The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove …
The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove a theorem about equality types of coequalizers and pushouts, reminiscent of an induction principle and without any restrictions on the truncation levels. This result makes it possible to reason directly about certain equality types and to streamline existing proofs by eliminating the necessity of auxiliary constructions. To demonstrate this, we give a very short argument for the calculation of the fundamental group of the circle (Licata and Shulman [1]), and for the fact that pushouts preserve embeddings. Further, our development suggests a higher version of the Seifert-van Kampen theorem, and the set-truncation operator maps it to the standard Seifert-van Kampen theorem (due to Favonia and Shulman [2]). We provide a formalization of the main technical results in the proof assistant Lean.
Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of …
Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of a cycle are not necessarily closed. This paper derives a principle reminiscent of induction for cycles for the case that the graph is given as the symmetric closure of a locally confluent and (co-)well-founded relation. We show that, assuming the property in question is sufficiently nice, it is enough to prove it for the empty cycle and for cycles given by local confluence.
The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove …
The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed.
We prove a theorem about equality types of coequalizers and pushouts, reminiscent of an induction principle and without any restrictions on the truncation levels. This result makes it possible to reason directly about certain equality types and to streamline existing proofs by eliminating the necessity of auxiliary constructions.
To demonstrate this, we give a very short argument for the calculation of the fundamental group of the circle (Licata and Shulman '13), and for the fact that pushouts preserve embeddings. Further, our development suggests a higher version of the Seifert-van Kampen theorem, and the set-truncation operator maps it to the standard Seifert-van Kampen theorem (due to Favonia and Shulman '16).
We provide a formalization of the main technical results in the proof assistant Lean.
Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. …
Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy basis, with a construction reminiscent of an argument by Craig C. Squier. We then go on to translate this construction to the setting of homotopy type theory, where managing equalities between paths is important in order to construct functions which are coherent with respect to higher dimensions. Eventually, we apply the result to approximate a series of open questions in homotopy type theory, such as the characterisation of the homotopy groups of the free group on a set and the pushout of 1-types. This paper expands on our previous conference contribution "Coherence via Wellfoundedness" (arXiv:2001.07655) by laying out the construction in the language of higher-dimensional rewriting.
These theories extend the existent proof of the first sylow theorem (written by Florian Kammueller and L. C. Paulson) by what is often called the second, third and fourth sylow …
These theories extend the existent proof of the first sylow theorem (written by Florian Kammueller and L. C. Paulson) by what is often called the second, third and fourth sylow theorem. These theorems state propositions about the number of Sylow p-subgroups of a group and the fact that they are conjugate to each other. The proofs make use of an implementation of group actions and their properties.
Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of …
Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of a cycle are not necessarily closed. This paper derives a principle reminiscent of induction for cycles for the case that the graph is given as the symmetric closure of a locally confluent and (co-)well-founded relation. We show that, assuming the property in question is sufficiently nice, it is enough to prove it for the empty cycle and for cycles given by local confluence.
Our motivation and application is in the field of homotopy type theory, which allows us to work with the higher-dimensional structures that appear in homotopy theory and in higher category theory, making coherence a central issue. This is in particular true for quotienting - a natural operation which gives a new type for any binary relation on a type and, in order to be well-behaved, cuts off higher structure (set-truncates). The latter makes it hard to characterise the type of maps from a quotient into a higher type, and several open problems stem from this difficulty.
We prove our theorem on cycles in a type-theoretic setting and use it to show coherence conditions necessary to eliminate from set-quotients into 1-types, deriving approximations to open problems on free groups and pushouts. We have formalised the main result in the proof assistant Lean.
The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove …
The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove a theorem about equality types of coequalizers and pushouts, reminiscent of an induction principle and without any restrictions on the truncation levels. This result makes it possible to reason directly about certain equality types and to streamline existing proofs by eliminating the necessity of auxiliary constructions. To demonstrate this, we give a very short argument for the calculation of the fundamental group of the circle (Licata and Shulman '13), and for the fact that pushouts preserve embeddings. Further, our development suggests a higher version of the Seifert-van Kampen theorem, and the set-truncation operator maps it to the standard Seifert-van Kampen theorem (due to Favonia and Shulman '16). We provide a formalization of the main technical results in the proof assistant Lean.
Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. …
Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy basis, with a construction reminiscent of an argument by Craig C. Squier. We then go on to translate this construction to the setting of homotopy type theory, where managing equalities between paths is important in order to construct functions which are coherent with respect to higher dimensions. Eventually, we apply the result to approximate a series of open questions in homotopy type theory, such as the characterisation of the homotopy groups of the free group on a set and the pushout of 1-types. This paper expands on our previous conference contribution "Coherence via Wellfoundedness" (arXiv:2001.07655) by laying out the construction in the language of higher-dimensional rewriting.
Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. …
Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy basis, with a construction reminiscent of an argument by Craig C. Squier. We then go on to translate this construction to the setting of homotopy type theory, where managing equalities between paths is important in order to construct functions which are coherent with respect to higher dimensions. Eventually, we apply the result to approximate a series of open questions in homotopy type theory, such as the characterisation of the homotopy groups of the free group on a set and the pushout of 1-types. This paper expands on our previous conference contribution "Coherence via Wellfoundedness" (arXiv:2001.07655) by laying out the construction in the language of higher-dimensional rewriting.
Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. …
Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy basis, with a construction reminiscent of an argument by Craig C. Squier. We then go on to translate this construction to the setting of homotopy type theory, where managing equalities between paths is important in order to construct functions which are coherent with respect to higher dimensions. Eventually, we apply the result to approximate a series of open questions in homotopy type theory, such as the characterisation of the homotopy groups of the free group on a set and the pushout of 1-types. This paper expands on our previous conference contribution "Coherence via Wellfoundedness" (arXiv:2001.07655) by laying out the construction in the language of higher-dimensional rewriting.
Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of …
Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of a cycle are not necessarily closed. This paper derives a principle reminiscent of induction for cycles for the case that the graph is given as the symmetric closure of a locally confluent and (co-)well-founded relation. We show that, assuming the property in question is sufficiently nice, it is enough to prove it for the empty cycle and for cycles given by local confluence.
Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of …
Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of a cycle are not necessarily closed. This paper derives a principle reminiscent of induction for cycles for the case that the graph is given as the symmetric closure of a locally confluent and (co-)well-founded relation. We show that, assuming the property in question is sufficiently nice, it is enough to prove it for the empty cycle and for cycles given by local confluence.
Our motivation and application is in the field of homotopy type theory, which allows us to work with the higher-dimensional structures that appear in homotopy theory and in higher category theory, making coherence a central issue. This is in particular true for quotienting - a natural operation which gives a new type for any binary relation on a type and, in order to be well-behaved, cuts off higher structure (set-truncates). The latter makes it hard to characterise the type of maps from a quotient into a higher type, and several open problems stem from this difficulty.
We prove our theorem on cycles in a type-theoretic setting and use it to show coherence conditions necessary to eliminate from set-quotients into 1-types, deriving approximations to open problems on free groups and pushouts. We have formalised the main result in the proof assistant Lean.
The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove …
The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove a theorem about equality types of coequalizers and pushouts, reminiscent of an induction principle and without any restrictions on the truncation levels. This result makes it possible to reason directly about certain equality types and to streamline existing proofs by eliminating the necessity of auxiliary constructions. To demonstrate this, we give a very short argument for the calculation of the fundamental group of the circle (Licata and Shulman [1]), and for the fact that pushouts preserve embeddings. Further, our development suggests a higher version of the Seifert-van Kampen theorem, and the set-truncation operator maps it to the standard Seifert-van Kampen theorem (due to Favonia and Shulman [2]). We provide a formalization of the main technical results in the proof assistant Lean.
The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove …
The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed.
We prove a theorem about equality types of coequalizers and pushouts, reminiscent of an induction principle and without any restrictions on the truncation levels. This result makes it possible to reason directly about certain equality types and to streamline existing proofs by eliminating the necessity of auxiliary constructions.
To demonstrate this, we give a very short argument for the calculation of the fundamental group of the circle (Licata and Shulman '13), and for the fact that pushouts preserve embeddings. Further, our development suggests a higher version of the Seifert-van Kampen theorem, and the set-truncation operator maps it to the standard Seifert-van Kampen theorem (due to Favonia and Shulman '16).
We provide a formalization of the main technical results in the proof assistant Lean.
The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove …
The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove a theorem about equality types of coequalizers and pushouts, reminiscent of an induction principle and without any restrictions on the truncation levels. This result makes it possible to reason directly about certain equality types and to streamline existing proofs by eliminating the necessity of auxiliary constructions. To demonstrate this, we give a very short argument for the calculation of the fundamental group of the circle (Licata and Shulman '13), and for the fact that pushouts preserve embeddings. Further, our development suggests a higher version of the Seifert-van Kampen theorem, and the set-truncation operator maps it to the standard Seifert-van Kampen theorem (due to Favonia and Shulman '16). We provide a formalization of the main technical results in the proof assistant Lean.
