Michael Shulman
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View article: The Univalence Principle
The Univalence Principle Open
The Univalence Principle is the statement that equivalent mathematical structures are indistinguishable. We prove a general version of this principle that applies to all set-based, categorical, and higher-categorical structures defined in …
View article: Displayed type theory and semi-simplicial types
Displayed type theory and semi-simplicial types Open
We introduce Displayed Type Theory (dTT) , a multi-modal homotopy type theory with discrete and simplicial modes. In the intended semantics, the discrete mode is interpreted by a model for an arbitrary $\infty$ -topos, while the simplicial…
View article: Strange new universes: Proof assistants and synthetic foundations
Strange new universes: Proof assistants and synthetic foundations Open
Existing computer programs called proof assistants can verify the correctness of mathematical proofs but their specialized proof languages present a barrier to entry for many mathematicians. Large language models have the potential to lowe…
View article: Internal Parametricity, without an Interval
Internal Parametricity, without an Interval Open
Parametricity is a property of the syntax of type theory implying, e.g., that there is only one function having the type of the polymorphic identity function. Parametricity is usually proven externally, and does not hold internally. Intern…
View article: Displayed Type Theory and Semi-Simplicial Types
Displayed Type Theory and Semi-Simplicial Types Open
We introduce Displayed Type Theory (dTT), a multi-modal homotopy type theory with discrete and simplicial modes. In the intended semantics, the discrete mode is interpreted by a model for an arbitrary $\infty$-topos, while the simplicial m…
View article: Semantics of multimodal adjoint type theory
Semantics of multimodal adjoint type theory Open
We show that contrary to appearances, Multimodal Type Theory (MTT) over a 2-category M can be interpreted in any M-shaped diagram of categories having, and functors preserving, M-sized limits, without the need for extra left adjoints. This…
View article: LNL polycategories and doctrines of linear logic
LNL polycategories and doctrines of linear logic Open
We define and study LNL polycategories, which abstract the judgmental structure of classical linear logic with exponentials. Many existing structures can be represented as LNL polycategories, including LNL adjunctions, linear exponential c…
View article: Semantics of multimodal adjoint type theory
Semantics of multimodal adjoint type theory Open
We show that contrary to appearances, Multimodal Type Theory (MTT) over a 2-category M can be interpreted in any M-shaped diagram of categories having, and functors preserving, M-sized limits, without the need for extra left adjoints. This…
View article: Univalent Monoidal Categories
Univalent Monoidal Categories Open
Univalent categories constitute a well-behaved and useful notion of category in univalent foundations. The notion of univalence has subsequently been generalized to bicategories and other structures in (higher) category theory. Here, we zo…
View article: AFFINE LOGIC FOR CONSTRUCTIVE MATHEMATICS
AFFINE LOGIC FOR CONSTRUCTIVE MATHEMATICS Open
We show that numerous distinctive concepts of constructive mathematics arise automatically from an “antithesis” translation of affine logic into intuitionistic logic via a Chu/Dialectica construction. This includes apartness relations, com…
View article: The directed plump ordering
The directed plump ordering Open
Based on Taylor's hereditarily directed plump ordinals, we define the directed plump ordering on W-types in Martin-Löf type theory. This ordering is similar to the plump ordering but comes equipped with non-empty finite joins in addition t…
View article: A Stratified Approach to Löb Induction
A Stratified Approach to Löb Induction Open
Guarded type theory extends type theory with a handful of modalities and constants to encode productive recursion. While these theories have seen widespread use, the metatheory of guarded type theories, particularly guarded dependent type …
View article: *-Autonomous Envelopes and Conservativity
*-Autonomous Envelopes and Conservativity Open
We prove 2-categorical conservativity for any {0,T}-free fragment of MALL\nover its corresponding intuitionistic version: that is, that the universal map\nfrom a closed symmetric monoidal category to the *-autonomous category that it\nfree…
View article: Magnitude homology of enriched categories and metric spaces
Magnitude homology of enriched categories and metric spaces Open
Magnitude is a numerical invariant of enriched categories, including in particular metric spaces as $[0,\infty)$-enriched categories. We show that in many cases magnitude can be categorified to a homology theory for enriched categories, wh…
