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Given a monotone Lagrangian $L$ in a compact symplectic manifold $X$, we construct a commutative diagram relating the closed-open string map $\mathcal{CO}_\lambda \colon \operatorname{QH}^*(X) \to \operatorname{HH}^*(\mathcal{F}…
Category theory is the language of homological algebra, allowing us to state broadly applicable theorems and results without needing to specify the details for every instance of analogous objects. However, authors often stray from the realm…
On a cotangent bundle $T\sp*G$ of a Lie group $G$ one can describe the standard Liouville form $\theta$ and the symplectic form $d \theta$ in terms of the right Maurer Cartan form and the left moment mapping (of the right action of $G$ on…
Given a category $\mathcal{E}$, we establish sufficient conditions on a faithful isofibration $\mathcal{E}\rightarrow\operatorname{Mon}(\mathcal{V})$ valued in the category of monoids internal to a monoidal additive category $\mathcal{V}$…
We use the terms "$\infty$-categories" and "$\infty$-functors" to mean the objects and morphisms in an "$\infty$-cosmos." Quasi-categories, Segal categories, complete Segal spaces, naturally marked simplicial sets, iterated complete Segal…
In the preceding part (I) of this paper, we showed that for any torsion pair (i.e., $t$-structure without the shift-closedness) in a triangulated category, there is an associated abelian category, which we call the heart. Two extremal cases…
Bayes' rule tells us how to invert a causal process in order to update our beliefs in light of new evidence. If the process is believed to have a complex compositional structure, we may observe that the inversion of the whole can be…
We show an organized form of quantum de Finetti theorem for Boolean independence. We define a Boolean analogue of easy quantum groups for the categories of interval partitions, which is a family of sequences of quantum semigroups. We…
We formulate a model-independent theory of co/cartesian morphisms and co/cartesian fibrations: that is, one which resides entirely *within the $\infty$-category of $\infty$-categories*. We prove this is suitably compatible with the…
We develop a technique for studying first-order codifferential calculi (FOCCs) initiated by Doi and Quillen in the context of cyclic cohomology. Their classification, for a given coalgebra, reduces to the classification of subbicomodules in…
It is well known that both the symplectic structure and the Poisson brackets of classical field theory can be constructed directly from the Lagrangian in a covariant way, without passing through the non-covariant canonical Hamiltonian…
We prove an equivalence between cocomplete Yoneda structures and certain proarrow equipments on a 2-category $\mathcal K$. In order to do this, we recognize the presheaf construction of a cocomplete Yoneda structure as a relative, lax…
We define a class of higher inductive types that can be constructed in the category of sets under the assumptions of Zermelo-Fraenkel set theory without the axiom of choice or the existence of uncountable regular cardinals. This class…
A general construction of Knop creates a symmetric monoidal category $\mathcal{T}(\mathcal{A},\delta)$ from any regular category $\mathcal{A}$ and a fixed degree function $\delta$. A special case of this construction are the Deligne…
We construct a cofibrantly generated Quillen model structure on the category of small differential graded categories. ----- Nous construisons une structure de categorie de modeles de Quillen a engendrement cofibrant sur la categorie des…
The covariant canonical formalism is a covariant extension of the traditional canonical formalism of fields. In contrast to the traditional canonical theory, it has a remarkable feature that canonical equations of gauge theories or gravity…
Motivated by the Grothendieck construction, we study the functorialities of the comma construction for strict $\omega$-categories. To state the most general functorialities, we use the language of Gray $\omega$-categories, that is,…
Coalgebras for an endofunctor provide a category-theoretic framework for modeling a wide range of state-based systems of various types. We provide an iterative construction of the reachable part of a given pointed coalgebra that is inspired…
The construction of a C*-algebra of a differential groupoid is presented. It is shown that it defines a covariant functor from the category of differential groupoids in a sense of S. Zakrzewski to the category of C*-algebras.
Recently, there has been renewed interest in the theory and applications of de Paiva's dialectica categories and their relationship to the category of polynomial functors. Both fall under the theory of generalized polynomial categories,…