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6
votes
1answer
246 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 ...
6
votes
0answers
135 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 ...
2
votes
0answers
132 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 ...
7
votes
0answers
352 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
303 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 ...
1
vote
1answer
572 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 ...
19
votes
6answers
2k 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
570 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 ...