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[MATH] It is crucial for the success of UF that we have a working definition and theory of [MATH] -categories. Even if they are not |
fundamental in the sense that everything is built out of them, they are still fundamental in the sense that they are a key tool in the development of modern higher algebra, geometry, and topology. |
The problem of defining [MATH] -categories is equivalent to the problem of defining the type of simplicial types, [MATH] . A simplicial type is just a functor [MATH] , so if we know how to define [MATH] and the type of functors between any two [MATH] , then we can define |
[MATH] . On the other hand, [MATH] itself can be defined as the subtype of [MATH] consisting of complete Segal types (also called Rezk types RiehlShulman2017 |
The problem of defining [MATH] can also be reduced to another, apparently simpler problem, namely that of defining the type of semi-simplicial types, [MATH] . A semi-simplicial type is a functor [MATH] , where [MATH] is the subcategory of [MATH] with the same objects but only injective functions. However, [MATH] is a d... |
set of objects, so we can give a more direct description as follows: A semi-simplicial type [MATH] consists of: a type of [MATH] -simplices [MATH] , and |
for every pair of [MATH] -simplices [MATH] , a type of [MATH] -simplices from [MATH] to [MATH] [MATH] , and for every triple of [MATH] -simplices [MATH] and |
[MATH] -simplices [MATH] [MATH] , and [MATH] , a type of [MATH] -simplices with boundary [MATH] and so on … Again we have the problem that it seems that infinitely much data is needed, but here it seems more plausible that an inductive (or coinductive) approach could work. Only, no-one has figured out how to do it, and... |
We can define [MATH] -categories in terms of semi-simplicial types, as the complete semi-Segal types CapriottiKraus2017 . This work was inspired by analogous work in the classical setting Harpaz2015 . Thus, the problems of defining |
[MATH] -categories, simplicial types, and semi-simplicial types are equivalent, but currently just beyond reach. It is interesting to contrast the case of [MATH] -categories with that of [MATH] -groups: An [MATH] -category structure on a pointed, connected type of objects is the same as an [MATH] -monoid (the type of w... |
[MATH] -groups is simply that of pointed, connected types, with the type of group elements being the identity type [MATH] , with [MATH] the designated point. |
While the above problems concern “large” types, there are also problems of higher structure concerning “small” types. For example, the three-sphere as a type [MATH] should carry the structure of an |
[MATH] -group, because it is the homotopy type of the Lie group [MATH] . Thus, we should be able to construct the homotopy type of the classifying space [MATH] with [MATH] , but so far we’ve not been able to do so. (We have the H-space structure, which is a first approximation BuchholtzRijke2018 .) In this case, howeve... |
An obvious approach would be to construct in the usual way a simplicial set whose homotopy type is [MATH] . If we then had the realization operation [MATH] that turns a simplicial set into the homotopy type it represents, then we’d be done. But such a realization operation itself seems impossible to construct in elemen... |
As a final, important, but more open-ended problem, let me mention the problem of developing the meta-theory of HoTT inside HoTT/UF. This has two sides, one relatively easy, and one quite hard. The relatively easy side is the syntactic one, but even here there are difficulties. We can represent extrinsic untyped syntax... |
The more difficult side is the semantic one. We want to define interpretations of the intrinsic syntax in inner models, first of all the canonical model where syntactic types [MATH] are mapped to types [MATH] , syntactic terms [MATH] are mapped to terms |
[MATH] , and so on. (For proof-theoretic reasons, we expect to only be able to represent the interpretation locally, for instance type theory with [MATH] universes inside the [MATH] st universe, or using stronger principles in the target type theory than are in the source type theory.) Shulman has called this problem “... |
suggests itself. The problem is that if we use the QIT intrinsic syntax, then everything syntactic is a (homotopy) set and the elimination rule will only allow us to eliminate into sets, whereas for the canonical model we’re eliminating into [MATH] . And if we try to formalize the intrinsic syntax using a non-truncated... |
Possible further means of construction Now that we have seen concretely both the range of constructions that are currently possible in (elementary) HoTT, and some prominent problems that seem out of reach, let us take stock. |
The first conclusion is that we’d very much like to prove that the problem of semi-simplicial types cannot be solved in elementary HoTT. But assuming that, the next conclusion is that elementary HoTT is by itself too incomplete to serve its foundational role as the basis for UF. Further means of constructions need to b... |
