Two high-level "pictures" of probability theory have emerged: one that takes as central the notion of random variable, and one that focuses on distributions and probability channels (Markov kernels). While the channel-based picture has been successfully axiomatized, and widely generalized, using the notion of Markov category, the categorical semantics of the random variable picture remain less clear. Simpson's probability sheaves are a recent approach, in which probabilistic concepts like random variables are allowed vary over a site of sample spaces. Simpson has identified rich structure on these sites, most notably an abstract notion of conditional independence, and given examples ranging from probability over databases to nominal sets. We aim bring this development together with the generality and abstraction of Markov categories: We show that for any suitable Markov category, a category of sample spaces can be defined which satisfies Simpson's axioms, and that a theory of probability sheaves can be developed purely synthetically in this setting. We recover Simpson's examples in a uniform fashion from well-known Markov categories, and consider further generalizations.
We exhibit a bridge between the theory of cellular categories, used in algebraic topology and homological algebra, and the model-theoretic notion of stable independence. Roughly speaking, we show that the combinatorial cellular categories (those where, in a precise sense, the cellular morphisms are generated by a set) are exactly those that give rise to stable independence notions. We give two applications: on the one hand, we show that the abstract elementary classes of roots of Ext studied by Baldwin-Eklof-Trlifaj are stable and tame. On the other hand, we give a simpler proof (in a special …
For most models of $(\infty,2)$-categories an embedding of the $\infty$-category of 2-categories into that of $(\infty,2)$-categories has been constructed in the form of a nerve construction of some flavor. We prove that all those nerve embeddings induce equivalent functors, modulo change of model. We also show that all the nerve embeddings realize the $\infty$-category of 2-categories as the sub-$\infty$-category of $(\infty,2)$-categories that are local with respect to a certain class of maps.
We provide several crucial technical extensions of the theory of stable independence notions in accessible categories. In particular, we describe circumstances under which a stable independence notion can be transferred from a subcategory to a category as a whole, and examine a number of applications to categories of groups and modules, extending results of [MAa]. We prove, too, that under the hypotheses of [LRV], a stable independence notion immediately yields higher-dimensional independence as in [SV].
We study concrete sheaf models for a call-by-value higher-order language with recursion. Our family of sheaf models is a generalization of many examples from the literature, such as models for probabilistic and differentiable programming, and fully abstract logical relations models. We treat recursion in the spirit of synthetic domain theory. We provide a general construction of a lifting monad starting from a class of admissible monomorphisms in the site of the sheaf category. In this way, we obtain a family of models parametrized by a concrete site and a class of monomorphisms, for which we …
We exhibit an equivalence between the model-theoretic framework of universal classes and the category-theoretic framework of locally multipresentable categories. We similarly give an equivalence between abstract elementary classes (AECs) admitting intersections and locally polypresentable categories. We use these results to shed light on Shelah's presentation theorem for AECs.
The primary contribution of this paper is to give a formal, categorical treatment to Penrose's abstract tensor notation, in the context of traced symmetric monoidal categories. To do so, we introduce a typed, sum-free version of an abstract tensor system and demonstrate the construction of its associated category. We then show that the associated category of the free abstract tensor system is in fact the free traced symmetric monoidal category on a monoidal signature. A notable consequence of this result is a simple proof for the soundness and completeness of the diagrammatic language for trac…
We present a fully abstract model of a call-by-value language with higher-order functions, recursion and natural numbers, as an exponential ideal in a topos. Our model is inspired by the fully abstract models of O'Hearn, Riecke and Sandholm, and Marz and Streicher. In contrast with semantics based on cpo's, we treat recursion as just one feature in a model built by combining a choice of modular components.
A generalization of incidence relations in abstract polytope has been explored, and parameterized surfaces are used as primers. The abstract orientable incidence structure is defined as an algebraic model of incidence relations, in which some algebraic properties in abtract polytope theory are generalized. The geometric interpretation of abstract orientable incidence structure are also discussed. The orientable incidence structure in a semi-regular normal CW complex are briefly investigated.
