arXiv subjects

Rune Haugseng

Publications and source records attributed to Rune Haugseng.

36 records · Page 2Linked to original sources

arXiv2021

Homotopy-coherent algebra via Segal conditions

Many homotopy-coherent algebraic structures can be described by Segal-type limit conditions determined by an "algebraic pattern", bywhich we mean an $\infty$-category equipped with a factorization system and a collection of "elementary" objects. Examples of structures that occur as such "Segal $\mathcal{O}$-spaces" for an algebraic pattern $\mathcal{O}$ include $\infty$-categories, $(\infty,n)$-categories, $\infty$-operads, $\infty$-properads, and algebras for an $\infty$-operad in spaces. In the first part of this paper we set up a general frameworkn for algebraic patterns and their Segal objects, including conditions under which the latter are preserved by left and right Kan extensions. In particular, we obtain necessary and sufficent conditions on a pattern $\mathcal{O}$ for free Segal $\mathcal{O}$-spaces to be described by an explicit colimit formula, in which case we say that $\mathcal{O}$ is "extendable". In the second part of the paper we explore the relationship between extendable algebraic patterns and polynomial monads, by which we mean cartesian monads on presheaf $\infty$-categories that are accessible and preserve weakly contractible limits. We first show that the free Segal $\mathcal{O}$-space monad for an extendable pattern $\mathcal{O}$ is always polynomial. Next, we prove an $\infty$-categorical version of Weber's Nerve Theorem for polynomial monads, and use this to define a canonical extendable pattern from any polynomial monad, whose Segal spaces are equivalent to the algebras of the monad. These constructions yield functors between polynomial monads and extendable algebraic patterns, and we show that these exhibit full subcategories of "saturated" algebraic patterns and "complete" polynomial monads as localizations, and moreover restrict to an equivalence between the $\infty$-categories of saturated patterns and complete polynomial monads.

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arXiv2021

On (co)ends in $\infty$-categories

In this short note we prove that two definitions of (co)ends in $\infty$-categories, via twisted arrow $\infty$-categories and via $\infty$-categories of simplices, are equivalent. We also show that weighted (co)limits, which can be defined as certain (co)ends, can alternatively be described as (co)limits over left and right fibrations, respectively.

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arXiv2021

On lax transformations, adjunctions, and monads in $(\infty,2)$-categories

We use the basic expected properties of the Gray tensor product of $(\infty,2)$-categories to study (co)lax natural transformations. Using results of Riehl-Verity and Zaganidis we identify lax transformations between adjunctions and monads with commutative squares of (monadic) right adjoints. We also identify the colax transformations whose components are equivalences (generalizing the "icons" of Lack) with the 2-morphisms that arise from viewing $(\infty,2)$-categories as simplicial $\infty$-categories. Using this characterization we identify the $\infty$-category of monads on a fixed object and colax morphisms between them with the $\infty$-category of associative algebras in endomorphisms.

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arXiv2020

Lax colimits and free fibrations in $\infty$-categories

We define and discuss lax and weighted colimits of diagrams in $\infty$-categories and show that the coCartesian fibration associated to a functor is given by its lax colimit. A key ingredient, of independent interest, is a simple characterization of the free Cartesian fibration associated to a a functor of $\infty$-categories. As an application of these results, we prove that lax representable functors are preserved under exponentiation, and also that the total space of a presentable Cartesian fibration between $\infty$-categories is presentable, generalizing a theorem of Makkai and Paré to the $\infty$-categorical setting. Lastly, in the appendix, we observe that pseudofunctors between (2,1)-categories give rise to functors between $\infty$-categories via the Duskin nerve.

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arXiv2020

Shifted Coisotropic Correspondences

We define (iterated) coisotropic correspondences between derived Poisson stacks, and construct symmetric monoidal higher categories of derived Poisson stacks where the $i$-morphisms are given by $i$-fold coisotropic correspondences. Assuming an expected equivalence of different models of higher Morita categories, we prove that all derived Poisson stacks are fully dualizable, and so determine framed extended TQFTs by the Cobordism Hypothesis. Along the way we also prove that the higher Morita category of $E_{n}$-algebras with respect to coproducts is equivalent to the higher category of iterated cospans.