These theories extend the existent proof of the first sylow theorem (written by Florian Kammueller and L. C. Paulson) by what is often called the second, third and fourth sylow …
These theories extend the existent proof of the first sylow theorem (written by Florian Kammueller and L. C. Paulson) by what is often called the second, third and fourth sylow theorem. These theorems state propositions about the number of Sylow p-subgroups of a group and the fact that they are conjugate to each other. The proofs make use of an implementation of group actions and their properties.
Given a type A in homotopy type theory (HoTT), we can define the free ∞-group on A as the loop space of the suspension of A + 1. Equivalently, this …
Given a type A in homotopy type theory (HoTT), we can define the free ∞-group on A as the loop space of the suspension of A + 1. Equivalently, this free higher group can be defined as a higher inductive type F(A) with constructors unit: F(A), cons: A~F(A)~F(A), and conditions saying that every cons(a) is an auto-equivalence on F(A). Assuming that A is a set (i.e. satisfies the principle of unique identity proofs), we are interested in the question whether F(A) is a set as well, which is very much related to an open problem in the HoTT book [22, Ex. 8.2]. We show an approximation to the question, namely that the fundamental groups of F(A) are trivial, i.e. that ||F(A)||1 is a set.
We present Voevodsky's construction of a model of univalent type theory in the category of simplicial sets. To this end, we first give a general technique for constructing categorical models …
We present Voevodsky's construction of a model of univalent type theory in the category of simplicial sets.
To this end, we first give a general technique for constructing categorical models of dependent type theory, using universes to obtain coherence. We then construct a (weakly) universal Kan fibration, and use it to exhibit a model in simplicial sets. Lastly, we introduce the Univalence Axiom, in several equivalent formulations, and show that it holds in our model.
As a corollary, we conclude that Martin-Lof type theory with one univalent universe (formulated in terms of contextual categories) is at least as consistent as ZFC with two inaccessible cardinals.
Homotopy type theory is a recent research area connecting type theory with homotopy theory by interpreting types as spaces. In particular, one can prove and mechanize type-theoretic analogues of homotopy-theoretic …
Homotopy type theory is a recent research area connecting type theory with homotopy theory by interpreting types as spaces. In particular, one can prove and mechanize type-theoretic analogues of homotopy-theoretic theorems, yielding homotopy theory. Here we consider the Seifert-van Kampen theorem, which characterizes the loop structure of spaces obtained by gluing. This is useful in homotopy theory because many spaces are constructed by gluing, and the loop structure helps distinguish distinct spaces. The synthetic proof showcases many new characteristics of synthetic homotopy theory, such as the encode-decode method, enforced homotopy-invariance, and lack of underlying sets.
This paper presents a novel connection between homotopical algebra and mathematical logic. It is shown that a form of intensional type theory is valid in any Quillen model category, generalizing …
This paper presents a novel connection between homotopical algebra and mathematical logic. It is shown that a form of intensional type theory is valid in any Quillen model category, generalizing the Hofmann-Streicher groupoid model of Martin-Loef type theory.
We develop category theory within Univalent Foundations, which is a foundational system for mathematics based on a homotopical interpretation of dependent type theory. In this system, we propose a definition …
We develop category theory within Univalent Foundations, which is a foundational system for mathematics based on a homotopical interpretation of dependent type theory. In this system, we propose a definition of ‘category’ for which equality and equivalence of categories agree. Such categories satisfy a version of the univalence axiom, saying that the type of isomorphisms between any two objects is equivalent to the identity type between these objects; we call them ‘saturated’ or ‘univalent’ categories. Moreover, we show that any category is weakly equivalent to a univalent one in a universal way. In homotopical and higher-categorical semantics, this construction corresponds to a truncated version of the Rezk completion for Segal spaces, and also to the stack completion of a prestack.
In homotopy type theory, we construct the propositional truncation as a colimit, using only non-recursive higher inductive types (HITs). This is a first step towards reducing recursive HITs to non-recursive …
In homotopy type theory, we construct the propositional truncation as a colimit, using only non-recursive higher inductive types (HITs). This is a first step towards reducing recursive HITs to non-recursive HITs. This construction gives a characterization of functions from the propositional truncation to an arbitrary type, extending the universal property of the propositional truncation. We have fully formalized all the results in a new proof assistant, Lean.