In some sense the situation is analogous to the question of new axioms for set theory, but there are two main differences: First, we want to use type theory as a programming language and that means that for any proposed extension, we should say how the new constructions |
compute when combined with the other constructions of type theory. The univalence axiom is a sore point in this regard, as it has been a long-standing open problem to give it a computational meaning. This is now close to being solved via various cubical type theories |
Angiuli-Harper-Wilson2017 Bezem-Coquand-Huber2014 CCHM2018 but there remains a question of whether the corresponding model structures on various categories of cubical sets model |
[MATH] -groupoids (we know that the test model structures do BuchholtzMorehouse2017 ). It is still completely open whether the propositional resizing axiom can be given computational meaning. |
Secondly, we want to use HoTT also in other models than [MATH] -groupoids. It is conjectured that elementary HoTT can be interpreted in any [MATH] -topos: a left-exact localization of the functor category [MATH] for a small [MATH] -category |
[MATH] . (It would take us too far afield to give the exact formulation and up-to-date status of this conjecture; see HoTTWikiTopos .) |
Some of the most interesting targets are given by cohesive [MATH] -toposes , whose objects can be thought of as geometrically structured [MATH] -groupoids. (See also Schreiber’s Chapter.) For example, in the cohesive [MATH] -topos of smooth [MATH] -groupoids, |
[MATH] , we find all smooth manifolds among the [MATH] -truncated objects. And we certainly want to be able to be able to reason about smooth [MATH] -categories using a HoTT interpreted in |
[MATH] . So it will not do to propose a construction of the type [MATH] that can’t be performed meaningfully in any [MATH] -topos. |
In contrast, for the problem of interpreting type theory in internal models, including these cohesive [MATH] -toposes, we need not require that the means of doing so themselves are available in arbitrary models. It seems sufficient to be able to do this “at the top level”. But however we solve this problem, it should p... |
Summing up, we expect there to be a stratification of homotopy type theories, • HoTT ua MLTT plus the univalence axiom and propositional resizing, |
• HoTT el HoTT ua plus pushouts, • HoTT el++ HoTT el plus constructions needed for [MATH] • HoTT UF HoTT el++ plus reflective constructions. |
Here, HoTT ua is the basis for Voevodsky’s UniMath formalization effort Voevodsky2015 UniMath . We know that HoTT el is strictly stronger (in the sense of having fewer models; not in the sense of proof-theoretic strength), because it is consistent with HoTT ua that the [MATH] th universe is an [MATH] -type, while if th... |
[MATH] nor for autophagy, it is still conceivable that we can take HoTT el++ and HoTT UF to be HoTT el For each of these we can consider adding classical axioms such as the law of the excluded middle (LEM) or the axiom of choice (AC). These can be seen as constructions that we don’t know how to perform in general, but ... |
We can also remove the resizing axiom to get (generalized) predicative systems, and for both the predicative and impredicative systems we may add various generalized inductive types to increase the proof-theoretic strength if needed, while keeping the systems constructive and without changing their class of [MATH] -top... |
With all that in mind, let us discuss some possible further means of construction that we might add either for HoTT el or HoTT UF |
4.1 Simplicial type theory For any [MATH] -topos [MATH] , we can consider the simplicial objects in [MATH] , i.e., functors [MATH] , and this is again an [MATH] -topos. As mentioned above, we find therein a full subcategory of [MATH] -categories relative to [MATH] . This is the basis for the suggestion by Riehl and Shu... |
for a synthetic type theory for [MATH] -categories. In their type theory, let’s call it sHoTT, types are interpreted as simplicial types , and they give definitions for Segal and Rezk types with the latter representing [MATH] -categories. They can also define a type of discrete simplicial types, representing ordinary t... |
Much work remains before we can judge how useful this type theory is for reasoning about [MATH] -categories. But from a philosophical point of view it cannot be satisfactory to view sHoTT as a foundational theory, for instance playing the role of HoTT UF . (And it is of course not intended as such!) Because simplicial ... |
4.2 Two-level type theories Another approach to solving the problem of defining simplicial types is to have another layer above the univalent type theory in which to reason about infinitary strict constructions, including (semi-)simplicial types. One proposal is Voevodsky’s Homotopy Type System (HTS) Voevodsky2013 . Th... |