Forking is a central notion of model theory, generalizing linear independence in vector spaces and algebraic independence in fields. We develop the theory of forking in abstract, category-theoretic terms, for reasons both practical (we require a characterization suitable for work in $μ$-abstract elementary classes, i.e. accessible categories with all morphisms monomorphisms) and expository (we hope, with this account, to make forking accessible - and useful - to a broader mathematical audience). In particular, we present an axiomatic definition of what we call a stable independence notion on a…
We introduce the CP*-construction on a dagger compact closed category as a generalisation of Selinger's CPM-construction. While the latter takes a dagger compact closed category and forms its category of "abstract matrix algebras" and completely positive maps, the CP*-construction forms its category of "abstract C*-algebras" and completely positive maps. This analogy is justified by the case of finite-dimensional Hilbert spaces, where the CP*-construction yields the category of finite-dimensional C*-algebras and completely positive maps. The CP*-construction fully embeds Selinger's CPM-constru…
We introduce the framework of AECats (abstract elementary categories), generalising both the category of models of some first-order theory and the category of subsets of models. Any AEC and any compact abstract theory ("cat", as introduced by Ben-Yaacov) forms an AECat. In particular, we find applications in positive logic and continuous logic: the category of (subsets of) models of a positive or continuous theory is an AECat. The Kim-Pillay theorem for first-order logic characterises simple theories by the properties dividing independence has. We prove a version of the Kim-Pillay theorem for …
We show that metric abstract elementary classes (mAECs) are, in the sense of [LR] (i.e. arXiv:1404.2528), coherent accessible categories with directed colimits, with concrete $\aleph_1$-directed colimits and concrete monomorphisms. More broadly, we define a notion of $κ$-concrete AEC---an AEC-like category in which only the $κ$-directed colimits need be concrete---and develop the theory of such categories, beginning with a category-theoretic analogue of Shelah's Presentation Theorem and a proof of the existence of an Ehrenfeucht-Mostowski functor in case the category is large. For mAECs in par…
The notion of a tensor product with projections or with inclusions is defined. It is shown that the definition of stochastic independence relies on such a structure and that independence can be defined in an arbitrary category with a tensor product with inclusions or projections. In this context, the classifications of quantum stochastic independence by Muraki, Ben Ghorbal, and Schürmann become classifications of the tensor products with inclusions for the categories of algebraic probability spaces and non-unital algebraic probability spaces. The notion of a reduction of one independence to an…
This is an introduction to the study of abstract homotopy theory by means of model categories and $(\infty,1)$-categories. The only prerequisites are very basic general topology and abstract algebra. None categorical background is needed. The final objective is to show that classical homotopy theory for topological spaces can be more naturally understood in terms of categorical language.
In this paper, we define what it means for an object in an abstract module category to be dualizable and we give a homological description of the direct limit closure of the dualizable objects. Our description recovers existing results of Govorov and Lazard, Oberst and R{ö}hrl, and Christensen and Holm. When applied to differential graded modules over a differential graded algebra, our description yields that a DG-module is semi-flat if and only if it can be obtained as a direct limit of finitely generated semi-free DG-modules. We obtain similar results for graded modules over graded rings and…
We study structures which have arisen in recent work by the present author and Bob Coecke on a categorical axiomatics for Quantum Mechanics; in particular, the notion of strongly compact closed category. We explain how these structures support a notion of scalar which allows quantitative aspects of physical theory to be expressed, and how the notion of strong compact closure emerges as a significant refinement of the more classical notion of compact closed category. We then proceed to an extended discussion of free constructions for a sequence of progressively more complex kinds of structured …
We define the notion of an indexed profunctor over a 2-category, and use it to develop an abstract theory of limits. The theory subsumes (conical) limits, weighted limits, ends and Kan extensions. Results include an abstract version of the theorem that right adjoint functors preserve limits, and an abstract account of the phenomenon that the comparison arrow implicated in the preservation of a limit by a functor is natural in any functorial variable the functor happens to depend on. These results make an extensive use of the data and axioms of an indexed profunctor over a 2-category.