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 …
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.
We define a notion of weak ω-category internal to a model of Martin-Löf's type theory, and prove that each type bears a canonical weak ω-category structure obtained from the tower …
We define a notion of weak ω-category internal to a model of Martin-Löf's type theory, and prove that each type bears a canonical weak ω-category structure obtained from the tower of iterated identity types over that type. We show that the ω-categories arising in this way are in fact ω-groupoids.
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, …
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.
The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove …
The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove a theorem about equality types of coequalizers and pushouts, reminiscent of an induction principle and without any restrictions on the truncation levels. This result makes it possible to reason directly about certain equality types and to streamline existing proofs by eliminating the necessity of auxiliary constructions. To demonstrate this, we give a very short argument for the calculation of the fundamental group of the circle (Licata and Shulman [1]), and for the fact that pushouts preserve embeddings. Further, our development suggests a higher version of the Seifert-van Kampen theorem, and the set-truncation operator maps it to the standard Seifert-van Kampen theorem (due to Favonia and Shulman [2]). We provide a formalization of the main technical results in the proof assistant Lean.
We consider the problem of defining the integers in Homotopy Type Theory (HoTT). We can define the type of integers as signed natural numbers (i.e., using a coproduct), but its …
We consider the problem of defining the integers in Homotopy Type Theory (HoTT). We can define the type of integers as signed natural numbers (i.e., using a coproduct), but its induction principle is very inconvenient to work with, since it leads to an explosion of cases. An alternative is to use set-quotients, but here we need to use set-truncation to avoid non-trivial higher equalities. This results in a recursion principle that only allows us to define function into sets (types satisfying UIP). In this paper we consider higher inductive types using either a small universe or bi-invertible maps. These types represent integers without explicit set-truncation that are equivalent to the usual coproduct representation. This is an interesting example since it shows how some coherence problems can be handled in HoTT. We discuss some open questions triggered by this work. The proofs have been formally verified using cubical Agda.
Higher inductive types (HITs) in homotopy type theory are a powerful generalization of inductive types. Not only can they have ordinary constructors to define elements, but also higher constructors to …
Higher inductive types (HITs) in homotopy type theory are a powerful generalization of inductive types. Not only can they have ordinary constructors to define elements, but also higher constructors to define equalities (paths). We say that a HIT H is non-recursive if its constructors do not quantify over elements or paths in H. The advantage of non-recursive HITs is that their elimination principles are easier to apply than those of general HITs.
This talk gave a sketch of the contents and background to a book with the title `Nonabelian algebraic topology' being written under support of a Leverhulme Emeritus Fellowship (2002-2004) by …
This talk gave a sketch of the contents and background to a book with the title `Nonabelian algebraic topology' being written under support of a Leverhulme Emeritus Fellowship (2002-2004) by the speaker and Rafael Sivera (Valencia). The aim is to give in one place a full account of work by R. Brown and P.J. Higgins since the 1970s which defines and applies crossed complexes and cubical higher homotopy groupoids in algebraic topology and group cohomology.
Homotopy Type Theory is a new field of mathematics based on the recently-discovered correspondence between Martin-Löf's constructive type theory and abstract homotopy theory. We have a powerful interplay between these …
Homotopy Type Theory is a new field of mathematics based on the recently-discovered correspondence between Martin-Löf's constructive type theory and abstract homotopy theory. We have a powerful interplay between these disciplines - we can use geometric intuition to formulate new concepts in type theory and, conversely, use type-theoretic machinery to verify and often simplify existing mathematical proofs.
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 …
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-Lof type theory with a natural numbers object, that is moreover closed under homotopy coequalizers.
Recent work on homotopy type theory exploits an exciting new correspondence between Martin-Lof's dependent type theory and the mathematical disciplines of category theory and homotopy theory. The mathematics suggests new …
Recent work on homotopy type theory exploits an exciting new correspondence between Martin-Lof's dependent type theory and the mathematical disciplines of category theory and homotopy theory. The mathematics suggests new principles to add to type theory, while the type theory can be used in novel ways to do computer-checked proofs in a proof assistant. In this paper, we formalize a basic result in algebraic topology, that the fundamental group of the circle is the integers. Our proof illustrates the new features of homotopy type theory, such as higher inductive types and Voevodsky's univalence axiom. It also introduces a new method for calculating the path space of a type, which has proved useful in many other examples.
