Abstract Partial difference operators for a large class of functors between presheaf categories are introduced, extending our previous work on the difference operator to the multivariable case. These combine into the Jacobian profunctor that provides the setting for a lax chain rule. We introduce a functorial version of multivariable Newton series whose aim is to recover a functor from its iterated differences. Not all functors are recovered; however, we get a best approximation in the form of a left adjoint, and the induced comonad is idempotent. Its fixed points are what we call soft analytic functors, a generalization of the well-studied multivariable analytic functors.

Abstract The results of Section 3.1 of the 2017 paper “Isomorphism Theorems between Models of Mixed Choice” need an additional assumption when $\bullet$ is “ $1$ .” If $\bullet$ is nothing or “ $\leq 1$ ,” no change is needed. Also, the mistake only applies to the angelic cases, namely to the maps $r_{{\mathtt {A}}{\mathtt {P}}}$ and $s^\bullet _{{\mathtt {A}}{\mathtt {P}}}$ ; the demonic cases $r_{{\mathtt {D}}{\mathtt {P}}}$ and $s^\bullet _{{\mathtt {D}}{\mathtt {P}}}$ are unaffected. If $\bullet$ is “ $1$ ,” and in the angelic cases, instead of just assuming that $\mathcal L X$ is locally convex, we need to additionally assume that $X$ is compact, or that $\mathcal L X$ is locally convex-compact, sober, and topological – for example, if $X$ is core-compact – or that $X$ is LCS-complete, namely, a homeomorph of a $G_\delta$ subspace of a locally compact sober space.

Abstract We study the complexity of deterministic and probabilistic inversions of partial computable functions on the reals.

Abstract Stone locales together with continuous maps form a coreflective subcategory of spectral locales and perfect maps. A proof in the internal language of an elementary topos was previously given by the second-named author. This proof can be easily translated to univalent type theory using resizing axioms. In this work, we show how to achieve such a translation without resizing axioms, by working with large, locally small, and small-complete frames with small bases. This requires predicative reformulations of several fundamental concepts of locale theory in predicative HoTT/UF, which we investigate systematically.

Abstract We introduce Displayed Type Theory (dTT) , a multi-modal homotopy type theory with discrete and simplicial modes. In the intended semantics, the discrete mode is interpreted by a model for an arbitrary $\infty$ -topos, while the simplicial mode is interpreted by Reedy fibrant augmented semi-simplicial diagrams in that model. This simplicial structure is represented inside the theory by a primitive notion of display or dependency , guarded by modalities, yielding a partially-internal form of unary parametricity. Using the display primitive, we then give a coinductive definition, at the simplicial mode, of a type of semi-simplicial types. Roughly speaking, a semi-simplicial type consists of a type together with, for each , a displayed semi-simplicial type over . This mimics how simplices can be generated geometrically through repeated cones, and is made possible by the display primitive at the simplicial mode. The discrete part of then yields the usual infinite indexed definition of semi-simplicial types, both semantically and syntactically. Thus, dTT enables working with semi-simplicial types in full semantic generality.

Abstract We extend the recently introduced setting of coherent differentiation by taking into account not only differentiation but also Taylor expansion in categories which are not necessarily (left) additive. The main idea consists in extending summability into an infinitary functor which intuitively maps any object to the object of its countable summable families. This functor is endowed with a canonical structure of a bimonad. In a linear logical categorical setting, Taylor expansion is then axiomatized as a distributive law between this summability functor and the resource comonad (aka. exponential). This distributive law allows to extend the summability functor into a bimonad on the coKleisli category of the resource comonad: this extended functor computes the Taylor expansion of the (nonlinear) morphisms of the coKleisli category. We also show how this categorical axiomatization of Taylor expansion can be generalized to arbitrary cartesian categories, leading to a general theory of Taylor expansion formally similar to that of cartesian differential categories, although it does not require the underlying cartesian category to be left additive. We provide several examples of concrete categories that arise in denotational semantics and feature such analytic structures.

