Autoformalizing the calculation of $\pi_3(S^2)$
We report our findings and the results of our experiment of autoformalizing the homotopy type theoretic computation of $\pi_3(S^2)$ in agda-unimath using Codex.
math.AT↗
arXiv subjects
Egbert Rijke
Publications and source records attributed to Egbert Rijke.
Central H-spaces and banded types
We introduce and study central types, which are generalizations of Eilenberg-Mac Lane spaces. A type is central when it is equivalent to the component of the identity among its own self-equivalences. From centrality alone we construct an infinite delooping in terms of a tensor product of banded types, which are the appropriate notion of torsor for a central type. Our constructions are carried out in homotopy type theory, and therefore hold in any $\infty$-topos. Even when interpreted into the $\infty$-topos of spaces, our approach to constructing these deloopings is new. Along the way, we further develop the theory of H-spaces in homotopy type theory, including their relation to evaluation fibrations and Whitehead products. These considerations let us, for example, rule out the existence of H-space structures on the $2n$-sphere for $n > 0$. We also give a novel description of the moduli space of H-space structures on an H-space. Using this description, we generalize a formula of Arkowitz-Curjel and Copeland for counting the number of path components of this moduli space. As an application, we deduce that the moduli space of H-space structures on the $3$-sphere is $Ω^6 \mathbb{S}^3$.
math.AT↗
Algebraic Presentations of Type Dependency
C-systems were defined by Cartmell as the algebraic structures that correspond exactly to generalised algebraic theories. B-systems were defined by Voevodsky in his quest to formulate and prove an initiality conjecture for type theories. They play a crucial role in Voevodsky's construction of a syntactic C-system from a term monad. In this work, we construct an equivalence between the category of C-systems and the category of B-systems, thus proving a conjecture by Voevodsky. We construct this equivalence as the restriction of an equivalence between more general structures, called CE-systems and E-systems, respectively. To this end, we identify C-systems and B-systems as "stratified" CE-systems and E-systems, respectively; that is, systems whose contexts are built iteratively via context extension, starting from the empty context.
math.CT↗
Epimorphisms and Acyclic Types in Univalent Foundations
We characterize the epimorphisms in homotopy type theory (HoTT) as the fiberwise acyclic maps and develop a type-theoretic treatment of acyclic maps and types in the context of synthetic homotopy theory as developed in univalent foundations. We present examples and applications in group theory, such as the acyclicity of the Higman group, through the identification of groups with 0-connected, pointed 1-types. Many of our results are formalized as part of the agda-unimath library.
cs.LO↗
Delooping the sign homomorphism in univalent mathematics
In univalent mathematics there are at least two equivalent ways to present the category of groups. Groups presented in their usual algebraic form are called abstract groups, and groups presented as pointed connected $1$-types are called concrete groups. Since these two descriptions of the category of groups are equivalent, we find that every algebraic group corresponds uniquely to a concrete group -- its delooping -- and that each abstract group homomorphisms corresponds uniquely to a pointed map between concrete groups. The $n$-th abstract symmetric group $S_n$ of all bijections $[n]\simeq [n]$, for instance, corresponds to the concrete group of all $n$-element types. The sign homomorphism from $S_n$ to $S_2$ should therefore correspond to a pointed map from the type $BS_n$ of all $n$-element types to the type $BS_2$ of all $2$-element types. Making use of the univalence axiom, we characterize precisely when a pointed map $BS_n\to_\ast BS_2$ is a delooping of the sign homomorphism. Then we proceed to give several constructions of the delooping of the sign homomorphism. Notably, the construction following a method of Cartier can be given without reference to the sign homomorphism. Our results are formalized in the agda-unimath library.
math.GR↗
Introduction to Homotopy Type Theory
This is an introductory textbook to univalent mathematics and homotopy type theory, a mathematical foundation that takes advantage of the structural nature of mathematical definitions and constructions. It is common in mathematical practice to consider equivalent objects to be the same, for example, to identify isomorphic groups. In set theory it is not possible to make this common practice formal. For example, there are as many distinct trivial groups in set theory as there are distinct singleton sets. Type theory, on the other hand, takes a more structural approach to the foundations of mathematics that accommodates the univalence axiom. This, however, requires us to rethink what it means for two objects to be equal. This textbook introduces the reader to Martin-Löf's dependent type theory, to the central concepts of univalent mathematics, and shows the reader how to do mathematics from a univalent point of view. Over 200 exercises are included to train the reader in type theoretic reasoning. The book is entirely self-contained, and in particular no prior familiarity with type theory or homotopy theory is assumed.