Homotopy type theory proposes higher inductive types (HITs) as a means of defining and reasoning about inductively-generated objects with higher-dimensional structure. As with the univalence axiom, however, homotopy type theory …
Homotopy type theory proposes higher inductive types (HITs) as a means of defining and reasoning about inductively-generated objects with higher-dimensional structure. As with the univalence axiom, however, homotopy type theory does not specify the computational behavior of HITs. Computational interpretations have now been provided for univalence and specific HITs by way of cubical type theories, which use a judgmental infrastructure of dimension variables. We extend the cartesian cubical computational type theory introduced by Angiuli et al. with a schema for indexed cubical inductive types (CITs), an adaptation of higher inductive types to the cubical setting. In doing so, we isolate the canonical values of a cubical inductive type and prove a canonicity theorem with respect to these values.
Higher inductive-inductive types (HIITs) generalize inductive types of dependent type theories in two ways. On the one hand they allow the simultaneous definition of multiple sorts that can be indexed …
Higher inductive-inductive types (HIITs) generalize inductive types of dependent type theories in two ways. On the one hand they allow the simultaneous definition of multiple sorts that can be indexed over each other. On the other hand they support equality constructors, thus generalizing higher inductive types of homotopy type theory. Examples that make use of both features are the Cauchy real numbers and the well-typed syntax of type theory where conversion rules are given as equality constructors. In this paper we propose a general definition of HIITs using a small type theory, named the theory of signatures. A context in this theory encodes a HIIT by listing the constructors. We also compute notions of induction and recursion for HIITs, by using variants of syntactic logical relation translations. Building full categorical semantics and constructing initial algebras is left for future work. The theory of HIIT signatures was formalised in Agda together with the syntactic translations. We also provide a Haskell implementation, which takes signatures as input and outputs translation results as valid Agda code.
Homotopy type theory is an extension of Martin-Löf type theory with principles inspired by category theory and homotopy theory. With these extensions, type theory can be used to construct proofs …
Homotopy type theory is an extension of Martin-Löf type theory with principles inspired by category theory and homotopy theory. With these extensions, type theory can be used to construct proofs of homotopy-theoretic theorems, in a way that is very amenable to computer-checked proofs in proof assistants such as Coq and Agda. In this paper, we give a computer-checked construction of Eilenberg-MacLane spaces. For an abelian group G, an Eilenberg-MacLane space K(G,n) is a space (type) whose nth homotopy group is G, and whose homotopy groups are trivial otherwise. These spaces are a basic tool in algebraic topology; for example, they can be used to build spaces with specified homotopy groups, and to define the notion of cohomology with coefficients in G. Their construction in type theory is an illustrative example, which ties together many of the constructions and methods that have been used in homotopy type theory so far.
We investigate inductive types in type theory, using the insights provided by homotopy type theory and univalent foundations of mathematics. We do so by introducing the new notion of a …
We investigate inductive types in type theory, using the insights provided by homotopy type theory and univalent foundations of mathematics. We do so by introducing the new notion of a homotopy-initial algebra. This notion is defined by a purely type-theoretic contractibility condition that replaces the standard, category-theoretic universal property involving the existence and uniqueness of appropriate morphisms. Our main result characterizes the types that are equivalent to W-types as homotopy-initial algebras.
In homotopy type theory, the truncation operator ||-||n (for a number n greater or equal to -1) is often useful if one does not care about the higher structure of …
In homotopy type theory, the truncation operator ||-||n (for a number n greater or equal to -1) is often useful if one does not care about the higher structure of a type and wants to avoid coherence problems. However, its elimination principle only allows to eliminate into n-types, which makes it hard to construct functions ||A||n -> B if B is not an n-type. This makes it desirable to derive more powerful elimination theorems. We show a first general result: If B is an (n+1)-type, then functions ||A||n -> B correspond exactly to functions A -> B that are constant on all (n+1)-st loop spaces. We give one elementary proof and one proof that uses a higher inductive type, both of which require some effort. As a sample application of our result, we show that we can construct set-based representations of 1-types, as long as they have braided loop spaces. The main result with one of its proofs and the application have been formalised in Agda.
We present a development of cellular cohomology in homotopy type theory. Cohomology associates to each space a sequence of abelian groups capturing part of its structure, and has the advantage …
We present a development of cellular cohomology in homotopy type theory. Cohomology associates to each space a sequence of abelian groups capturing part of its structure, and has the advantage over homotopy groups in that these abelian groups of many common spaces are easier to compute. Cellular cohomology is a special kind of cohomology designed for cell complexes: these are built in stages by attaching spheres of progressively higher dimension, and cellular cohomology defines the groups out of the combinatorial description of how spheres are attached. Our main result is that for finite cell complexes, a wide class of cohomology theories (including the ones defined through Eilenberg-MacLane spaces) can be calculated via cellular cohomology. This result was formalized in the Agda proof assistant.
