basic constructions:
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Vopěnka’s principle is a large cardinal axiom which implies a good deal of simplification in the theory of locally presentable categories.
It is fairly strong as large cardinal axioms go: its consistency follows from the existence of huge cardinal?s, and it implies the existence of arbitrarily large measurable cardinals.
Vopěnka’s principle has many equivalent statements. Here are a few:
The VP is equivalent to the statement:
Every discrete full subcategory of a locally presentable category is small.
The VP is equivalent to the statement:
For every proper class sequence $\langle M_\alpha | \alpha \in Ord\rangle$ of first-order structures, there is a pair of ordinals $\alpha\lt\beta$ for which $M_\alpha$ embeds elementarily into $M_\beta$.
The VP is equivalent to the statement:
For $C$ a locally presentable category, every full subcategory $D \hookrightarrow C$ which is closed under colimits is a coreflective subcategory.
This is (AdamekRosicky, theorem 6.28).
The VP is equivalent to the statement:
Every cofibrantly generated model category (in a slightly more general sense than usual) is a combinatorial model category.
This is in (Rosicky)
If one insists on the traditional stricter definition of cofibrant generated model category, then the VP still implies that these are all combinatorial. But the VP is slightly stronger than this statement.
The VP is equivalent to both of the statements:
This is in (BCMR).
The Vopěnka principle implies the weak Vopěnka principle.
The weak VP is equivalent to the statement:
For $C$ a locally presentable category, every full subcategory $D \hookrightarrow C$ which is closed under limits is a reflective subcategory.
This is AdamekRosicky, theorem 6.22 and example 6.23
Vopěnka’s principle can be relativized to levels of the Lévy hierarchy by restricting the complexity of the (definable) classes to which it is applied. The following theorems are from (BCMR).
For any $n\ge 1$, the following statements are equivalent.
The “$n=0$ case” of this is:
For any $n\ge 1$, the following statements are equivalent.
Many more refined results can be found in (BCMR).
From a category-theoretic perspective, Vopěnka’s principle can be motivated by applications and consequences, but it can also be argued for somewhat a priori, on the basis that large discrete categories are rather pathological objects. We can’t avoid them entirely (at least, not without restricting the rest of mathematics fairly severely), but maybe at least we can prevent them from occurring in some nice situations, such as full subcategories of locally presentable categories. See this MO answer.
The VP implies the statement:
Let $C$ be a left proper combinatorial model category and $Z \in Mor(C)$ a class of morphisms. Then the left Bousfield localization $L_Z W$ exists.
This is theorem 2.3 in (RosickyTholen)
The VP implies the statement:
Let $C$ be a locally presentable (∞,1)-category and $Z$ a class of morphisms in $C$. Then the reflective localization of $C$ at $W$ extsts.
By the facts discussed at locally presentable (∞,1)-category and combinatorial model category and Bousfield localization of model categories we have that every locally presentable $(\infty,1)$-category is presented by a combinatorial model category and that under this correspondence reflective localizations correspond to left Bousfield localizations. The claim then follows with the (above theorem).
As usually stated, Vopěnka’s principle is not formalizable in first-order ZF set theory, because it involves a “second-order” quantification over proper classes (“…there does not exist a large discrete subcategory…”). It can, however, be formalized in this way in a class-set theory such as NBG.
On the other hand, it can be formalized in ZF as a first-order axiom schema consisting of one axiom for each class-defining formula $\phi$, stating that “$\phi$ does not define a class which is a large discrete subcategory…” We might call this axiom schema the Vopěnka axiom scheme. As in most situations of this sort, the first-order Vopěnka scheme is appreciably weaker than the second-order Vopěnka principle. See, for instance, this MO question and answer.
Unlike some large cardinal axioms, Vopěnka’s principle does not appear to be merely an assertion that “there exist very large cardinals” but rather an assertion about the precise size of the “universe” (the “boundary” between sets and proper classes). In other words, the universe could be “too big” for Vopěnka’s principle to hold, in addition to being “too small.”
(The equivalence of Vopěnka’s principle with the existence of C(n)-extendible cardinals may appear to contradict this. However, the property of being $C(n)$-extendible itself “depends on the size of the whole universe” in a sense.)
More precisely, if $\kappa$ is a cardinal such that $V_\kappa$ satisfies ZFC + Vopěnka’s principle, then knowing that $\lambda\gt\kappa$ does not necessarily imply that $V_\lambda$ also satifies Vopěnka’s principle. By contrast, if $V_\kappa$ satisfies ZFC + “there exists a measurable cardinal” (say), then there must be a measurable cardinal less than $\kappa$, and that measurable cardinal will still exist in $V_\lambda$ for any $\lambda\gt\kappa$. On the other hand, large cardinal axioms such as “there exist arbitrarily large measurable cardinals” have the same property that Vopěnka’s principle does: even if measurable cardinals are unbounded below $\kappa$, they will not be unbounded below $\lambda$ if $\lambda$ is the next greatest inaccessible cardinal after $\kappa$.
Relativizing Vopěnka’s principle to cardinals also raises the same first- versus second-order issues as above. We say that a Vopěnka cardinal is one where Vopěnka’s principle holds “in $V_\kappa$” where the quantification over classes is interpreted as quantification over all subsets of $V_\kappa$. By contrast, we could define an almost-Vopěnka cardinal to be one where $V_\kappa$ satisfies the first-order Vopěnka scheme. Then one can show, using the Mahlo reflection principle (see here again), that every Vopěnka cardinal $\kappa$ is a limit of $\kappa$-many almost-Vopěnka cardinals, and in particular the smallest almost-Vopěnka cardinal cannot be Vopěnka. Thus, being Vopěnka is much stronger than being almost-Vopěnka.
If Vopěnka’s principle fails, then there exist counterexamples to all of its equivalent statements, such as a large discrete full subcategory of a locally presentable category. If Vopěnka’s principle fails but the first-order Vopěnka scheme holds, then no such counterexamples can be explicitly definable.
On the other hand, if the Vopěnka scheme also fails, then there will be explicit finite formulas one can write down which define counterexamples. However, there is no “universal” counterexample, in the following sense: if Vopěnka’s principle is consistent, then for any class-defining formula $\phi$, there is a model of set theory in which Vopěnka’s principle fails (and even in which the first-order Vopěnka scheme fails), but in which $\phi$ does not define a counterexample to it. See here yet again.
The relation to the theory of locally presentable categories is the contents of chapter 6 of
The relation to combinatorial model categories is discussed in
The implication of VP on homotopy theory, model categories and cohomology localization are discussed in the following articles
Carles Casacuberta, Dirk Scevenels, Jeff Smith, Implications of large-cardinal principles in homotopical localization Advances in Mathematics Volume 197, Issue 1, 20 October 2005, Pages 120-139
Joan Bagaria, Carles Casacuberta, Adrian Mathias, Jiri Rosicky Definable orthogonality classes in accessible categories are small, arXiv