Searcharxiv⌕ Search

arXiv subjects

Nima Rasekh

Publications and source records attributed to Nima Rasekh.

At least 19 recordsLinked to original sources
arXiv2026

Fractured Structures in Condensed Mathematics

We construct a fractured structure, in the sense of Lurie, on the $\infty$-topos of condensed anima. This fractured structure allows us to better comprehend various properties of condensed anima - we use it to exhibit an explicit collection of jointly conservative points for condensed anima. To rule out further candidates for fractured structures, we analyze limits in the category of extremally disconnected spaces. In particular, we show that it does not admit all fibers, answering a question from Clausen.

math.AT↗
arXiv2025

Filter Quotient Model Structures

The filter quotient construction is a particular instance of a filtered colimit of categories. It has primarily been considered in the context of categorical logic, where it has been used effectively to construct non-trivial models, for example new models of set theory. In this work we prove that given a model category and a suitable notion of filter of subterminal objects, the filter quotient construction will preserve the model structure. We also show that this new model structure inherits certain important properties (such as being simplicial or proper), but not all (such as being cofibrantly generated). Finally, we show it is compatible with the construction of filter quotient $\infty$-categories.

math.CT↗
arXiv2025

Non-Standard Models of Homotopy Type Theory

Homotopy type theory is a modern foundation for mathematics that introduces the univalence axiom and is particularly suitable for the study of homotopical mathematics and its formalization via proof assistants. In order to better comprehend the mathematical implications of homotopy type theory, a variety of models have been constructed and studied. Here a model is understood as a model category with suitable properties implementing the various type theoretical constructors and axioms. A first example is the simplicial model due to Kapulkin--Lumsdaine--Voevodsky. By now, many other models have been constructed, due to work of Arndt, Kapulkin, Lumsdaine, Warren and particularly Shulman, culminating in a proof that every Grothendieck $\infty$-topos can be obtained as the underlying $\infty$-category of a model category that models homotopy type theory. In this paper we propose the filter quotient construction as a new method to construct further models of homotopy type theory. Concretely, we prove that with minor assumptions, the filter quotient construction preserves all model categorical properties individually that implement various type theoretical constructors and axioms. On the other hand, the filter quotient construction does not preserve many external properties that are of set theoretical nature, such as cocompleteness, local presentability or cofibrant generation. Combining these, the filter quotient construction preserves models of homotopy type theory and can result in models that have not been considered before and exhibit behaviors that diverge from any of the established models.

math.CT↗
arXiv2025

Simplicial Homotopy Type Theory is not just Simplicial: What are $\infty$-Categories?

$\infty$-category theory was originally developed in the context of classical homotopy theory using standard set theoretical assumptions, but has since been extended to a variety of mathematical foundations. One such successful effort, primarily due to Martini and Wolf, introduced a theory of $\infty$-categories internal to the foundation of an arbitrary Grothendieck $\infty$-topos, meaning they used categorical foundations. Another approach, due to Riehl and Shulman, developed a theory of $\infty$-categories internal to their own type theory: simplicial homotopy type theory (sHoTT), meaning they employed a (homotopy) type theoretic foundation. One aspect of developing a theory of $\infty$-categories in different foundations consists of introducing ways to translate from one foundation to another. Concretely, as part of their work, Riehl and Shulman prove that $\infty$-categories internal to Grothendieck $\infty$-topoi give us categorical models of sHoTT. In fact the name ``simplicial'' in sHoTT suggests that all categorical models of sHoTT should be given by simplicial objects in suitable $\infty$-categories. In this paper we prove that contrary to this expectation, there are models of sHoTT that are not simply simplicial objects. This suggests that in a general foundations, the notion of $\infty$-category is more general than previously assumed.

math.CT↗
arXiv2025

Cosmological Unstraightening

The unstraightening construction due to Lurie establishes an equivalence between presheaves and fibrations, using one prominent model of $(\infty,1)$-categories, namely quasi-categories. In this work we generalize this result by proving that for all $\infty$-cosmoi of $(\infty,1)$-categories in the sense of Riehl and Verity, which includes quasi-categories but also complete Segal spaces or $1$-complicial sets, their corresponding notions of fibrations and presheaves are biequivalent $\infty$-cosmoi via a natural zig-zag of cosmological biequivalences. The major idea that makes this possible is a lift of the quasi-categorical unstraightening construction to a cosmological biequivalence.

math.CT↗
arXiv2024

$(\infty,n)$-Limits II: Comparison across models

We show that the notion of $(\infty,n)$-limit defined using the enriched approach and the one defined using the internal approach coincide. We also give explicit constructions of various double $(\infty,n-1)$-categories implementing various join constructions, slice constructions and cone constructions, and study their properties. We further prove that key examples of $(\infty,n)$-categories are (co)complete.