We show that the category of abstract elementary classes (AECs) and concrete functors is closed under constructions of "limit type," which generalizes the approach of Mariano, Zambrano and Villaveces away from the syntactically oriented framework of institutions. Moreover, we provide a broader view of this closure phenomenon, considering a variety of categories of accessible categories with additional structure, and relaxing the assumption that the morphisms be concrete functors.
Treewidth is a well-known graph invariant with multiple interesting applications in combinatorics. On the practical side, many NP-complete problems are polynomial-time (sometimes even linear-time) solvable on graphs of bounded treewidth. On the theoretical side, treewidth played an essential role in the proof of the celebrated Robertson-Seymour graph minor theorem. While defining treewidth-like invariants on graphs and treewidth analogues on other sorts of combinatorial objects (incl. hypergraphs, digraphs) has been a fruitful avenue of research, a direct, categorial description capturing mult…
Batanin and Markl's operadic categories are categories in which each map is endowed with a finite collection of "abstract fibres" -- also objects of the same category -- subject to suitable axioms. We give a reconstruction of the data and axioms of operadic categories in terms of the décalage comonad D on small categories. A simple case involves unary operadic categories -- ones wherein each map has exactly one abstract fibre -- which are exhibited as categories which are, first of all, coalgebras for the comonad D, and, furthermore, algebras for the monad induced on the category of D-coalgebr…
We develop Markov categories as a framework for synthetic probability and statistics, following work of Golubtsov as well as Cho and Jacobs. This means that we treat the following concepts in purely abstract categorical terms: conditioning and disintegration; various versions of conditional independence and its standard properties; conditional products; almost surely; sufficient statistics; versions of theorems on sufficient statistics due to Fisher--Neyman, Basu, and Bahadur. Besides the conceptual clarity offered by our categorical setup, its main advantage is that it provides a uniform trea…
We introduce $μ$-Abstract Elementary Classes ($μ$-AECs) as a broad framework for model theory that includes complete boolean algebras and Dirichlet series, and begin to develop their classification theory. Moreover, we note that $μ$-AECs correspond precisely to accessible categories in which all morphisms are monomorphisms, and begin the process of reconciling these divergent perspectives: not least, the preliminary classification-theoretic results for μ-AECs transfer directly to accessible categories with monomorphisms.
Game semantics has provided adequate models for a variety of programming languages, in which types are interpreted as two-player games and programs as strategies. Melliès (2018) suggested that such categories of games and strategies may be obtained as instances of a simple abstract construction on weak double categories. However, in the particular case of simple games, his construction slightly differs from the standard category. We refine the abstract construction using factorisation systems, and show that the new construction yields the standard category of simple games and strategies. Anoth…
We give a self-contained introduction to accessible categories and how they shed light on both model- and set-theoretic questions. We survey for example recent developments on the study of presentability ranks, a notion of cardinality localized to a given category, as well as stable independence, a generalization of pushouts and model-theoretic forking that may interest mathematicians at large. We give many examples, including recently discovered connections with homotopy theory and homological algebra. We also discuss concrete versions of accessible categories (such as abstract elementary cla…
The notes were prepared for a series of talks that I gave in Hagen in late June and early July 2003, and, with some changes, in the University of La Laguña, the Canary Islands, in September, 2003. They aim (i) to revisit some oldish material on abstract homotopy and simplicial ly enriched categories, that seems to be being used in today's resurgence of interest in the area and to try to view it in a new light, or perhaps from new directions; (ii) to introduce Segal categories and various other tools used by the Nice-Toulouse group of abstract homotopy theorists and link them into some of the o…
Adobe Research（奥多比研究院）；University of Massachusetts, Amherst（马萨诸塞大学阿默斯特分校）
Conditional independence has been widely used in AI, causal inference, machine learning, and statistics. We introduce categoroids, an algebraic structure for characterizing universal properties of conditional independence. Categoroids are defined as a hybrid of two categories: one encoding a preordered lattice structure defined by objects and arrows between them; the second dual parameterization involves trigonoidal objects and morphisms defining a conditional independence structure, with bridge morphisms providing the interface between the binary and ternary structures. We illustrate categoro…
A semi-localization of a category is a full reflective subcategory with the property that the reflector is semi-left-exact. In this article we first determine an abstract characterization of the categories which are semi-localizations of an exact Mal'tsev category, by specializing a result due to S. Mantovani. We then turn our attention to semi-abelian categories, where a special type of semi-localizations are known to coincide with torsion-free subcategories. A new characterisation of protomodular categories in terms of binary relations is obtained, inspired by the one discovered in the point…
We prove that the homotopy prederivator of a cofibration category is equivalent to the homotopy prederivator of its associated quasi-category of frames, as introduced by Szumiło. We use this comparison result to deduce various abstract properties of the obtained prederivators.
