15
votes
Non-small objects in categories
In the opposite category of the category of sets, and of many algebraic categories, the only small objects are the empty set and the singleton. A conceptual reason for this is Freyd's (or Gabriel and ...
- 3,411
13
votes
Non-small objects in categories
In the category $\mathsf{Top}$ of topological spaces and continuous maps the only $\lambda$-presentable objects are discrete spaces. This appears 1.14(6) in Locally presentable and Accessible ...
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13
votes
Accepted
Does the homotopy category of spaces admit a weak generating set?
This paper by Kevin Carlson and Dan Christensen says that the answer to question one is no: No set of spaces detects isomorphisms in the homotopy category, arXiv:1910.04141.
- 561
11
votes
Accepted
In a locally presentable category, is every object (a retract of) the colimit of a chain of smaller objects?
The last Remark in my joint paper gives a positive answer to the Question.
- 2,160
10
votes
Accepted
What was Burroni's sketch for topological spaces?
If you read french, you can look at page 6 here. It is far from formal but it gives a good idea of the mixed sketch. In my understanding, there is a unique inductive cone in that mixed sketch, which ...
- 218
10
votes
Accepted
Can the dual of a finitely-accessible category be accessible?
In Accessible Categories: The Foundations of Categorical Model Theory by Makkai and Paré, there is the example of a finitely accessible self-dual category. Apparently the example is due to Isbell. ...
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9
votes
Accepted
Example of non accessible model categories
I don't know whether set-theoretic hypotheses are necessary to answer this question. But if we assume the negation of Vopěnka's principle, here is an example: by Example 6.12 of Adamek-Rosicky ...
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9
votes
Accepted
What are the reflective subcategories of the category of presentable categories?
Some ideas, building off of Simon Henry and Ivan di Liberti's remarks:
$Pr^L$ is in fact essentially a large (not huge) category (in either the ordinary or $\infty$ context).
That is, let $\lambda$ ...
- 63.9k
9
votes
Accepted
Raising the index of accessibility
Under GCH, if $\lambda < \mu$ are regular cardinals, then $\lambda \lhd \mu$ implies $\lambda \ll\mu$. The proof uses the following standard fact:
Lemma. Suppose $\lambda \leq \gamma$ are ...
- 8,099
9
votes
Accepted
presentability rank of categories of coalgebras
The case of algebras for a monad is discussed explicitly in Gregory Bird's thesis (see theorem 6.9). The case of the categories of algebras for an endofunctor or pointed endofunctor can be deduced ...
- 42.4k
9
votes
Accepted
Why are compactly generated $\infty$-categories closed under limits in $\operatorname{Cat}_{\infty}$?
By lemma 5.4.5.5., the forgetful functor $Pr^R_\omega\to Cat_\infty$ preserves pullbacks : the projection functors in the pullback preserve filtered colimits.
It's easy to prove that the same thing ...
- 15.8k
8
votes
What was Burroni's sketch for topological spaces?
The category of topological spaces is the category of models of a relational universal strict Horn theory $T$ without equality, i.e. the axioms are of the form $(\forall x)(\phi(x) \rightarrow \psi(x))...
- 3,901
8
votes
Adjusting the definition of a well-powered category to category theory with universes: size issues
Your dilemma can be resolved by Scott's trick, if your universes are cumulative von Neumann universes.
Briefly, given an equivalence relation $E \subseteq C \times C$ on a class $C$ and $x \in C$, ...
- 48.8k
8
votes
Accepted
Which abelian groups are $\aleph_1$-filtered colimits of finitely-generated abelian groups?
Yes. This can be rephrased as: let $G$ be an abelian group [resp. group] with a chain of f.g. subgroups $G_\alpha$ for $\alpha<\omega_1$. Is $G$ f.g.?
The answer is yes for abelian groups:
The ...
- 63.9k
8
votes
Accepted
When is the Eilenberg-Moore category of a monad on an ind-category itself an ind-category?