math.AT↗
arXiv2024

Insights From Univalent Foundations: A Case Study Using Double Categories

Category theory unifies mathematical concepts, aiding comparisons across structures by incorporating objects and morphisms, which capture their interactions. It has influenced areas of computer science such as automata theory, functional programming, and semantics. Certain objects naturally exhibit two classes of morphisms, leading to the concept of a double category, which has found applications in computing science (e.g., ornaments, profunctor optics, denotational semantics). The emergence of diverse categorical structures motivated a unified framework for category theory. However, unlike other mathematical objects, classification of categorical structures faces challenges due to various relevant equivalences. This poses significant challenges when pursuing the formalization of categories and restricts the applicability of powerful techniques, such as transport along equivalences. This work contends that univalent foundations offers a suitable framework for classifying different categorical structures based on desired notions of equivalences, and remedy the challenges when formalizing categories. The richer notion of equality in univalent foundations makes the equivalence of a categorical structure an inherent part of its structure. We concretely apply this analysis to double categorical structures. We characterize and formalize various definitions in Coq UniMath, including (pseudo) double categories and double bicategories, up to chosen equivalences. We also establish univalence principles, making chosen equivalences part of the double categorical structure, analyzing strict double setcategories (invariant under isomorphisms), pseudo double setcategories (invariant under isomorphisms), univalent pseudo double categories (invariant under vertical equivalences) and univalent double bicategories (invariant under gregarious equivalences).

math.CT↗
arXiv2023

$(\infty,n)$-Limits I: Definition and first consistency results

We give a model-independent definition of limits for diagrams valued in an $(\infty,n)$-category. We show that this definition is compatible with the existing notion of homotopy 2-limits for 2-categories, with the existing notion of $(\infty,1)$-limits for $(\infty,1)$-categories, and with itself across different values of $n$.

math.AT↗
arXiv2023

Univalent Double Categories

Category theory is a branch of mathematics that provides a formal framework for understanding the relationship between mathematical structures. To this end, a category not only incorporates the data of the desired objects, but also "morphisms", which capture how different objects interact with each other. Category theory has found many applications in mathematics and in computer science, for example in functional programming. Double categories are a natural generalization of categories which incorporate the data of two separate classes of morphisms, allowing a more nuanced representation of relationships and interactions between objects. Similar to category theory, double categories have been successfully applied to various situations in mathematics and computer science, in which objects naturally exhibit two types of morphisms. Examples include categories themselves, but also lenses, petri nets, and spans. While categories have already been formalized in a variety of proof assistants, double categories have received far less attention. In this paper we remedy this situation by presenting a formalization of double categories via the proof assistant Coq, relying on the Coq UniMath library. As part of this work we present two equivalent formalizations of the definition of a double category, an unfolded explicit definition and a second definition which exhibits excellent formal properties via 2-sided displayed categories. As an application of the formal approach we establish a notion of univalent double category along with a univalence principle: equivalences of univalent double categories coincide with their identities

math.CT↗
arXiv2023

An $(\infty,n)$-categorical straightening-unstraightening construction

We provide an $(\infty,n)$-categorical version of the straightening-unstraightening construction, asserting an equivalence between the $(\infty,n)$-category of double $(\infty,n-1)$-right fibrations over an $(\infty,n)$-category $\mathcal{C}$ and that of the $(\infty,n)$-functors from $\mathcal{C}$ valued in $(\infty,n-1)$-categories. We realize this in the form of a Quillen equivalence between appropriate model structures; on the one hand, a model structure for double $(\infty,n-1)$-right fibrations over a generic precategory object $W$ in $(\infty,n-1)$-categories and, on the other hand, a model structure for $(\infty,n)$-functors from its homotopy coherent categorification $\mathfrak{C} W$ valued in $(\infty,n-1)$-categories.

math.AT↗
arXiv2022

A homotopy coherent nerve for $(\infty,n)$-categories

In the case of $(\infty,1)$-categories, the homotopy coherent nerve gives a right Quillen equivalence between the models of simplicially enriched categories and of quasi-categories. This shows that homotopy coherent diagrams of $(\infty,1)$-categories can equivalently be defined as functors of quasi-categories or as simplicially enriched functors out of the homotopy coherent categorifications. In this paper, we construct a homotopy coherent nerve for $(\infty,n)$-categories. We show that it realizes a right Quillen equivalence between the models of categories strictly enriched in $(\infty,n-1)$-categories and of Segal category objects in $(\infty,n-1)$-categories. This similarly enables us to define homotopy coherent diagrams of $(\infty,n)$-categories equivalently as functors of Segal category objects or as strictly enriched functors out of the homotopy coherent categorifications.