Cubical type theory provides a constructive justification to certain aspects of homotopy type theory such as Voevodsky's univalence axiom. This makes many extensionality principles, like function and propositional extensionality, directly …
Cubical type theory provides a constructive justification to certain aspects of homotopy type theory such as Voevodsky's univalence axiom. This makes many extensionality principles, like function and propositional extensionality, directly provable in the theory. This paper describes a constructive semantics, expressed in a presheaf topos with suitable structure inspired by cubical sets, of some higher inductive types. It also extends cubical type theory by a syntax for the higher inductive types of spheres, torus, suspensions, truncations, and pushouts. All of these types are justified by the semantics and have judgmental computation rules for all constructors, including the higher dimensional ones, and the universes are closed under these type formers.
Category theory in homotopy type theory is intricate as categorical laws can only be stated "up to homotopy", and thus require coherences. The established notion of a univalent category (as …
Category theory in homotopy type theory is intricate as categorical laws can only be stated "up to homotopy", and thus require coherences. The established notion of a univalent category (as introduced by Ahrens et al.) solves this by considering only truncated types, roughly corresponding to an ordinary category. This fails to capture many naturally occurring structures, stemming from the fact that the naturally occurring structures in homotopy type theory are not ordinary, but rather higher categories. Out of the large variety of approaches to higher category theory that mathematicians have proposed, we believe that, for type theory, the simplicial strategy is best suited. Work by Lurie and Harpaz motivates the following definition. Given the first ( n +3) levels of a semisimplicial type S , we can equip S with three properties: first, contractibility of the types of certain horn fillers; second, a completeness property; and third, a truncation condition. We call this a complete semi-Segal n -type . This is very similar to an earlier suggestion by Schreiber. The definition of a univalent (1-) category by Ahrens et al. can easily be extended or restricted to the definition of a univalent n -category (more precisely, ( n ,1)-category) for n ∈ {0,1,2}, and we show that the type of complete semi-Segal n -types is equivalent to the type of univalent n -categories in these cases. Thus, we believe that the notion of a complete semi-Segal n -type can be taken as the definition of a univalent n -category. We provide a formalisation in the proof assistant Agda using a completely explicit representation of semi-simplicial types for levels up to 4.
One of the most used algorithm in rewriting theory is the Knuth-Bendix completion procedure which starts from a terminating rewriting system and iteratively adds rules to it, trying to produce …
One of the most used algorithm in rewriting theory is the Knuth-Bendix completion procedure which starts from a terminating rewriting system and iteratively adds rules to it, trying to produce an equivalent convergent rewriting system. It is in particular used to study presentations of monoids, since normal forms of the rewriting system provide canonical representatives of words modulo the congruence generated by the rules. Here, we are interested in extending this procedure in order to retrieve information about the low-dimensional homotopy properties of a monoid. We therefore consider the notion of coherent presentation, which is a generalization of rewriting systems that keeps track of the cells generated by confluence diagrams. We extend the Knuth-Bendix completion procedure to this setting, resulting in a homotopical completion procedure. It is based on a generalization of Tietze transformations, which are operations that can be iteratively applied to relate any two presentations of the same monoid. We also explain how these transformations can be used to remove useless generators, rules, or confluence diagrams in a coherent presentation, thus leading to a homotopical reduction procedure. Finally, we apply these techniques to the study of some examples coming from representation theory, to compute minimal coherent presentations for them: braid, plactic and Chinese monoids.
Homotopy theory can be developed synthetically in homotopy type theory, using types to describe spaces, the identity type to describe paths in a space, and iterated identity types to describe …
Homotopy theory can be developed synthetically in homotopy type theory, using types to describe spaces, the identity type to describe paths in a space, and iterated identity types to describe higher-dimensional paths.While some aspects of homotopy theory have been developed synthetically and formalized in proof assistants, some seemingly easy examples have proved difficult because the required manipulations of paths becomes complicated.In this paper, we describe a cubical approach to developing homotopy theory within type theory.The identity type is complemented with higher-dimensional cube types, such as a type of squares, dependent on four points and four lines, and a type of three-dimensional cubes, dependent on the boundary of a cube.Path-over-a-path types and higher generalizations are used to describe cubes in a fibration over a cube in the base.These higher-dimensional cube and path-over types can be defined from the usual identity type, but isolating them as independent conceptual abstractions has allowed for the formalization of some previously difficult examples.