math.LO↗
Characterizations of modalities and lex modalities
A reflective subuniverse in homotopy type theory is an internal version of the notion of a localization in topology or in the theory of $\infty$-categories. Working in homotopy type theory, we give new characterizations of the following conditions on a reflective subuniverse $L$: (1) the associated subuniverse $L'$ of $L$-separated types is a modality; (2) $L$ is a modality; (3) $L$ is a lex modality; and (4) $L$ is a cotopological modality. In each case, we give several necessary and sufficient conditions. Our characterizations involve various families of maps associated to $L$, such as the $L$-étale maps, the $L$-equivalences, the $L$-local maps, the $L$-connected maps, the unit maps $η_X$, and their left and/or right orthogonal complements. More generally, our main theorem gives an overview of how all of these classes related to each other. We also give examples that show that all of the inclusions we describe between these classes of maps can be strict.
math.CT↗
The long exact sequence of homotopy $n$-groups
Working in homotopy type theory, we introduce the notion of $n$-exactness for a short sequence $F\to E\to B$ of pointed types, and show that any fiber sequence $F\hookrightarrow E \twoheadrightarrow B$ of arbitrary types induces a short sequence $\|F\|_{n-1} \to \|E\|_{n-1} \to \|B\|_{n-1}$ that is $n$-exact at $\|E\|_{n-1}$. We explain how the indexing makes sense when interpreted in terms of $n$-groups, and we compare our definition to the existing definitions of an exact sequence of $n$-groups for $n=1,2$. As the main application, we obtain the long $n$-exact sequence of homotopy $n$-groups of a fiber sequence.
math.AT↗
Modal Descent
Any modality in homotopy type theory gives rise to an orthogonal factorization system of which the left class is stable under pullbacks. We show that there is a second orthogonal factorization system associated to any modality, of which the left class is the class of $\bigcirc$-equivalences and the right class is the class of $\bigcirc$-étale maps. This factorization system is called the reflective factorization system of a modality, and we give a precise characterization of the orthogonal factorization systems that arise as the reflective factorization system of a modality. In the special case of the $n$-truncation the reflective factorization system has a simple description: we show that the $n$-étale maps are the maps that are right orthogonal to the map $\mathbf{1} \to \mathbf{S}^{n+1}$. We use the $\bigcirc$-étale maps to prove a modal descent theorem: a map with modal fibers into $\bigcirc X$ is the same thing as a $\bigcirc$-étale map into a type $X$. We conclude with an application to real-cohesive homotopy type theory and remarks how $\bigcirc$-étale maps relate to the formally etale maps from algebraic geometry.
math.CT↗
Localization in Homotopy Type Theory
We study localization at a prime in homotopy type theory, using self maps of the circle. Our main result is that for a pointed, simply connected type $X$, the natural map $X \to X_{(p)}$ induces algebraic localizations on all homotopy groups. In order to prove this, we further develop the theory of reflective subuniverses. In particular, we show that for any reflective subuniverse $L$, the subuniverse of $L$-separated types is again a reflective subuniverse, which we call $L'$. Furthermore, we prove results establishing that $L'$ is almost left exact. We next focus on localization with respect to a map, giving results on preservation of coproducts and connectivity. We also study how such localizations interact with other reflective subuniverses and orthogonal factorization systems. As key steps towards proving the main theorem, we show that localization at a prime commutes with taking loop spaces for a pointed, simply connected type, and explicitly describe the localization of an Eilenberg-Mac Lane space $K(G,n)$ with $G$ abelian. We also include a partial converse to the main theorem.
math.AT↗
Modalities in homotopy type theory
Univalent homotopy type theory (HoTT) may be seen as a language for the category of $\infty$-groupoids. It is being developed as a new foundation for mathematics and as an internal language for (elementary) higher toposes. We develop the theory of factorization systems, reflective subuniverses, and modalities in homotopy type theory, including their construction using a "localization" higher inductive type. This produces in particular the ($n$-connected, $n$-truncated) factorization system as well as internal presentations of subtoposes, through lex modalities. We also develop the semantics of these constructions.
math.CT↗
Classifying Types
The study of homotopy theoretic phenomena in the language of type theory is sometimes loosely called `synthetic homotopy theory'. Homotopy theory in type theory is only one of the many aspects of homotopy type theory, which also includes the study of the set theoretic semantics (models of homotopy type theory and univalence in a meta-theory of sets or categories), type theoretic semantics (internal models of homotopy type theory), and computational semantics, as well as the study of various questions in the internal language of homotopy type theory which are not necessarily motivated by homotopy theory, or questions related to the development of formalized libraries of mathematics based on homotopy type theory. This thesis concerns the development of synthetic homotopy theory.
