We introduce in this paper a new formalisation of positive opetopes where faces are organised in a poset. Then we show that our definition is equivalent to that of positives opetopes as given by Marek Zawadowski.
We show that the E_1-equivalence C^∙(S^2) ≃ H^∙(S^2) does not intertwine the inclusion of constant loops into the free loop space S^2 → LS^2. That is, the isomorphism HH_∙(H^∙(S^2)) ≅ H^∙(LS^2) does not preserve the obvious maps to H^∙(S^2) that exist on both sides. We give an explicit computation of the defect in terms of the E_∞-structure on C^∙(S^2). Finally, we relate our calculation to recent work of Poirier-Tradler on the string topology of S^2.
This is a sequel of our previous work, arXiv:2209.09686, on the development of derived contact geometry, in which we formally introduced shifted contact structures on derived stacks and proved some results for $k$-shifted contact derived schemes, with $k<0$. In this paper, we extend these results from derived schemes to derived Artin stacks. In brief, we first show that for $k<0$, every $k$-shifted contact derived Artin stack admits a contact Darboux atlas. Secondly, we canonically describe the symplectification of a derived Artin stack equipped with a $k$-shifted contact structure, where $k<0$. Lastly, we give several constructions of contact derived stacks using certain cotangent stacks and shifted prequantization structures.
We give explicit formulas for the asymptotic growth rate of the number of summands in tensor powers in certain monoidal categories with finitely many indecomposable objects, and related structures.
This note unifies, in the framework of categories, the constructions named localizations of categories and algebraic dilatations of rings.
We develop the theory of Yoneda Ext groups over a ring in homotopy type theory (HoTT) and describe their interpretation into an $\infty$-topos. This is an abstract approach to Ext groups which does not require projective or injective resolutions. While it produces group objects that are a priori large, we show that the $\operatorname{Ext}^1$ groups are equivalent to small groups, leaving open the question of whether the higher Ext groups are essentially small as well. We also show that the $\operatorname{Ext}^1$ groups take on the usual form as a product of cyclic groups whenever the input modules are finitely presented and the ring is a PID (in the constructive sense). When interpreted into an $\infty$-topos of sheaves on a 1-category, our Ext groups recover (and give a resolution-free approach to) sheaf Ext groups, which arise in algebraic geometry. (These are also called "local" Ext groups.) We may therefore interpret results about Ext from HoTT and apply them to sheaf Ext. To show this, we prove that injectivity of modules in HoTT interprets to internal injectivity in these models. It follows, for example, that sheaf Ext can be computed using resolutions which are projective or injective in the sense of HoTT, when they exist, and we give an example of this in the projective case. We also discuss the relation between internal $\mathbb{Z} G$-modules (for a $0$-truncated group object $G$) and abelian groups in the slice over $BG$, and study the interpretation of our Ext groups in both settings.
We record two facts on spaces of derived maps between the operads E_d of little d-cubes. Firstly, these mapping spaces are equivalent to the mapping spaces between the non-unitary versions of E_d. Secondly, all endomorphisms of E_d are automorphisms. We also discuss variants for localisations of E_d and for versions with tangential structures.
We develop a ready-to-use comprehensive theory for (super) 2-vector bundles over smooth manifolds. It is based on the bicategory of (super) algebras, bimodules, and intertwiners as a model for 2-vector spaces. We discuss symmetric monoidal structures and the corresponding notions of dualizability, and we derive a classification in terms of Cech cohomology with values in a crossed module. One important feature of our 2-vector bundles is that they contain bundle gerbes as well as ordinary algebra bundles as full sub-bicategories, and hence provide a unifying framework for these so far distinct objects. We provide several examples of isomorphisms between bundle gerbes and algebra bundles, coming from representation theory, twisted K-theory, and spin geometry.
We extend Bourke and Garner's idempotent adjunction between monads and pretheories to the framework of $\infty$-categories and we use this to prove many classical results about monads in the $\infty$-categorical framework. Amongst other things, we show that the category of algebras for an accessible monads on a locally presentable $\infty$-category $\mathcal{E}$ is again locally presentable, and that a diagram of accessible monads on a locally presentable $\infty$-category admits a colimit. Our results also provide a new and simpler way to construct and describe monads in terms of theories.