In this paper, we study algebras over the little cubes operads introduced by Boardman and Vogt, using the formalism of higher category theory.
In this paper, we study algebras over the little cubes operads introduced by Boardman and Vogt, using the formalism of higher category theory.
String rewriting systems have proved very useful to study monoids. In good cases, they give finite presentations of monoids, allowing computations on those and their manipulation by a computer. Even …
String rewriting systems have proved very useful to study monoids. In good cases, they give finite presentations of monoids, allowing computations on those and their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide one with a notion of canonical representative of the elements of the presented monoid. Polygraphs are a higher-dimensional generalization of this notion of presentation, from the setting of monoids to the much more general setting of n-categories. One of the main purposes of this article is to give a progressive introduction to the notion of higher-dimensional rewriting system provided by polygraphs, and describe its links with classical rewriting theory, string and term rewriting systems in particular. After introducing the general setting, we will be interested in proving local confluence for polygraphs presenting 2-categories and introduce a framework in which a finite 3-dimensional rewriting system admits a finite number of critical pairs.
We introduce homotopical methods based on rewriting on higher-dimensional categories to prove coherence results in categories with an algebraic structure. We express the coherence problem for (symmetric) monoidal categories as …
We introduce homotopical methods based on rewriting on higher-dimensional categories to prove coherence results in categories with an algebraic structure. We express the coherence problem for (symmetric) monoidal categories as an asphericity problem for a track category and use rewriting methods on polygraphs to solve it. The setting is extended to more general coherence problems, viewed as 3-dimensional word problems in a track category, including the case of braided monoidal categories.
This paper presents a dependently-typed construction of semi-simplicial sets in a type theory where sets are taken to be types. This addresses an open question raised on the wiki of …
This paper presents a dependently-typed construction of semi-simplicial sets in a type theory where sets are taken to be types. This addresses an open question raised on the wiki of the special year on Univalent Foundations at the Institute of Advanced Study (2012–2013).
Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of …
Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and beautiful "univalence axiom" implies that isomorphic structures can be identified. On the other hand, "higher inductive types" provide direct, logical descriptions of some of the basic spaces and constructions of homotopy theory. Both are impossible to capture directly in classical set-theoretic foundations, but when combined in homotopy type theory, they permit an entirely new kind of "logic of homotopy types". This suggests a new conception of foundations of mathematics, with intrinsic homotopical content, an "invariant" conception of the objects of mathematics -- and convenient machine implementations, which can serve as a practical aid to the working mathematician. This book is intended as a first systematic exposition of the basics of the resulting "Univalent Foundations" program, and a collection of examples of this new style of reasoning -- but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant.
This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments at the …
This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments at the interface with deformation quantization.
We report on the development of the HoTT library, a formalization of homotopy type theory in the Coq proof assistant. It formalizes most of basic homotopy type theory, including univalence, …
We report on the development of the HoTT library, a formalization of homotopy type theory in the Coq proof assistant. It formalizes most of basic homotopy type theory, including univalence, higher inductive types, and significant amounts of synthetic homotopy theory, as well as category theory and modalities. The library has been used as a basis for several independent developments. We discuss the decisions that led to the design of the library, and we comment on the interaction of homotopy type theory with recently introduced features of Coq, such as universe polymorphism and private inductive types.
This paper presents a type theory in which it is possible to directly manipulate n-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in …
This paper presents a type theory in which it is possible to directly manipulate n-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system. Further, Voevodsky's univalence axiom is provable in this system. We also explain an extension with some higher inductive types like the circle and propositional truncation. Finally we provide semantics for this cubical type theory in a constructive meta-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 …
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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , as the quotient space identifying antipodal points of the n-sphere, does not translate directly to homotopy type theory. Instead, we define ℝP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> by induction on n simultaneously with its tautological bundle of 2-element sets. As the base case, we take ℝP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> to be the empty type. In the inductive step, we take ℝP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n+1</sup> to be the mapping cone of the projection map of the tautological bundle of ℝP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , and we use its universal property and the univalence axiom to define the tautological bundle on ℝP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n+1</sup> . By showing that the total space of the tautological bundle of ℝP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> is the n-sphere S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , we retrieve the classical description of ℝP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n+1</sup> as ℝP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> with an (n + 1)-disk attached to it. The infinite dimensional real projective space ℝP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> , defined as the sequential colimit of ℝP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> 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.
