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Jan Steinebrunner

Publications and source records attributed to Jan Steinebrunner.

16 recordsLinked to original sources
arXiv2026

The prime decomposition fibre sequence for moduli spaces of reducible 3-manifolds

We study the moduli space $B\textrm{Diff}^+(M)$, for $M$ a reducible, oriented 3-manifold with irreducible prime factors $P_1,\ldots,P_n$. A programme of C\'esar de S\'a-Rourke, Hendriks-Laudenbach, and Hendriks-McCullough studies the homotopy type of $\textrm{Diff}^+(M)$ in terms of the $\textrm{Diff}^+(P_i)$. Inspired by a delooping proposed by Hatcher, we construct a map from $B\textrm{Diff}^+(M)$ to $B\textrm{Diff}^+(P_1 \sqcup \dots \sqcup P_n)$, called the splitting map, that yields a prime decomposition fibre sequence. The fibre $H_g(P_1, \dots, P_n)$ is a space of $1$-handle attachments which we describe geometrically as a homotopy colimit of certain configuration spaces on the $P_i$. Firstly, this allows us to show that for $n>0$ the fibre is equivalent to a finite, connected cell complex. Secondly, this makes the fibre sequence an effective tool for computations, which we illustrate by computing the rational cohomology ring of $B\textrm{Diff}^+\!\left((S^1\times S^2)^{\sharp 2}\right)$.

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arXiv2025

Open 2D TFTs admit initial open-closed extensions

We show that any open 2-dimensional topological field theory valued in a symmetric monoidal $\infty$-category (with suitable colimits) extends canonically to an open-closed field theory whose value at the circle is the Hochschild homology object of its value at the disk. As a corollary, we obtain an action of the moduli spaces of surfaces on the Hochschild homology object of $E_1$-Calabi-Yau algebras. This provides a space level refinement of previous work of Costello over $\mathbb{Q}$ and Wahl-Westerland and Wahl over $\mathbb{Z}$, and serves as a crucial ingredient to Lurie's "non-compact cobordism hypothesis" in dimension 2. As part of the proof we also give a description of slice categories of the d-dimensional bordism category with boundary, which may be of independent interest.

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arXiv2025

2-dimensional TFTs via modular $\infty$-operads

This lecture series is based on joint work in progress with Shaul Barkan, as well as work in progress of the author. The five sections of these notes correspond to the five lectures, but more details have been added. $2$-dimensional topological field theories ($2$D TFTs) valued in vector spaces are commutative Frobenius algebras. The goal of this lecture series is to generalize from the $1$-category of vector spaces to any symmetric monoidal $\infty$-category $\mathcal{C}$, i.e. to study symmetric monoidal functors $\mathrm{Bord}_2 \to \mathcal{C}$. Choosing $\mathcal{C}$ to be the $(2,1)$-category of linear categories, this recovers a definition of modular functors, and choosing it to be the derived category of a ring yields a notion closely related to cohomological field theories. We will introduce a notion of modular $\infty$-operads and algebras over them, construct the modular $\infty$-operad of surfaces $\mathcal{M}$, and show that algebras over $\mathcal{M}$ in $\mathcal{C}$ are exactly $2$D TFTs valued in $\mathcal{C}$. Along the way we will encounter variants modular $\infty$-operads (such as cyclic $\infty$-operads and $\infty$-properads) as well as a proof of the $1$D cobordism hypothesis with singularities. This uses some (mild) $\infty$-category theory, but no familiarity with ($\infty$-)operads will be assumed. The main goal will be to filter $\mathcal{M}$ by genus to obtain an obstruction-theoretic description of $2$D TFTs with general target. Applying this to invertible TFTs one can construct a new spectral sequence exhibiting relations between the cohomology groups of moduli spaces of curves.

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arXiv2025

Fully faithful functors and pushouts of $\infty$-categories

We study stability properties of fully faithful functors, and compute mapping anima in pushouts of $\infty$-categories along fully faithful functors. We provide applications of these calculations to pushouts along Dwyer functors and Reedy categories.