math.AT↗
arXiv2022

Twisted Arrow Construction for Segal Spaces

We give an explicit description of the twisted arrow construction for simplicial spaces and demonstrate individually that it preserves the defining properties of a complete Segal space. Moreover, we show that for a Segal space, the natural projection from the twisted arrow Segal space is a left fibration.

math.CT↗
arXiv2022

A Model for the Higher Category of Higher Categories

We use fibrations of complete Segal spaces to construct four complete Segal spaces: Reedy fibrant simplicial spaces, Segal spaces, complete Segal spaces, and spaces. Moreover, we show each one comes with a universal fibration that classifies Reedy left fibrations, Segal coCartesian fibrations, coCartesian fibrations and left fibrations and prove these are representable fibrations. Finally, we use equivalences between quasi-categories and complete Segal spaces to present analogous constructions using fibrations of quasi-categories.

math.CT↗
arXiv2022

A Theory of Elementary Higher Toposes

We define an elementary $\infty$-topos that simultaneously generalizes an elementary topos and Grothendieck $\infty$-topos. We then prove it satisfies the expected topos theoretic properties, such as descent, local Cartesian closure, locality and classification of univalent morphisms, generalizing results by Lurie and Gepner-Kock. We also define $\infty$-logical functors and show the resulting $\infty$-category is closed under limits and filtered colimits, generalizing the analogous result for elementary toposes and Grothendieck $\infty$-toposes. Moreover, we give an alternative characterization of elementary $\infty$-toposes and their $\infty$-logical functors via their ind-completions. Finally we generalize these results by discussing the case of elementary (n,1)-toposes and give various examples and non-examples.

math.CT↗
arXiv2021

Shadows are Bicategorical Traces

Hochschild homology has proved to be an important invariant in algebra and homotopy theory, in particular due to its relevance in algebraic $K$-theory and fixed point theory, leading to the development of numerous variants of the original construction. Ponto introduced a bicategorical axiomatization of Hochschild homology-type invariants, called a shadow, which captures the essential common properties of all known variants of Hochschild homology, such as Morita invariance. In this paper we clarify the relationship between shadows and Hochschild homology. After extending the notion of Hochschild homology to bicategories in a natural manner, we prove the existence of a universal shadow on any bicategory $\mathscr{B}$, taking values in the Hochschild homology of $\mathscr{B}$, through which all other shadows on $\mathscr{B}$ factor. Shadows are thus co-represented by a bicategorical version of Hochschild homology. Using the universal shadow on the free adjunction bicategory, we can then establish a universal Morita invariance theorem, of which all known cases are immediate corollaries. Building on this understanding of shadows on bicategories, we propose an $\infty$-categorical generalization of shadows as functors out of Hochschild homology of an $(\infty,2)$-category in the sense of Berman. As a first step towards constructing relevant examples of $\infty$-categorical shadows, we define the Hochschild homology of enriched $\infty$-categorical bimodules and prove that they assemble into a shadow. As part of this work we compute the Hochschild homology of several important $2$-categories (such as the free adjunction), which can be of independent interest.

math.CT↗
arXiv2021

Quasi-Categories vs. Segal Spaces: Cartesian Edition

We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: On marked simplicial sets, on bisimplicial spaces, on bisimplicial sets, on marked simplicial spaces. The main way to prove these equivalences is by using the Quillen equivalences between quasi-categories and complete Segal spaces as defined by Joyal-Tierney and the straightening construction due to Lurie.

math.CT↗
arXiv2021

Yoneda Lemma for $\mathcal{D}$-Simplicial Spaces

For a small category $\mathcal{D}$ we define fibrations of simplicial presheaves on the category $\mathcal{D}\timesΔ$, which we call localized $\mathcal{D}$-left fibration. We show these fibrations can be seen as fibrant objects in a model structure, the localized $\mathcal{D}$-covariant model structure, that is Quillen equivalent to a category of functors valued in simplicial presheaves on $\mathcal{D}$, where the Quillen equivalence is given via a generalization of the Grothendieck construction. We use our understanding of this construction to give a detailed characterization of fibrations and weak equivalences in this model structure and in particular obtain a Yoneda lemma. We apply this general framework to study Cartesian fibrations of $(\infty,n)$-categories, for models of $(\infty,n)$-categories that arise via simplicial presheaves, such as $n$-fold complete Segal spaces. This, in particular, results in the Yoneda lemma and Grothendieck construction for Cartesian fibrations of $(\infty,n)$-categories.

math.CT↗