Projects

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.

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 ∞-operads, for which we also derive an explicit formula

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

We generalize Lurie’s construction of the symmetric monoidal envelope of an ∞-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 ∞-operads over various algebraic patterns. In particular, we show that the G-∞-operads of Nardin-Shah are equivalent to “fibrous patterns” over the (2,1)-category Span(FG) of spans of finite G-sets. When G is trivial this means that Lurie’s ∞-operads can equivalently be defined over Span(F) instead of F∗.

We give a simple proof that complete Segal animae are equivalent to categories.

Dagger categories are an essential tool for categorical descriptions of quantum physics, but pose a challenge to category theorists as their definition is in tension with the “principle of equivalence” that lies at the heart of category theory. In this note we propose the alternative, coherent definition of an “involutive category with a notion of positivity” and show that the 2-category formed by these is biequivalent to the 2-category of dagger categories.

We define a tracelike transformation to be a natural family of conjugation invariant maps Tx,C:homC(x,x)→homC(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)∈k to any endomorphism f:V→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↦Tr(fn). Unlikefor Tr the relevant connected component of the moduli space is not contractible, but ratherequivalent to BZ/nZ or BS1 for n=0. As a result we obtain a Z/nZ-action on Tr(fn) as well as a circle action on Tr(idx).

We compute the classifying space of the surface category Cob2 whose objects are closed 1-manifolds and whose morphisms are diffeomorphism classes of 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 C2 studied by Galatius-Madsen-Tillmann-Weiss. However, we also show that for the wide subcategory Cobχ≤02⊂Cob2 that contains all morphisms without disks or spheres, the classifying space BCobχ≤02 is surprisingly large. Its rational homotopy groups contain the homology of all moduli spaces of tropical curves Δg 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 labelled cospan categories. We also use this to show that the (2,1)-category of cospans of finite sets has a contractible classifying space.

The homotopy category of the bordism category hBordd 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 hBordd for d=1, exhibiting it as a circle bundle over CP∞−1. As part of our proof we construct a quotient Bordred1 of the cobordism category where circles are deleted. We show that this category has classifying space Ω∞−2CP∞−1 and moreover that, if one equips these bordisms with a map to a simply connected space X, the resulting Bordred1(X) can be thought of as a cobordism model for the topological cyclic homology TC(S[ΩX]). In the second part of the paper we construct an infinite loop space map B(hBordred1)→Q(Σ2CP∞+) in this model and use it to derive combinatorial formulas for rational cocycles on Bordred1 representing the Miller-Morita-Mumford classes κi∈H2i+2((B(hBord1);Q).

We show that the conditions in Steimle’s ‘additivity theorem for cobordism categories’ can be weakened to only require \emph{locally} (co)Cartesian fibrations, making it applicable to a larger class of functors. As an application we compute the difference in classifying spaces between the infinity category of cospans of finite sets and its homotopy category.