Tree-width is an invaluable tool for computational problems on graphs. But often one would like to compute on other kinds of objects (e.g. decorated graphs or even algebraic structures) where there is no known tree-width analogue. Here we define an abstract analogue of tree-width which provides a uniform definition of various tree-width-like invariants including graph tree-width, hypergraph tree-width, complemented tree-width and even new constructions such as the tree-width of modular quotients. We obtain this generalization by developing a general theory of categories that admit abstract ana…
Higher dimensional central extensions of groups were introduced by G. Janelidze as particular instances of the abstract notion of covering morphism from categorical Galois theory. More recently, the notion has been extended to and studied in arbitrary semi-abelian categories. In this article, we further extend the scope to exact Mal'tsev categories and beyond.
A (closed) dynamical system is a notion of how things can be, together with a notion of how they may change given how they are. The idea and mathematics of closed dynamical systems has proven incredibly useful in those sciences that can isolate their object of study from its environment. But many changing situations in the world cannot be meaningfully isolated from their environment - a cell will die if it is removed from everything beyond its walls. To study systems that interact with their environment, and to design such systems in a modular way, we need a robust theory of open dynamical sys…
We develop abstract nonsense for module categories over monoidal categories (this is a straightforward categorification of modules over rings). As applications we show that any semisimple monoidal category with finitely many simple objects is equivalent to the category of representations of a weak Hopf algebra (theorem of T. Hayashi) and classify module categories over the fusion category of $\hat{sl}(2)$ at a positive integer level where we meet once again ADE classification pattern.
In this dissertation, we compare the "classical" homology of an $ω$-category (defined as the homology of its Street nerve) with its polygraphic homology. More precisely, we prove that both homologies generally do not coincide and call homologically coherent the particular strict $ω$-categories for which polygraphic homology and homology of the nerve do coincide. The goal pursued is to find abstract and concrete criteria to detect homologically coherent $ω$-categories. For example, we prove that all (small) categories, considered as strict $ω$-categories with unit cells above dimension 1, are h…
In this paper we study categories $(F,\mathbf{C},\mathbf{D})$ and $(\mathbb{F},\mathbf{C},\mathbf{Set})$ and prove them to be fibred on $\mathbf{C}$. Then we examine Grothendieck construction in the context of an ordinary functor $F: \mathbf{C} \rightarrow \mathbf{D}$ through the concept of trivial categorification, using an appropriate functor $\mathbf{F}: \mathbf{C} \xrightarrow{F} \mathbf{D} \xrightarrow{I} \mathbf{Cat}$ to construct $\int_{\mathbf{C}^{op}} \bar{\mathbf{F}}$. This category characterizes a functor as an abstract right category action while its dual $\mathcal{X} \rtimes_{\mat…
We show that a number of results on abstract elementary classes (AECs) hold in accessible categories with concrete directed colimits. In particular, we prove a generalization of a recent result of Boney on tameness under a large cardinal assumption. We also show that such categories support a robust version of the Ehrenfeucht-Mostowski construction. This analysis has the added benefit of producing a purely language-free characterization of AECs, and highlights the precise role played by the coherence axiom.