Let $T$ be a monad on an accessible category (i.e. an $\mathbf{Ind}$-category). If the underlying endofunctor of $T$ is finitary (i.e. preserves filtered colimits), then the Eilenberg–Moore category ...
- 10.6k
7
votes
Raising the index of accessibility
More generally, the following is Fact 2.5 of Lieberman, Rosický, and Vasey - Internal sizes in $\mu$-abstract elementary classes:
Theorem. Assume $\lambda$ and $\mu$ are regular cardinals and $2^{<...
7
votes
Example: Accessible category without colimits
The category Hil of Hilbert spaces, considered as a full subcategory of Ban is $\aleph_1$-accessible but not locally presentable, in fact it is self dual.
The category Lin of linear orders and ...
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7
votes
What was Burroni's sketch for topological spaces?
The reference for Burroni's sketch of topological spaces is Comptes rendus de l'académie des sciences, 27 juillet 1970 (tome 271), which amazingly can be found online.
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7
votes
Example of non accessible model categories
An example which does not depend on set theory is the equivariant model structure on the category of maps of spaces by Emmanuel Farjoun.
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7
votes
Adjusting the definition of a well-powered category to category theory with universes: size issues
One simple way is to read the traditional definitions (à la Grothendieck, etc) but replacing existence conditions with chosen structure:
A category $C$ is $U$-locally-small if it is equipped with a $...
- 21.3k
7
votes
Accepted
About the Yoneda objects of a locally presentable category
The usual name for "Yoneda objects" is "tiny" or "small-projective".
In general the tiny objects in a presheaf category are the retracts of representables. In particular, if $A$ is Cauchy-complete, ...
- 66.7k
7
votes
About the Yoneda objects of a locally presentable category
These objects are usually called absolutely presentable. In $Set^\mathcal A$, they are precisely retracts of representables. Presheaf categories are characterized as cocomplete categories having a ...
- 2,160
7
votes
Accepted
Is every accessible category well-powered?
It seems to me that every category with a small set of dense generator is well powered. In particular accessible categories are well powered.
Dense generator means that you have a small full ...
- 42.4k
7
votes
Accepted
Almost combinatorial accessible model categories
Actually, you can, and you don't need accessibility (local presentability of the underlying category is enough).
Under your assumption, for each $i:A \to B$ a generating cofibration, take $B \coprod_A ...
- 42.4k
7
votes
Accepted
Relation between Ind-completion and "additive"-ind-completion
The point is what Ivan hints at in his last paragraph, that additivity is a property rather than an extra structure.
In fact, we have:
Suppose $C$ is an additive category. Then the forgetful functor $...
- 15.8k
7
votes
Relation between Ind-completion and "additive"-ind-completion
The equivalence you mention holds more generally whenever your base of enrichment has a finitely presentable unit. This certainly includes $Ab$ but also many other examples: $Cat$, $sSet$, $GAb$, $...
- 499
6
votes
Accepted
Can I check the accessibility of a functor on directed colimits of presentable objects?
Yes, and this doesn't require any assumption on $\mathcal C$. This follows from the following basic fact: if $I$ is filtered and $(A_i)_{i\in I}$ is a diagram of categories with colimit $A$, then the ...
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5
votes
Accepted
Does $\mathsf{Ind}_\lambda^\mu(\mathsf{Ind}_\kappa^\lambda(\mathcal C)) = \mathsf{Ind}_\kappa^\mu(\mathcal C)$?
I asked myself the same questions while working on this paper [TAC].
It seemed to me that the way to answer questions like this is to embed $\textbf{Ind}_\kappa^\lambda (\mathcal{C})$ into the usual $\...
- 15.9k
4
votes
Is every accessible category well-powered?
Some considerations, not a full answer (yet).
In Accessible categories and models for linear logic, at page 2, Barr claims that every accessible category is well powered. He even claims that is ...
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4
votes
About the Yoneda objects of a locally presentable category
Question 1: The Yoneda objects are precisely retracts of representables. There are quite a few buzzwords which are relevant here (Cauchy completion, idempotent splitting completion, Karoubi envelope, ...
- 53.3k
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