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arXiv2020

On the equivalence between $Θ_{n}$-spaces and iterated Segal spaces

We give a new proof of the equivalence between two of the main models for $(\infty,n)$-categories, namely the $n$-fold Segal spaces of Barwick and the $Θ_{n}$-spaces of Rezk, by proving that these are algebras for the same monad on the $\infty$-category of $n$-globular spaces. The proof works for a broad class of $\infty$-categories that includes all $\infty$-topoi.

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arXiv2020

Two models for the homotopy theory of $\infty$-operads

We compare two models for $\infty$-operads: the complete Segal operads of Barwick and the complete dendroidal Segal spaces of Cisinski and Moerdijk. Combining this with comparison results already in the literature, this implies that all known models for $\infty$-operads are equivalent - for instance, it follows that the homotopy theory of Lurie's $\infty$-operads is equivalent to that of dendroidal sets and that of simplicial operads.

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arXiv2020

Rectification of enriched infinity-categories

We prove a rectification theorem for enriched infinity-categories: If V is a nice monoidal model category, we show that the homotopy theory of infinity-categories enriched in V is equivalent to the familiar homotopy theory of categories strictly enriched in V. It follows, for example, that infinity-categories enriched in spectra or chain complexes are equivalent to spectral categories and dg-categories. A similar method gives a comparison result for enriched Segal categories, which implies that the homotopy theories of n-categories and (infinity,n)-categories defined by iterated infinity-categorical enrichment are equivalent to those of more familiar versions of these objects. In the latter case we also include a direct comparison with complete n-fold Segal spaces. Along the way we prove a comparison result for fibrewise simplicial localizations potentially of independent use.

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arXiv2020

On a spectral sequence for the cohomology of infinite loop spaces

We study the mod-2 cohomology spectral sequence arising from delooping the Bousfield-Kan cosimplicial space giving the 2-nilpotent completion of a connective spectrum $X$. Under good conditions its $E_{2}$-term is computable as certain non-abelian derived functors evaluated at $H^*(X)$ as a module over the Steenrod algebra, and it converges to the cohomology of $Ω^\infty X$. We provide general methods for computing the $E_{2}$-term, including the construction of a multiplicative spectral sequence of Serre type for cofibration sequences of simplicial commutative algebras. Some simple examples are also considered; in particular, we show that the spectral sequence collapses at $E_{2}$ when $X$ is a suspension spectrum.

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arXiv2020

The higher Morita category of $E_n$-algebras

We introduce simple models for associative algebras and bimodules in the context of non-symmetric $\infty$-operads, and use these to construct an $(\infty,2)$-category of associative algebras, bimodules, and bimodule homomorphisms in a monoidal $\infty$-category. By working with $\infty$-operads over $Δ^{n,\text{op}}$ we iterate these definitions and generalize our construction to get an $(\infty,n+1)$-category of $E_{n}$-algebras and iterated bimodules in an $E_{n}$-monoidal $\infty$-category. Moreover, we show that if $\mathcal{C}$ is an $E_{n+k}$-monoidal $\infty$-category then the $(\infty,n+1)$-category of $E_{n}$-algebras in $\mathcal{C}$ has a natural $E_{k}$-monoidal structure. We also identify the mapping $(\infty,n)$-categories between two $E_{n}$-algebras, which allows us to define interesting non-connective deloopings of the Brauer space of a commutative ring spectrum.

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arXiv2020

Bimodules and natural transformations for enriched $\infty$-categories

We introduce a notion of bimodule in the setting of enriched $\infty$-categories, and use this to construct a double $\infty$-category of enriched $\infty$-categories where the two kinds of 1-morphisms are functors and bimodules. We then consider a natural definition of natural transformations in this context, and show that in the underlying $(\infty,2)$-category of enriched $\infty$-categories with functors as 1-morphisms the 2-morphisms are given by natural transformations.

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arXiv2020

$\infty$-Operads as Analytic Monads

We develop an $\infty$-categorical version of the classical theory of polynomial and analytic functors, initial algebras, and free monads. Using this machinery, we provide a new model for $\infty$-operads, namely $\infty$-operads as analytic monads. We justify this definition by proving that the $\infty$-category of analytic monads is equivalent to that of dendroidal Segal spaces, known to be equivalent to the other existing models for $\infty$-operads.

