We describe a $\C$-linear additive *-autonomous category of Banach spaces. Please note that a correction has been appended to the original version 1 which is maintained here for reference. Also, a proposed example of a *-autonomous category of topological $\C$-linear spaces has been added to version 2.
We prove that the category of Banach $L^0$-modules over a given $σ$-finite measure space is both complete and cocomplete, which means that it admits all small limits and colimits.
We prove that the forgetful functors from the categories of $C^*$- and $W^*$-algebras to Banach $*$-algebras, Banach algebras or Banach spaces are all monadic, answering a question of J.Rosický, and that the categories of unital (commutative) $C^*$-algebras are not locally-isometry $\aleph_0$-generated either as plain or as metric-enriched categories, answering a question of I. Di Liberti and Rosický. We also prove a number of negative presentability results for the category of von Neumann algebras: not only is that category not locally presentable, but in fact its only presentable objects are…
Properties of categories enriched over the category of metric spaces are investigated and applied to a study of constructions known from that category and the category of Banach spaces. For every class of morphisms satisfying a mild smallness condition we prove the corresponding approximate-injectivity class is weakly reflective, and we study the properties of the reflection morphisms. As an application we present a new categorical proof of the essential uniqueness of the Gurarii space.
我们定义了一个单子M一类可测量的代数集，我们展示了如何产生这个单子的理论向量值的整合，是有关的概念佩蒂斯积分。我们证明了这个单子的代数X是一个代数学上的局部凸向量空间，它被赋予了将X中的向量\int f dm与传入映射f：T --> X和T上的测度m相关联的运算。证明了只要Banach空间中的每一个有界弱可测函数都有Pettis积分，则Banach空间是M-代数。因此，所有可分的巴拿赫空间和所有自反的巴拿赫空间都是M-代数。
Let $\mathcal C$ be a concrete category. We prove that if $\mathcal{C}$ admits a universally free object $\mathsf F$, then there is a projectively universal morphism $u\colon \mathsf F\to \mathsf F$, i.e., a morphism $u$ such that for any $B\in \mathcal{C}$ and $τ\in {\rm Mor}(B)$ there exists an epimorphism $π\in {\rm Mor}(\mathsf F, B)$ such that $πτ= u π$. This builds upon and extends various ideas by Darji and Matheron (Proc. Am. Math. Soc. 145 (2017)) who proved such a result for the category of separable Banach spaces with contractive operators as well as certain classes of dynamical sys…
We prove 2-categorical conservativity for any {0,T}-free fragment of MALL over its corresponding intuitionistic version: that is, that the universal map from a closed symmetric monoidal category to the *-autonomous category that it freely generates is fully faithful, and similarly for other doctrines. This implies that linear logics and graphical calculi for *-autonomous categories can also be interpreted canonically in closed symmetric monoidal categories. In particular, every closed symmetric monoidal category can be fully embedded in a *-autonomous category, preserving both tensor products …
It is known that the quantale of sup-preserving maps from a complete lattice to itself is a Frobenius quantale if and only if the lattice is completely distributive. Since completely distributive lattices are the nuclear objects in the autonomous category of complete lattices and sup-preserving maps, we study the above statement in a categorical setting. We introduce the notion of Frobenius structure in an arbitrary autonomous category, generalizing that of Frobenius quantale. We prove that the monoid of endomorphisms of a nuclear object has a Frobenius structure. If the environment category i…
We prove that the category of dg-modules and dg-algebras in a Grothendieck quasi-abelian category are endowed with a Quillen model structure. This allows some flexibility in setting up a theory of derived algebraic geometry in the infinite dimensional setting. For example, the category of complete bornological vector spaces, or equivalently, convenient vector spaces, is a Grothendieck quasi-abelian category. Closely related is the Grothendieck quasi-abelian category of ind-Banach spaces whose associated model category is shown to be Quillen equivalent. Applications include the Chevalley-Eilenb…
Effectus theory is a relatively new approach to categorical logic that can be seen as an abstract form of generalized probabilistic theories (GPTs). While the scalars of a GPT are always the real unit interval [0,1], in an effectus they can form any effect monoid. Hence, there are quite exotic effectuses resulting from more pathological effect monoids. In this paper we introduce sigma-effectuses, where certain countable sums of morphisms are defined. We study in particular sigma-effectuses where unnormalized states can be normalized. We show that a non-trivial sigma-effectus with normalization…
We define and study a probability monad on the category of complete metric spaces and short maps. It assigns to each space the space of Radon probability measures on it with finite first moment, equipped with the Kantorovich-Wasserstein distance. This monad is analogous to the Giry monad on the category of Polish spaces, and it extends a construction due to van Breugel for compact and for 1-bounded complete metric spaces. We prove that this Kantorovich monad arises from a colimit construction on finite power-like constructions, which formalizes the intuition that probability measures are limit…