math.LO↗
Higher Groups in Homotopy Type Theory
We present a development of the theory of higher groups, including infinity groups and connective spectra, in homotopy type theory. An infinity group is simply the loops in a pointed, connected type, where the group structure comes from the structure inherent in the identity types of Martin-Löf type theory. We investigate ordinary groups from this viewpoint, as well as higher dimensional groups and groups that can be delooped more than once. A major result is the stabilization theorem, which states that if an $n$-type can be delooped $n+2$ times, then it is an infinite loop type. Most of the results have been formalized in the Lean proof assistant.
cs.LO↗
The real projective spaces in homotopy type theory
Homotopy type theory is a version of Martin-Löf type theory taking advantage of its homotopical models. In particular, we can use and construct objects of homotopy theory and reason about them using higher inductive types. In this article, we construct the real projective spaces, key players in homotopy theory, as certain higher inductive types in homotopy type theory. The classical definition of RP(n), as the quotient space identifying antipodal points of the n-sphere, does not translate directly to homotopy type theory. Instead, we define RP(n) by induction on n simultaneously with its tautological bundle of 2-element sets. As the base case, we take RP(-1) to be the empty type. In the inductive step, we take RP(n+1) to be the mapping cone of the projection map of the tautological bundle of RP(n), and we use its universal property and the univalence axiom to define the tautological bundle on RP(n+1). By showing that the total space of the tautological bundle of RP(n) is the n-sphere, we retrieve the classical description of RP(n+1) as RP(n) with an (n+1)-cell attached to it. The infinite dimensional real projective space, defined as the sequential colimit of the RP(n) with the canonical inclusion maps, is equivalent to the Eilenberg-MacLane space K(Z/2Z,1), which here arises as the subtype of the universe consisting of 2-element types. Indeed, the infinite dimensional projective space classifies the 0-sphere bundles, which one can think of as synthetic line bundles. These constructions in homotopy type theory further illustrate the utility of homotopy type theory, including the interplay of type theoretic and homotopy theoretic ideas.
math.AT↗
The join construction
In homotopy type theory we can define the join of maps as a binary operation on maps with a common co-domain. This operation is commutative, associative, and the unique map from the empty type into the common codomain is a neutral element. Moreover, we show that the idempotents of the join of maps are precisely the embeddings, and we prove the `join connectivity theorem', which states that the connectivity of the join of maps equals the join of the connectivities of the individual maps. We define the image of a map $f:A\to X$ in $U$ via the join construction, as the colimit of the finite join powers of $f$. The join powers therefore provide approximations of the image inclusion, and the join connectivity theorem implies that the approximating maps into the image increase in connectivity. A modified version of the join construction can be used to show that for any map $f:A\to X$ in which $X$ is only assumed to be locally small, the image is a small type. We use the modified join construction to give an alternative construction of set-quotients, the Rezk completion of a precategory, and we define the $n$-truncation for any $n:\mathbb{N}$. Thus we see that each of these are definable operations on a univalent universe for Martin-Löf type theory with a natural numbers object, that is moreover closed under homotopy coequalizers.
math.CT↗
The Cayley-Dickson Construction in Homotopy Type Theory
We define in the setting of homotopy type theory an H-space structure on $\mathbb S^3$. Hence we obtain a description of the quaternionic Hopf fibration $\mathbb S^3\hookrightarrow\mathbb S^7\twoheadrightarrow\mathbb S^4$, using only homotopy invariant tools.
math.AT↗
Sets in homotopy type theory
Homotopy Type Theory may be seen as an internal language for the $\infty$-category of weak $\infty$-groupoids which in particular models the univalence axiom. Voevodsky proposes this language for weak $\infty$-groupoids as a new foundation for mathematics called the Univalent Foundations of Mathematics. It includes the sets as weak $\infty$-groupoids with contractible connected components, and thereby it includes (much of) the traditional set theoretical foundations as a special case. We thus wonder whether those `discrete' groupoids do in fact form a (predicative) topos. More generally, homotopy type theory is conjectured to be the internal language of `elementary' $\infty$-toposes. We prove that sets in homotopy type theory form a $ΠW$-pretopos. This is similar to the fact that the $0$-truncation of an $\infty$-topos is a topos. We show that both a subobject classifier and a $0$-object classifier are available for the type theoretical universe of sets. However, both of these are large and moreover, the $0$-object classifier for sets is a function between $1$-types (i.e. groupoids) rather than between sets. Assuming an impredicative propositional resizing rule we may render the subobject classifier small and then we actually obtain a topos of sets.
math.CT↗