Homotopy limits and colimits are homotopical replacements for the usual limits and colimits of category theory, which can be approached either using classical explicit constructions or the modern abstract machinery of derived functors. Our first goal in this paper is expository: we explain both approaches and a proof of their equivalence. Our second goal is to generalize this result to enriched categories and homotopy weighted limits, showing that the classical explicit constructions still give the right answer in the abstract sense. This result partially bridges the gap between classical homo…
Let $\mathcal{A}$ be a locally noetherian Grothendieck category. In this paper we define and study the section functor on $\mathcal{A}$ with respect to an open subset of ASpec$\mathcal{A}$. Next we define and study local cohomology theory in $\mathcal{A}$ in terms of the section functors. Finally we study abstract local cohomology functor on the derived category $\mathcal{D}^+(\mathcal{A})$.
By the Lefschetz fixed point theorem, if an endomorphism of a topological space is fixed-point-free, then its Lefschetz number vanishes. This necessary condition is not usually sufficient, however; for that we need a refinement of the Lefschetz number called the Reidemeister trace. Abstractly, the Lefschetz number is a trace in a symmetric monoidal category, while the Reidemeister trace is a trace in a bicategory; in this paper we relate these contexts using indexed symmetric monoidal categories. In particular, we will show that for any symmetric monoidal category with an associated indexed sy…
In this note we interpret Voevodsky's Univalence Axiom in the language of (abstract) model categories. We then show that any posetal locally Cartesian closed model category $Qt$ in which the mapping $Hom^{(w)}(Z\times B,C):Qt\longrightarrow Sets$ is functorial in $Z$ and represented in $Qt$ satisfies our homotopy version of the Univalence Axiom, albeit in a rather trivial way. This work was motivated by a question reported in [Ob], asking for a model of the Univalence Axiom not equivalent to the standard one.
In the present work, a natural sequel to \cite{MaPi1}, we further discuss the existence of adjunctions between categories of institutions and of $π$-institutions. This is done at both a foundational and an applied level. Firstly, we reformulate and conceptually clarify such adjunctions in terms of the $2$-categorical data involved in the construction of categories of institution-like structures. More precisely, we remark that the process used for passing from rooms to institutions (\cite{Diac2}) can be extended, due to its $2$-functoriality, to more general room-like and institution-like struc…
We investigate algebraic and compositional properties of abstract multiway rewriting systems, which are archetypical structures underlying the formalism of the Wolfram model. We demonstrate the existence of higher homotopies in this class of rewriting systems, where homotopical maps are induced by the inclusion of appropriate rewriting rules taken from an abstract rulial space of all possible such rules. Furthermore, we show that a multiway rewriting system with homotopies up to order $n$ may naturally be formalized as an $n$-fold category, such that (upon inclusion of appropriate inverse morp…
In this paper we show that both the homotopy category of strict $n$-categories, $1\leqslant n \leqslant \infty$, and the homotopy category of Steiner's augmented directed complexes are equivalent to the category of homotopy types. In order to do so, we first prove an abstract result, based on a strategy of Fritsch and Latch, giving sufficient conditions for a nerve functor with values in simplicial sets to induce an equivalence at the level of homotopy categories. We then apply this result to strict $n$-categories and augmented directed complexes, for which the hypothesis of our theorem were e…
We develop aspects of functional analysis in an abstract axiomatic setting, through monoidal and enriched category theory. We work in a given closed category, whose objects we call spaces, and we study R-module objects therein (or algebras of a commutative monad), which we call linear spaces. Building on ideas of Lawvere and Kock, we study functionals on the space of scalar-valued maps, including compactly-supported Radon measures and Schwartz distributions. We develop an abstract theory of vector-valued integration with respect to these scalar functionals and their relatives. We study three a…