Another proposal is the two-level type theory (2LTT) AltenkirchCapriottiKraus2016 AnnenkovCapriottiKraus2017 . This is similar to HTS in that it distinguishes between fibrant and non-fibrant types (the latter are called pretypes , but instead of the reflection rule for the strict equality type it adds the rule for uniq... |
In order to define simplicial types in 2LTT an extra principle beyond the basic set-up is needed. This can be the assumption that the fibrant and non-fibrant natural numbers coincide, or the more technical assumption that Reedy fibrant diagrams of fibrant types indexed by a strict Reedy category have fibrant limits. Th... |
However useful the two-level type theories may turn out to be, they also seem unsatisfactory from a foundational perspective. Because what is a pretype? Pretypes can only be motivated via the models of HoTT as described in set-theoretic mathematics, where they arise as the objects of a model category presenting an [MAT... |
4.3 Computational type theories If we limit ourselves to one constructive model, then there is a principled way of making sense of new constructions. This is via the paradigm of computational type theories in the Nuprl tradition Nuprl-book . Here we consider a particular model to give meaning explanations for the judgm... |
4.4 Presentation axioms It may have perhaps occurred to some readers that the problems discussed in Sect. should be solved in the same way that they are solved in homotopical mathematics based in set-theory, namely by working with set-based presentations. |
We already mentioned geometric realization, an operation that produces the underlying homotopy type of a given simplicial set or topological space. We could consider adding [MATH] as a basic construction and the axiom stating that [MATH] is surjective, meaning that every type is merely equivalent to the geometric reali... |
Something like this may indeed be appropriate at the level of HoTT UF if it could be given a computational meaning. But certainly not at the level of HoTT el++ , because the axioms would severely restrict the range of models (they are constructive-homotopical taboos). |
Even the much weaker axiom sets cover (SC), stating that every type admits a surjection from a set admits a simple counter-model |
[MATH] -topos nLabTypesCover On the other hand, SC (or something like it) is necessary in order to describe the semantics of HoTT (with universes) in presheaf toposes. Indeed, the universe in a presheaf topos is built from certain sets covering the [MATH] -types of presheaves of small sets. |
Conclusion Higher structures are at once the raison d’être and so far, the Achilles’ heel, of HoTT/UF from a foundational perspective. HoTT can handle with ease many important higher structures, such as the |
[MATH] -type of sets and the [MATH] -type of categories, that can only imperfectly be represented in other foundational systems. But so far it cannot define the (untruncated) type of [MATH] -categories, and this is a major impediment to the foundational aspirations of HoTT/UF. Of course, HoTT can be (and has been) used... |
To be foundational, however, we need to find a compelling construction of, and theory of, [MATH] -categories, and of the semantics of HoTT-DSLs, inside homotopy type theory itself. It appears that new methods of construction are needed, but it is at this time not clear what they should be. |
A dramatic possibility, not mentioned in Sect. , is that we should take [MATH] -categories to be fundamental after all, and build a synthetic type theory where the types are |
[MATH] -categories rather than [MATH] -groupoids. This would be a directed type theory. Such a thing would undoubtedly be quite complicated due to the need to keep track of variances (see LicataHarper2011 Nuyts2015 for some preliminary attempts), and it would represent a return to the old ways of thinking about categor... |
[MATH] -categories all of whose morphisms are invertible rather than trying to build [MATH] -categories out of [MATH] -groupoids. In any case, directed type theories should be useful also as DSLs for reasoning about [MATH] -toposes. |
Personally, I think we’ll find some solution that allows us to stay at the level of [MATH] -groupoids for the foundational theory. Perhaps there is a kind of two-level type theory that allows us to capture the strict nature of the [MATH] -category of types without postulating a bunch of meaningless pretypes. |
An analogy can perhaps be made with the foundations of stable homotopy theory. The [MATH] -category of spectra is a symmetric monoidal stable [MATH] -category, and from a foundational point of view, this is the correct viewpoint, since spectra should be identified when they are weakly equivalent. However, it was discov... |
[MATH] -category can be presented by symmetric monoidal Quillen model categories (i.e., very strict structures), and this has been very important in facilitating computations in stable homotopy theory EKMM1995 MMSS2001 . (I should mention that there is work-in-progress by Finster-Licata-Morehouse-Riley on developing a ... |