Abstract A topological space has a domain model if it is homeomorphic to the maximal point space $\mbox{Max}(P)$ of a domain $P$ . Lawson proved that every Polish space $X$ has an $\omega$ -domain model $P$ and for such a model $P$ , $\mbox{Max}(P)$ is a $G_{\delta }$ -set of the Scott space of $P$ . Martin (2003) then asked whether it is true that for every $\omega$ -domain $Q$ , $\mbox{Max}(Q)$ is $G_{\delta }$ -set of the Scott space of $Q$ . In this paper, we give a negative answer to Martin’s long-standing open problem by constructing a counterexample. The counterexample here actually shows that the answer is no even for $\omega$ -algebraic domains. In addition, we also construct an $\omega$ -ideal domain $\widetilde{Q}$ for the constructed $Q$ such that their maximal point spaces are homeomorphic. Therefore, $\textrm{Max}(Q)$ is a $G_\delta$ -set of the Scott space of the new model $\widetilde{Q}$ .

Abstract The authors’ primary goal in this paper is to enhance the study of $T_0$ topological spaces by using the order of specialization of a $T_0$ -space to introduce the lower topology (with a subbasis of closed sets $\mathord{\uparrow } x$ ) and studying the interaction of the original topology and the lower topology. Using the lower topology, one can define and study new properties of the original space that provide deeper insight into its structure. One focus of study is the property R, which asserts that if the intersection of a family of finitely generated sets $\mathord{\uparrow } F$ , $F$ finite, is contained in an open set $U$ , then the same is true for finitely many of the family. We first show that property R is equivalent to several other interesting properties, for example, the property that all closed subsets of the original space are compact in the lower topology. We then find conditions under which these spaces are compact, well-filtered, and coherent, a weaker variant of stably compact spaces. We also investigate what have been called strong $d$ -spaces, develop some of their basic properties, and make connections with the earlier considerations involving spaces satisfying property R. Two key results we obtain are that if a dcpo $P$ with the Scott topology is a strong $d$ -space, then it is well-filtered, and if additionally the Scott topology of the product $P\times P$ is the product of the Scott topologies of the factors, then the Scott space of $P$ is sober. We also exhibit connections of this work with de Groot duality.

Abstract We present algorithms and a C code to reveal quantum contextuality and evaluate the contextuality degree (a way to quantify contextuality) for a variety of point-line geometries located in binary symplectic polar spaces of small rank. With this code we were not only able to recover, in a more efficient way, all the results of a recent paper by de Boutray et al. [(2022). Journal of Physics A: Mathematical and Theoretical55 475301], but also arrived at a bunch of new noteworthy results. The paper first describes the algorithms and the C code. Then it illustrates its power on a number of subspaces of symplectic polar spaces whose rank ranges from 2 to 7. The most interesting new results include: (i) non-contextuality of configurations whose contexts are subspaces of dimension 2 and higher, (ii) non-existence of negative subspaces of dimension 3 and higher, (iii) considerably improved bounds for the contextuality degree of both elliptic and hyperbolic quadrics for rank 4, as well as for a particular subgeometry of the three-qubit space whose contexts are the lines of this space, (iv) proof for the non-contextuality of perpsets and, last but not least, (v) contextual nature of a distinguished subgeometry of a multi-qubit doily, called a two-spread, and computation of its contextuality degree. Finally, in the three-qubit polar space we correct and improve the contextuality degree of the full configuration and also describe finite geometric configurations formed by unsatisfiable/invalid constraints for both types of quadrics as well as for the geometry whose contexts are all 315 lines of the space.

Abstract On a locally $\lambda$ -presentable symmetric monoidal closed category $\mathcal {V}$ , $\lambda$ -ary enriched equational theories correspond to enriched monads preserving $\lambda$ -filtered colimits. We introduce discrete $\lambda$ -ary enriched equational theories where operations are induced by those having discrete arities (equations are not required to have discrete arities) and show that they correspond to enriched monads preserving preserving $\lambda$ -filtered colimits and surjections. Using it, we prove enriched Birkhof-type theorems for categories of algebras of discrete theories. This extends known results from metric spaces and posets to general symmetric monoidal closed categories.