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arXiv2024

Moduli spaces of 3-manifolds with boundary are finite

We study the classifying space B Diff(M) of the diffeomorphism group of a connected, compact, orientable 3-manifold M. In the case that M is reducible we build a contractible space parametrising the systems of reducing spheres. We use this to prove that if M has non-empty boundary, then B Diff(M rel boundary) has the homotopy type of a finite CW complex. This was conjectured by Kontsevich and appears on the Kirby problem list as Problem 3.48. As a consequence, we are able to show that for every compact, orientable 3-manifold M, B Diff(M) has finite type.

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arXiv2023

Dagger categories via anti-involutions and positivity

Dagger categories are an essential tool for categorical descriptions of quantum physics, for example in categorical quantum mechanics and unitary topological field theory. Their definition however is in tension with the ``principle of equivalence'' that lies at the heart of category theory, thereby inhibiting generalizations to higher categories. In this note we propose an alternative, coherent description of dagger categories based on the well-studied notion of anti-involutions $d\colon \mathcal{C} \to \mathcal{C}^{op}$, which coherently square to the identity functor $\eta\colon d^2 \cong \operatorname{id}_{\mathcal{C}}$. A general anti-involution need not be the identity on objects, but we instead consider certain isomorphisms $dx \cong x$, which we call Hermitian fixed points as they generalize the notion of a Hermitian inner product on a vector space. We define a ``positivity notion" on $(\mathcal{C},d, \eta)$ in terms of such Hermitian fixed points. This terminology is motivated by the dagger category of Hilbert spaces, in which case the positivity notion consists of the positive definite pairings. Our main result is that the $2$-category of anti-involutive categories with a positivity notion is biequivalent to the $2$-category of dagger categories.

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arXiv2023

Segalification and the Boardman-Vogt tensor product

We develop an analog of Dugger and Spivak's necklace formula providing an explicit description of the Segal space generated by an arbitrary simplicial space. We apply this to obtain a formula for the Segalification of $n$-fold simplicial spaces, a new proof of the invariance of right fibrations, and a new construction of the Boardman-Vogt tensor product of $\infty$-operads, for which we also derive an explicit formula.

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arXiv2022

The equifibered approach to $\infty$-properads

We define a notion of $\infty$-properads that generalises $\infty$-operads by allowing operations with multiple outputs. Specializing to the case where each operation has a single output provides a simple new perspective on $\infty$-operads, but at the same time the extra generality allows for examples such as bordism categories. We also give an interpretation of our $\infty$-properads as Segal presheaves on a category of graphs by comparing them to the Segal $\infty$-properads of Hackney-Robertson-Yau. Combining these two approaches yields a flexible tool for doing higher algebra with operations that have multiple inputs and outputs. Crucially, this allows for a definition of algebras over an $\infty$-properad such that, for example, topological field theories are algebras over the bordism $\infty$-properad. The key ingredient to this paper is the notion of an equifibered map between $E_\infty$-monoids, which is a well-behaved generalisation of free maps. We also use this to prove facts about free $E_\infty$-monoids, for example that free $E_\infty$-monoids are closed under pullbacks along arbitrary maps.

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arXiv2022

Envelopes for Algebraic Patterns

We generalize Lurie's construction of the symmetric monoidal envelope of an $\infty$-operad to the setting of algebraic patterns. This envelope becomes fully faithful when sliced over the envelope of the terminal object, and we characterize its essential image. Using this, we prove a comparison result that allows us to compare analogues of $\infty$-operads over various algebraic patterns. In particular, we show that the $G$-$\infty$-operads of Nardin-Shah are equivalent to "fibrous patterns" over the $(2, 1)$-category $\mathrm{Span}(\mathbb{F}_G)$ of spans of finite $G$-sets. When $G$ is trivial this means that Lurie's $\infty$-operads can equivalently be defined over $\mathrm{Span}(\mathbb{F})$ instead of $\mathbb{F}_*$.