View article: Generalized stability for abstract homotopy theories
Generalized stability for abstract homotopy theories Open
We show that a derivator is stable if and only if homotopy finite limits and homotopy finite colimits commute, if and only if homotopy finite limit functors have right adjoints, and if and only if homotopy finite colimit functors have left…
View article: LNL polycategories and doctrines of linear logic
LNL polycategories and doctrines of linear logic Open
We define and study LNL polycategories, which abstract the judgmental structure of classical linear logic with exponentials. Many existing structures can be represented as LNL polycategories, including LNL adjunctions, linear exponential c…
View article: The derivator of setoids
The derivator of setoids Open
Without the axiom of choice, the free exact completion of the category of sets (i.e. the category of setoids) may not be complete or cocomplete. We will show that nevertheless, it can be enhanced to a derivator: the formal structure of cat…
View article: Construction of the circle in UniMath
Construction of the circle in UniMath Open
We show that the type $\mathrm{T}\mathbb{Z}$ of $\mathbb{Z}$-torsors has the dependent universal property of the circle, which characterizes it up to a unique homotopy equivalence. The construction uses Voevodsky's Univalence Axiom and pro…
View article: Categories of Nets
Categories of Nets Open
We present a unified framework for Petri nets and various variants, such as pre-nets and Kock's whole-grain Petri nets. Our framework is based on a less well-studied notion that we call $Σ$-nets, which allow finer control over whether toke…
View article: Categories of Nets
Categories of Nets Open
We present a unified framework for Petri nets and various variants, such as pre-nets and Kock's whole-grain Petri nets. Our framework is based on a less well-studied notion that we call $\Sigma$-nets, which allow finer control over whether…
View article: A practical type theory for symmetric monoidal categories
A practical type theory for symmetric monoidal categories Open
We give a natural-deduction-style type theory for symmetric monoidal categories whose judgmental structure directly represents morphisms with tensor products in their codomain as well as their domain. The syntax is inspired by Sweedler not…
View article: autonomous envelopes
autonomous envelopes Open
We show the doctrine of $\ast$-autonomous categories is 2-conservative over the doctrine of closed symmetric monoidal categories, i.e. the universal map from a closed symmetric monoidal category to the $\ast$-autonomous category that it fr…
View article: *-autonomous envelopes and 2-conservativity of duals
*-autonomous envelopes and 2-conservativity of duals Open
We show the doctrine of $\ast$-autonomous categories is 2-conservative over the doctrine of closed symmetric monoidal categories, i.e. the universal map from a closed symmetric monoidal category to the $\ast$-autonomous category that it fr…
View article: Modalities in homotopy type theory
Modalities in homotopy type theory Open
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 t…
View article: Modalities in homotopy type theory
Modalities in homotopy type theory Open
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 t…
View article: The 2-Chu-Dialectica construction and the polycategory of multivariable adjunctions
The 2-Chu-Dialectica construction and the polycategory of multivariable adjunctions Open
Cheng, Gurski, and Riehl constructed a cyclic double multicategory of multivariable adjunctions. We show that the same information is carried by a double polycategory, in which opposite categories are polycategorical duals. Moreover, this …
View article: A practical type theory for symmetric monoidal categories
A practical type theory for symmetric monoidal categories Open
We give a natural-deduction-style type theory for symmetric monoidal categories whose judgmental structure directly represents morphisms with tensor products in their codomain as well as their domain. The syntax is inspired by Sweedler not…
View article: Constructing symmetric monoidal bicategories functorially
Constructing symmetric monoidal bicategories functorially Open
We present a method of constructing monoidal, braided monoidal, and symmetric monoidal bicategories from corresponding types of monoidal double categories that satisfy a lifting condition. Many important monoidal bicategories arise natural…
View article: All $(\infty,1)$-toposes have strict univalent universes
All $(\infty,1)$-toposes have strict univalent universes Open
We prove the conjecture that any Grothendieck $(\infty,1)$-topos can be presented by a Quillen model category that interprets homotopy type theory with strict univalent universes. Thus, homotopy type theory can be used as a formal language…