higher Topos Theory, and I can't make sense of a few proofs in the sections about accessible $\infty$-categories.
In Proposition 5.4.4.3, Lemma 5.4.4.2 is used, but I don't see ...
2
votes
2answers
140 views
Example: Accessible category without colimits
I am looking for intuitive examples of the way(s) that colimits may fail to exist in the category of (Set-valued) models for a limit/colimit sketch.
Bonus points if the sketch and/or the colimit ...
5
votes
0answers
124 views
When is $Ind(C)$ small?
Let $C$ be a small category. Then $Ind(C)$ is the free completion of $C$ under filtered colimits. My sense is that typically, $Ind(C)$ is a large category. But sometimes it is small. For example, if $...
11
votes
2answers
603 views
What are the reflective subcategories of the category of presentable categories?
I am actually interested in the $\infty$-categorical case, but the same question is meaningful in the $1$-categorical situation as well.
A nice property of presentable $\infty$-categories is that if ...
5
votes
1answer
157 views
Can I check the accessibility of a functor on directed colimits of presentable objects?
Let $F: \mathcal{K} \to \mathcal{C}$ be a functor between $\lambda$-accessible categories, you can assume $\mathcal{C}$ to be Set if needed.
Is it true that $F$ is $\lambda$-accessible if and only if ...
7
votes
1answer
185 views
Saturated classes, generation by a set and pullbacks of categories
Assume that we have a pullback square
$$
\begin{array}{ccc}
A & \rightarrow & B \\
\downarrow & & \downarrow \\
C & \rightarrow & D \\
\end{array}
$$
with all functors ...
9
votes
4answers
786 views
What was Burroni's sketch for topological spaces?
In a 1981 talk, René Guitart cites Albert Burroni as having given "A first interesting example of a mixed sketch...for the category of topological spaces" in 1970. This was apparently done in Burroni'...
5
votes
1answer
88 views
About small $\omega$-orthogonality classes and Gabriel-Ulmer duality
I am reading the paper http://www.numdam.org/article/CTGDC_2001__42_1_51_0.pdf fixing the implication $(ii)\Rightarrow (i)$ of Theorem 1.39 of Adamek-Rosicky's book. The correct statement is: if $\...
4
votes
1answer
160 views
About small-orthogonality classes of a locally presentable category
Let $\mathcal{A} \subset \mathcal{K}$ be two locally presentable
categories. $\mathcal{A}$ reflective and closed under filtered
colimits. Then $\mathcal{A}$ is a small-orthogonality class. Let
$...
4
votes
3answers
287 views
About the Yoneda objects of a locally presentable category
This question is a follow-up of Extending functors defined on dense subcategories.
Let $\mathcal{K}$ be a locally presentable category. An object $X$ of
$\mathcal{K}$ is called a Yoneda object if ...
4
votes
1answer
201 views
Intuition behind $\lambda$-pure subobjects
While reading about accessible categories in Locally Presentable and Accessible Categories I came accross the notion of $\lambda$-pure subobjects, which seem to be important while characterising ...
8
votes
1answer
263 views
On the cardinal arithmetic of accessible categories
If $\lambda, \mu$ are regular cardinals, say that $\lambda \trianglelefteq \mu$ if $\lambda \leq \mu$ and
$$\forall X, \, |X| < \mu \implies \mathrm{cf} (P_\lambda(X)) < \mu$$
Here $P_\lambda(X)...
7
votes
1answer
369 views
What is known about the large cardinal strength of Shelah's categoricity conjecture?
Shelah's categoricity conjecture states that for every Abstract Elementary Class $\mathcal{K}$ there is a cardinal $\mu$ depending only on $\operatorname{LS}(K)$ (i.e. the Löwenheim–Skolem number of $\...
7
votes
0answers
177 views
When are the categories of algebras over props (co)complete?
Suppose P is a (colored) prop in a closed symmetric monoidal locally presentable category C. Is the category Alg_P of algebras over P in C locally presentable?
It seems that one can relatively easy ...
3
votes
0answers
168 views
Preimages of accessible full subcategories
My question is ultimately about the model theory of $L_{\infty, \omega}$, but it is more convenient to phrase it in terms of category theory. Suppose I have finitely accessible categories $\mathcal{C},...
10
votes
0answers
449 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. ...
3
votes
1answer
323 views
Equivalence of the two definitions of k-compactness/k-presentability
We say that an object $X$ of a category $C$ is $\kappa$-compact (also $\kappa$-presentable and $\kappa$-accessible) for a cardinal $\kappa$ if $h^X(\cdot):=Hom(X,\cdot)$ commutes with all $\kappa$-...
1
vote
1answer
612 views
The poset of k-small downward-closed subposets of a poset P is k-filtered when k is a regular cardinal?
Let $\kappa$ be a cardinal, and let $P$ be a poset. Let $\mathcal{P}_\kappa(P)$ denote the poset of $\kappa$-small subposets of $P$ and let $\mathcal{P}_\kappa^\downarrow(P)\subseteq\mathcal{P}_\...
26
votes
6answers
4k views
Reasons to believe Vopenka's principle/huge cardinals are consistent
There are a number of informal heuristic arguments for the consistency of ZFC, enough that I am happy enough to believe that ZFC is consistent. This is true for even some of the more tame large ...
7
votes
1answer
651 views
K-good trees and K-compactness of colimits over K-small downwards-closed subposets (500 point bounty if answered by Midnight EST))
Question:
Let $D:A\to (X\downarrow C)$ be a $\kappa$-good $S$-tree rooted at $X$ for a collection of morphisms $S$ in $C$, where $\kappa$ is a fixed uncountable regular cardinal. Then according to ...