We define and develop two-level type theory (2LTT), a version of Martin-Lof type theory which combines two different type theories. We refer to them as the inner and the outer …
We define and develop two-level type theory (2LTT), a version of Martin-Lof type theory which combines two different type theories. We refer to them as the inner and the outer type theory. In our case of interest, the inner theory is homotopy type theory (HoTT) which may include univalent universes and higher inductive types. The outer theory is a traditional form of type theory validating uniqueness of identity proofs (UIP). One point of view on it is as internalised meta-theory of the inner type theory.
There are two motivations for 2LTT. Firstly, there are certain results about HoTT which are of meta-theoretic nature, such as the statement that semisimplicial types up to level $n$ can be constructed in HoTT for any externally fixed natural number $n$. Such results cannot be expressed in HoTT itself, but they can be formalised and proved in 2LTT, where $n$ will be a variable in the outer theory. This point of view is inspired by observations about conservativity of presheaf models.
Secondly, 2LTT is a framework which is suitable for formulating additional axioms that one might want to add to HoTT. This idea is heavily inspired by Voevodsky's Homotopy Type System (HTS), which constitutes one specific instance of a 2LTT. HTS has an axiom ensuring that the type of natural numbers behaves like the external natural numbers, which allows the construction of a universe of semisimplicial types. In 2LTT, this axiom can be stated simply be asking the inner and outer natural numbers to be isomorphic.
After defining 2LTT, we set up a collection of tools with the goal of making 2LTT a convenient language for future developments. As a first such application, we develop the theory of Reedy fibrant diagrams in the style of Shulman. Continuing this line of thought, we suggest a definition of (infinity,1)-category and give some examples.
Homotopy type theory is a version of Martin-Lof type theory taking advantage of its homotopical models. In particular, we can use and construct objects of homotopy theory and reason about …
Homotopy type theory is a version of Martin-Lof 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 RPn, as the quotient space identifying antipodal points of the n-sphere, does not translate directly to homotopy type theory. Instead, we define RPn 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 RPn+1 to be the mapping cone of the projection map of the tautological bundle of RPn, and we use its universal property and the univalence axiom to define the tautological bundle on RPn+1.By showing that the total space of the tautological bundle of RPn is the n-sphere Sn, we retrieve the classical description of RPn+1 as RPn with an (n + 1)-disk attached to it. The infinite dimensional real projective space RP∞, defined as the sequential colimit of RPn 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.
This paper presents a type theory in which it is possible to directly manipulate $n$-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in …
This paper presents a type theory in which it is possible to directly manipulate $n$-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system. Further, Voevodsky's univalence axiom is provable in this system. We also explain an extension with some higher inductive types like the circle and propositional truncation. Finally we provide semantics for this cubical type theory in a constructive meta-theory.
Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of …
Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of a cycle are not necessarily closed. This paper derives a principle reminiscent of induction for cycles for the case that the graph is given as the symmetric closure of a locally confluent and (co-)well-founded relation. We show that, assuming the property in question is sufficiently nice, it is enough to prove it for the empty cycle and for cycles given by local confluence.
We develop normalisation by evaluation (NBE) for dependent types based on presheaf categories. Our construction is formulated in the metalanguage of type theory using quotient inductive types. We use a …
We develop normalisation by evaluation (NBE) for dependent types based on presheaf categories. Our construction is formulated in the metalanguage of type theory using quotient inductive types. We use a typed presentation hence there are no preterms or realizers in our construction, and every construction respects the conversion relation. NBE for simple types uses a logical relation between the syntax and the presheaf interpretation. In our construction, we merge the presheaf interpretation and the logical relation into a proof-relevant logical predicate. We prove normalisation, completeness, stability and decidability of definitional equality. Most of the constructions were formalized in Agda.
We compute coherent presentations of Artin monoids, that is, presentations by generators, relations, and relations between the relations. For that, we use methods of higher-dimensional rewriting that extend Squier’s and …
We compute coherent presentations of Artin monoids, that is, presentations by generators, relations, and relations between the relations. For that, we use methods of higher-dimensional rewriting that extend Squier’s and Knuth–Bendix’s completions into a homotopical completion–reduction, applied to Artin’s and Garside’s presentations. The main result of the paper states that the so-called Tits–Zamolodchikov 3-cells extend Artin’s presentation into a coherent presentation. As a byproduct, we give a new constructive proof of a theorem of Deligne on the actions of an Artin monoid on a category.