Jonathan A. Campbell, Kate Ponto
While not obvious from its initial motivation in linear algebra, there are many context where iterated traces can be defined. In this paper we prove a very general theorem about iterated 2-categorical traces. We show that many Lefschetz-type theorems in the literature are consequences of this result and the new perspective we provide allows for immediate spectral generalizations.
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Lefschetz fixed-object and fixed-morphism theorems for acyclic categories
本文为有限无环范畴上的自函子提出两种互补的 Lefschetz 数，并借助单纯结构与重心细分函子证明固定对象及固定态射链定理。
We develop a notion of iterated monoidal category and show that this notion corresponds in a precise way to the notion of iterated loop space. Specifically the group completion of the nerve of such a category is an iterated loop space and free iterated monoidal categories give rise to finite simplicial operads of the same homotopy type as the classical little cubes operads used to parametrize the higher H-space structure of iterated loop spaces. Iterated monoidal categories encompass, as a special case, the notion of braided tensor categories, as used in the theory of quantum groups.
The Lefschetz fixed point theorem follows easily from the identification of the Lefschetz number with the fixed point index. This identification is a consequence of the functoriality of the trace in symmetric monoidal categories. There are refinements of the Lefschetz number and the fixed point index that give a converse to the Lefschetz fixed point theorem. An important part of this theorem is the identification of these different invariants. We define a generalization of the trace in symmetric monoidal categories to a trace in bicategories with shadows. We show the invariants used in the con…
We set up operadic foundations for equivariant iterated loop space theory. We start by building up from a discussion of the approximation theorem and recognition principle for V-fold loop G-spaces to several avatars of a recognition principle for infinite loop G-spaces. We then explain what genuine permutative G-categories are and, more generally, what E_{\infty} G-categories are, giving examples showing how they arise. As an application, we prove the equivariant Barratt-Priddy-Quillen theorem as a statement about genuine G-spectra and use it to give a new, categorical, proof of the tom Dieck …
Topological field theories associated with Calabi-Yau categories
本文构造迭代 Calabi-Yau cospan 的对称幺半高阶范畴，并给出适用于逗号范畴的一般函子性程序，从而将 AKSZ 完全扩展 TFT 分解为取值于迭代 Calabi-Yau cospan 的完全扩展 TFT。
Operads were originally defined as V-operads, that is, enriched in a symmetric or braided monoidal category V. The symmetry or braiding in V is required in order to describe the associativity axiom the operads must obey, as well as the associativity that must be a property of the action of an operad on any of its algebras. After a review of the role of operads in loop space theory and higher categories we go over definitions of iterated monoidal categories and introduce a large family of simple examples. Then we generalize the definition of operad by defining n-fold operads and their algebras …
In this paper, we show that two constructions form stacks: Firstly, as one varies the $\infty$-topos, $\mathcal{X}$, Lurie's homotopy theory of higher categories internal to $\mathcal{X}$ varies in such a way as to form a stack over the $\infty$-category of all $\infty$-topoi. Secondly, we show that Haugseng's construction of the higher category of iterated spans in a given $\infty$-topos (equipped with local systems) can be used to define various stacks over that $\infty$-topos. As a prerequisite to these results, we discuss properties which limits of $\infty$-categories inherit from the $\in…
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.
The 2-category V-Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V. The exception is the case in which V is symmetric, which leads to V-Cat being symmetric as well. This paper describes how these facts are related to a categorical analogue of topological delooping. It seems that the analogy of loop spaces is a good guide for how to define the concept of enrichment over various types of monoidal objects, including k-fold monoidal categories and their higher dimensional counterparts. The main result is that for V a…
The structure of a $k$-fold monoidal category as introduced by Balteanu, Fiedorowicz, Schwänzl and Vogt can be seen as a weaker structure than a symmetric or even braided monoidal category. In this paper we show that it is still sufficient to permit a good definition of ($n$-fold) operads in a $k$-fold monoidal category which generalizes the definition of operads in a braided category. Furthermore, the inheritance of structure by the category of operads is actually an inheritance of iterated monoidal structure, decremented by at least two iterations. We prove that the category of $n$-fold oper…
Joyal and Street note in their paper on braided monoidal categories [Braided tensor categories, Advances in Math. 102(1993) 20-78] that the 2-category V-Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V. The exception that they mention is the case in which V is symmetric, which leads to V-Cat being symmetric as well. The symmetry in V-Cat is based upon the symmetry of V. The motivation behind this paper is in part to describe how these facts relating V and V-Cat are in turn related to a categorical analogue of to…
We extend Thomason's homotopy colimit construction in the category of permutative categories to categories of algebras over an arbitrary $\Cat$ operad and analyze its properties. We then use this homotopy colimit to prove that the classifying space functor induces an equivalence between the category of $n$-fold monoidal categories and the category of $\mathcal{C}_n$-spaces after formally inverting certain classes of weak equivalences, where $\mathcal{C}_n$ is the little $n$-cubes operad. As a consequence we obtain an equivalence of the categories of $n$-fold monoidal categories and the categor…
Iterated magnitude homology
本文提出迭代量级同调，融合量级同调与几何神经，适用于高阶丰富范畴，在严格2-范畴中恢复分类空间同调，并探讨偏序群、赋范群等情形。
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.
We show that in any symmetric monoidal category, if a weight for colimits is absolute, then the resulting colimit of any diagram of dualizable objects is again dualizable. Moreover, in this case, if an endomorphism of the colimit is induced by an endomorphism of the diagram, then its trace can be calculated as a linear combination of traces on the objects in the diagram. The formal nature of this result makes it easy to generalize to traces in homotopical contexts (using derivators) and traces in bicategories. These generalizations include the familiar additivity of the Euler characteristic an…
We propose a recursive definition of V-n-categories and their morphisms. We show that for V k-fold monoidal the structure of a (k-n)-fold monoidal strict (n+1)-category is possessed by V-n-Cat. This article is a completion of the work begun in math.CT/0306086, and the initial sections duplicate the beginning of that paper.
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 iterate…
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