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arXiv2021

The surface category and tropical curves

We compute the classifying space of the surface category $h\mathrm{Bord}_2$ whose objects are closed oriented $1$-manifolds and whose morphisms are diffeomorphism classes of oriented surface bordisms, and show that it is rationally equivalent to a circle. It is hence much smaller than the classifying space of the topologically enriched surface category $\mathrm{Bord}_2$ studied by Galatius-Madsen-Tillmann-Weiss. However, we also show that for the wide subcategory $h\mathrm{Bord}_2^{\chi\le 0} \subset h\mathrm{Bord}_2$ that contains all morphisms without disks or spheres, the classifying space $B(h\mathrm{Bord}_2^{\chi\le0})$ is surprisingly large. Its rational homotopy groups contain the homology of all moduli spaces of tropical curves $\Delta_g$ as a summand. The technical key result shows that a version of positive boundary surgery applies to a large class of discrete symmetric monoidal categories, which we call \emph{labelled cospan categories}. We also use this to show that the $(2,1)$-category of cospans of finite sets has a contractible classifying space.

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arXiv2021

The space of traces in symmetric monoidal infinity categories

We define a tracelike transformation to be a natural family of conjugation invariant maps $T_{x,C}: hom_C(x,x) \to hom_C(1,1)$ for all dualisable objects $x$ in any symmetric monoidal infinity-category $C$. This generalises the trace from linear algebra that assigns a scalar $Tr(f) \in k$ to any endomorphism $f:V \to V$ of a finite-dimensional $k$-vector space. Our main theorem computes the moduli space of tracelike transformations using the one-dimensional cobordism hypothesis with singularities. As a consequence we show that the trace $Tr$ can be uniquely extended to a tracelike transformation up to a contractible space of choices. This allows us to give several model-independent characterisations of the infinity-categorical trace. Restricting our notion of tracelike transformations from endomorphisms to automorphisms we in particular recover a theorem of Toën and Vezzosi. Other examples of tracelike transformations are for instance given by $f \mapsto Tr(f^n)$. Unlikefor $Tr$ the relevant connected component of the moduli space is not contractible, but ratherequivalent to $B\mathbb{Z}/n\mathbb{Z}$ or $BS^1$ for $n=0$. As a result we obtain a $\mathbb{Z}/n\mathbb{Z}$-action on $Tr(f^n)$ as well as a circle action on $Tr(id_x)$.

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arXiv2020

The classifying space of the one-dimensional bordism category and a cobordism model for TC of spaces

The homotopy category of the bordism category $hBord_d$ has as objects closed oriented $(d-1)$-manifolds and as morphisms diffeomorphism classes of $d$-dimensional bordisms. Using a new fiber sequence for bordism categories, we compute the classifying space of $hBord_d$ for $d = 1$, exhibiting it as a circle bundle over $\mathbb{CP}^\infty_{-1}$. As part of our proof we construct a quotient $Bord_1^{red}$ of the cobordism category where circles are deleted. We show that this category has classifying space $Ω^{\infty-2}\mathbb{CP}^\infty_{-1}$ and moreover that, if one equips these bordisms with a map to a simply connected space $X$, the resulting $Bord_1^{red}(X)$ can be thought of as a cobordism model for the topological cyclic homology $TC(\mathbb{S}[ΩX])$. In the second part of the paper we construct an infinite loop space map $B(hBord_1^{red}) \to Q(Σ^2 \mathbb{CP}^\infty_+)$ in this model and use it to derive combinatorial formulas for rational cocycles on $Bord_1^{red}$ representing the Miller-Morita-Mumford classes $κ_i \in H^{2i+2}((B(hBord_1); \mathbb{Q})$.

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arXiv2016

Homotopy- and Cohomology Groups of Kan Complexes

This article shows several new methods for proofs on Kan complexes while using them to give a compact introduction to the homotopy groups of these complexes. Then more advanced objects are studied starting with homology and the Hurewicz homomorphism. Eilenberg-Mac Lane-spaces are constructed explicitly and can then be used to define spectral cohomology. In the end the equivalence to the usual simplicial cohomology is shown.

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