[MATH] -topos of parametrized spectra: this captures the strict monoidal structure of spectra in a type theory.) And so it may be, that in order to realize the foundational potential of HoTT/UF, we shall need to capture the strict structure of type theory itself, perhaps by reflecting more of judgmental structure at th... |
I’m confident that a good solution will be eventually found. The field is still young, and it will be exciting to see what the future brings. |
# Source: arxiv 1807.02207 # Title: Weakly remarkable cardinals, Erdős cardinals, and the generic Vopěnka principle # Sections: all # Downloaded: 2026-03-03T02:37:12.792780+00:00 |
Weakly remarkable cardinals, Erdős cardinals, and the generic Vopěnka principle Abstract. We consider a weak version of Schindler’s remarkable cardinals that may fail to be [MATH] -reflecting. We show that the [MATH] -reflecting weakly remarkable cardinals are exactly the remarkable cardinals, and we show that the exis... |
1. Remarkability and weak remarkability Many large cardinal properties can be defined in terms of elementary embeddings between set-sized structures. For example, extendibility is defined in terms of elementary embeddings between rank initial segments of [MATH] , and supercompactness admits a similar characterization b... |
.) The first virtual large cardinals to be studied were the virtually supercompact cardinals, also known as the remarkable cardinals: |
Definition 1.1 (Schindler A cardinal [MATH] is remarkable if for every ordinal [MATH] there is an ordinal [MATH] and a generic elementary embedding [MATH] such that [MATH] |
We will consider a weak form of remarkability obtained by removing the condition [MATH] analogous to the weak form of virtual extendibility defined by Gitman and Hamkins , Definition 6] . We work in ZFC unless otherwise stated. |
Definition 1.2 A cardinal [MATH] is weakly remarkable if for every ordinal [MATH] there is an ordinal [MATH] and a generic elementary embedding [MATH] such that [MATH] |
In terms of consistency strength, remarkable cardinals and weakly remarkable cardinals are between ineffable cardinals and [MATH] -Erdős cardinals. If there is an [MATH] -Erdős cardinal then there is a transitive set model of ZFC + “there is a remarkable cardinal” by Schindler 11 , Lemma 1.2] , and if [MATH] is weakly ... |
The consistency strength of remarkable cardinals and weakly remarkable cardinals can be described more precisely in terms of the hierarchy of [MATH] -iterable cardinals defined by Gitman |
: they are between 1-iterable cardinals and 2-iterable cardinals. See Gitman and Welch for more information on [MATH] -iterable cardinals. |
A cardinal [MATH] is called [MATH] -reflecting if it is inaccessible and [MATH] . This definition is particularly natural in the case [MATH] : the [MATH] statements about a parameter [MATH] are the statements that can be expressed in the form “there is an ordinal [MATH] such that [MATH] ” where [MATH] is a formula in t... |
If [MATH] is a remarkable cardinal then for every ordinal [MATH] and every set [MATH] there is an ordinal [MATH] and a generic elementary embedding [MATH] such that [MATH] and having the additional property that [MATH] : see Bagaria, Gitman, and Schindler , Propositions 2.3 and 3.2] . The same argument establishes the ... |
The following result, proved in Section , says that the [MATH] -reflecting weakly remarkable cardinals are precisely the remarkable cardinals: |
Theorem 1.3 For every cardinal [MATH] , the following statements are equivalent. (1) [MATH] is remarkable. (2) [MATH] is weakly remarkable and [MATH] -reflecting. |
By contrast, the existence of a non- [MATH] -reflecting weakly remarkable cardinal has higher consistency strength than the existence of a remarkable cardinal: we will show that it is equiconsistent with the existence of an [MATH] -Erdős cardinal. (This is an unusual situation. More typically for a large cardinal prope... |
Following Baumgartner , we say that an infinite cardinal [MATH] is [MATH] -Erdős if for every club [MATH] in [MATH] and every function [MATH] that is regressive, meaning that [MATH] for all [MATH] in the domain of [MATH] , there is a subset [MATH] of order type [MATH] that is homogeneous for [MATH] , meaning that [MATH... |
We will not directly use the definition of [MATH] -Erdős cardinals in terms of club sets and regressive functions, only the following consequences of the definition. First, every [MATH] -Erdős cardinal is inaccessible. Second, if [MATH] is an [MATH] -Erdős cardinal then [MATH] for every cardinal [MATH] . Third, if [MAT... |
The following two results describe the relationship between [MATH] -Erdős cardinals and non- [MATH] -reflecting weakly remarkable cardinals. They will also be proved in Section |