This paper introduces the notion of $n$-morphisms between two $A_\infty$-algebras, such that 0-morphisms correspond to standard $A_\infty$-morphisms and 1-morphisms correspond to $A_\infty$-homotopies between $A_\infty$-morphisms. The set of higher morphisms between two $A_\infty$-algebras then defines a simplicial set which has the property of being an algebraic $\infty$-category. The operadic structure of $n-A_\infty$-morphisms is also encoded by new families of polytopes, which we call the $n$-multiplihedra and which generalize the standard multiplihedra. These are constructed from the standard simplices and multiplihedra by lifting the Alexander-Whitney map to the level of simplices. Rich combinatorics arise in this context, as conveniently described in terms of overlapping partitions. Shifting from the $A_\infty$ to the $\Omega B As$ framework, we define the analogous notion of $n$-morphisms between $\Omega B As$-algebras, which are again encoded by the $n$-multiplihedra, endowed with a refined cell decomposition by stable gauged ribbon tree type. We then realize this higher algebra of $A_\infty$ and $\Omega B As$-algebras in Morse theory. Given two Morse functions $f$ and $g$, we construct $n-\Omega B As$-morphisms between their respective Morse cochain complexes endowed with their $\Omega B As$-algebra structures, by counting perturbed Morse gradient trees associated to an admissible simplex of perturbation data. We moreover show that the simplicial set consisting of higher morphisms defined by a count of perturbed Morse gradient trees is a contractible Kan complex.
Weakly globular double categories are a model of weak $2$-categories based on the notion of weak globularity, and they are known to be suitably equivalent to Tamsamani $2$-categories. Fair $2$-categories, introduced by J. Kock, model weak $2$-categories with strictly associative compositions and weak unit laws. In this paper we establish a direct comparison between weakly globular double categories and fair $2$-categories and prove they are equivalent after localisation with respect to the $2$-equivalences. This comparison sheds new light on weakly globular double categories as encoding a strictly associative, though not strictly unital, composition, as well as the category of weak units via the weak globularity condition.
We define a notion of unitarity for pseudonatural transformations between unitary pseudofunctors on pivotal dagger 2-categories. We prove that the category Fun(C,D) of unitary pseudofunctors C -> D, unitary pseudonatural transformations and modifications is dagger with left and right duals, and furthermore pivotal dagger upon restriction to pivotal functors.
The spaces of configurations of non-k-overlapping discs have been studied as a bimodule over the little discs operad. In fact, the spaces form a filtered operad. We define and study the induced structure on the homology.
We develop a number of basic concepts in the theory of categories internal to an ∞-topos. We discuss adjunctions, limits and colimits as well as Kan extensions for internal categories, and we use these results to prove the universal property of internal presheaf categories. We furthermore construct the free cocompletion of an internal category by colimits that are indexed by an arbitrary class of diagram shapes.
We provide a reference for basic categorial properties of the categories of (possibly non-unital) $\mathbb{C}$-linear $*$-categories or $C^{*}$-categories, and (not necessarily unit-preserving) functors. Generalizing the classical case of algebras with $G$-action, we extend the construction of crossed products to categories with $G$-action. We will show that the crossed product functor preserves exact sequences and excisive squares and sends weak equivalences to equivalences.
We give a new proof of homological stability with the best known isomorphism range for mapping class groups of surfaces with respect to genus. The proof uses the framework of Randal-Williams-Wahl and Krannich applied to disk stabilization in the category of bidecorated surfaces, using the Euler characteristic instead of the genus as a grading. The monoidal category of bidecorated surfaces does not admit a braiding, distinguishing it from previously known settings for homological stability. Nevertheless, we find that it admits a suitable Yang-Baxter element, which we show is sufficient structure for homological stability arguments.
We define a tensor product for permutative categories and prove a number of key properties. We show that this product makes the 2-category of permutative categories closed symmetric monoidal as a bicategory.
It is known that the Grothendieck group of the category of Schur functors is the ring of symmetric functions. This ring has a rich structure, much of which is encapsulated in the fact that it is a"plethory": a monoid in the category of birings with its substitution monoidal structure. We show that similarly the category of Schur functors is a"2-plethory", which descends to give the plethory structure on symmetric functions. Thus, much of the structure of symmetric functions exists at a higher level in the category of Schur functors.