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arXiv2020

Segal spaces, spans, and semicategories

We show that Segal spaces, and more generally category objects in an $\infty$-category $\mathcal{C}$, can be identified with associative algebras in the double $\infty$-category of spans in $\mathcal{C}$. We use this observation to prove that "having identities" is a property of a non-unital $(\infty,n)$-category.

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arXiv2020

Linear Batalin-Vilkovisky quantization as a functor of $\infty$-categories

We study linear Batalin-Vilkovisky (BV) quantization, which is a derived and shifted version of the Weyl quantization of symplectic vector spaces. Using a variety of homotopical machinery, we implement this construction as a symmetric monoidal functor of $\infty$-categories. We also show that this construction has a number of pleasant properties: It has a natural extension to derived algebraic geometry, it can be fed into the higher Morita category of $E_n$-algebras to produce a "higher BV quantization" functor, and when restricted to formal moduli problems, it behaves like a determinant. Along the way we also use our machinery to give an algebraic construction of $E_n$-enveloping algebras for shifted Lie algebras.

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arXiv2019

Enriched $\infty$-operads

In this paper we initiate the study of enriched $\infty$-operads. We introduce several models for these objects, including enriched versions of Barwick's Segal operads and the dendroidal Segal spaces of Cisinski and Moerdijk, and show these are equivalent. Our main results are a version of Rezk's completion theorem for enriched $\infty$-operads: localization at the fully faithful and essentially surjective morphisms is given by the full subcategory of complete objects, and a rectification theorem: the homotopy theory of $\infty$-operads enriched in the $\infty$-category arising from a nice symmetric monoidal model category is equivalent to the homotopy theory of strictly enriched operads.

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arXiv2019

Enriched $\infty$-categories via non-symmetric $\infty$-operads

We set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal $\infty$-category $\mathcal{V}$. Our theory of enriched $\infty$-categories has many desirable properties; for instance, if the enriching $\infty$-category $\mathcal{V}$ is presentably symmetric monoidal then $\mathrm{Cat}^\mathcal{V}_\infty$ is as well. These features render the theory useful even when an $\infty$-category of enriched $\infty$-categories comes from a model category (as is often the case in examples of interest, e.g. dg-categories, spectral categories, and $(\infty,n)$-categories). This is analogous to the advantages of $\infty$-categories over more rigid models such as simplicial categories - for example, the resulting $\infty$-categories of functors between enriched $\infty$-categories automatically have the correct homotopy type. We construct the homotopy theory of $\mathcal{V}$-enriched $\infty$-categories as a certain full subcategory of the $\infty$-category of "many-object associative algebras" in $\mathcal{V}$. The latter are defined using a non-symmetric version of Lurie's $\infty$-operads, and we develop the basics of this theory, closely following Lurie's treatment of symmetric $\infty$-operads. While we may regard these "many-object" algebras as enriched $\infty$-categories, we show that it is precisely the full subcategory of "complete" objects (in the sense of Rezk, i.e. those whose space of objects is equivalent to its space of equivalences) which are local with respect to the class of fully faithful and essentially surjective functors. Lastly, we present some applications of our theory, most notably the identification of associative algebras in $\mathcal{V}$ as a coreflective subcategory of pointed $\mathcal{V}$-enriched $\infty$-categories as well as a proof of a strong version of the Baez-Dolan stabilization hypothesis.

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arXiv2018

Iterated spans and classical topological field theories

We construct higher categories of iterated spans, possibly equipped with extra structure in the form of "local systems", and classify their fully dualizable objects. By the Cobordism Hypothesis, these give rise to framed topological quantum field theories, which are the framed versions of the "classical" TQFTs considered in the quantization programme of Freed-Hopkins-Lurie-Teleman. Using this machinery, we also construct an infinity-category of Lagrangian correspondences between symplectic derived algebraic stacks and show that all its objects are fully dualizable.

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arXiv2014

The Becker-Gottlieb Transfer Is Functorial

The Becker-Gottlieb transfer gives a wrong-way map on suspension spectra for maps of spaces whose homotopy fibres are retracts of finite complexes. We prove that this construction is contravariantly functorial on the homotopy category level.

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