Theorem 1.4 Every [MATH] -Erdős cardinal is a limit of non- [MATH] -reflecting weakly remarkable cardinals. Theorem 1.5 If [MATH] is a non- [MATH] -reflecting weakly remarkable cardinal, then some ordinal greater than [MATH] is an [MATH] -Erdős cardinal in [MATH] |
We obtain the following immediate consequence: Corollary 1.6 The following statements are equiconsistent modulo ZFC and are equivalent modulo ZFC + [MATH] |
(1) There is an [MATH] -Erdős cardinal. (2) There is a non- [MATH] -reflecting weakly remarkable cardinal. I don’t know if the two statements in Corollary 1.6 are equivalent in ZFC: |
Question 1.7 Does the existence of a non- [MATH] -reflecting weakly remarkable cardinal imply the existence of an [MATH] -Erdős cardinal, provably in ZFC? |
Because the existence of an [MATH] -Erdős cardinal has higher consistency strength than the existence of a remarkable cardinal, it follows from Theorems 1.3 and 1.5 that the two theories ZFC + “there is a weakly remarkable cardinal” and ZFC + “there is a remarkable cardinal” are equiconsistent. The following result sho... |
Corollary 1.8 The following statements are equiconsistent modulo ZFC. (1) There is an [MATH] -Erdős cardinal. (2) There is a weakly remarkable cardinal and there is no [MATH] -reflecting cardinal. |
(3) There is a weakly remarkable cardinal and there is no remarkable cardinal. Proof. [MATH] implies [MATH] Assume there is an [MATH] -Erdős cardinal. Passing from [MATH] to [MATH] where [MATH] is the least [MATH] -reflecting cardinal if it exists, we may assume there is no [MATH] -reflecting cardinal. Because the exis... |
Statement implies statement because remarkable cardinals are [MATH] -reflecting. [MATH] implies [MATH] If statement holds then there is a weakly remarkable cardinal that is not remarkable, and therefore is not [MATH] -reflecting by Theorem 1.3 , so there is an [MATH] -Erdős cardinal in [MATH] by Theorem 1.5 |
In Section we will prove Theorems 1.3 1.4 , and 1.5 . In Section we will give an application involving the generic Vopěnka principle defined by Bagaria, Gitman, and Schindler |
2. Proof of Theorems 1.3 1.4 , and 1.5 We will need the following local forms of remarkabilty and weak remarkability. Definition 2.1 |
Let [MATH] be a cardinal and let [MATH] be an ordinal. (1) [MATH] is [MATH] -remarkable if there is an ordinal [MATH] and a generic elementary embedding [MATH] such that [MATH] |
(2) [MATH] is weakly [MATH] -remarkable if there is an ordinal [MATH] and a generic elementary embedding [MATH] such that [MATH] |
By definition, [MATH] is remarkable if and only if it is [MATH] -remarkable for every ordinal [MATH] , and [MATH] is weakly remarkable if and only if it is weakly [MATH] -remarkable for every ordinal [MATH] |
Definition 2.2 Let [MATH] be a cardinal and let [MATH] be an ordinal. (1) [MATH] is [MATH] -remarkable if it is [MATH] -remarkable for every ordinal [MATH] |
with [MATH] (2) [MATH] is weakly [MATH] -remarkable if it is weakly [MATH] -remarkable for every ordinal [MATH] with [MATH] By a well-known absoluteness lemma (see Bagaria, Gitman, and Schindler , Lemma 2.6] ) if an elementary embedding [MATH] such that [MATH] exists in some generic extension of [MATH] and [MATH] is a ... |
Proof of Theorem 1.3 It is clear that every remarkable cardinal is weakly remarkable and [MATH] -reflecting. Conversely, suppose that [MATH] is weakly remarkable and [MATH] -reflecting. We will show that [MATH] is [MATH] -remarkable for every ordinal [MATH] by induction on [MATH] . Let [MATH] and assume that [MATH] is ... |
[EQUATION] where [MATH] . If [MATH] then the restriction [MATH] witnesses that [MATH] is [MATH] -remarkable and we are done. Therefore we suppose that [MATH] |
The fact that [MATH] is [MATH] -remarkable is absolute to [MATH] , so by the elementary of [MATH] the model [MATH] satisfies “ [MATH] is [MATH] -remarkable” and it follows that [MATH] really is [MATH] -remarkable. Then [MATH] is [MATH] -remarkable because [MATH] . Equivalently, [MATH] is remarkable in [MATH] . Because ... |
Next we will prove Theorem 1.4 . In fact we will prove a stronger result in terms of the following definition. Definition 2.3 (Gitman and Hamkins , Definition 6] |
Let [MATH] be a cardinal and let [MATH] be a class. Then [MATH] is weakly virtually [MATH] -extendible if for every ordinal [MATH] there is an ordinal [MATH] and a generic elementary embedding [MATH] with [MATH] |
This definition can be used in the context of GB + AC, meaning Gödel–Bernays set theory with the axiom of choice but without the axiom of global choice. Any model of ZFC together with its definable (from parameters) classes gives a model of GB + AC, but there may be models of GB + AC with classes that are not definable... |
A cardinal is called weakly virtually extendible if it is weakly virtually [MATH] -extendible, meaning simply that for every ordinal [MATH] there is an ordinal [MATH] and a generic elementary embedding [MATH] with [MATH] . The following lemma is similar to the fact that every extendible cardinal is supercompact, which ... |
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