Bicategories, Pseudofunctors, Shadows: a Cheat Sheet
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Relations: Categories, Monoidal Categories, and Props
Logical Methods in Computer Science Vol. 14(3:14)2018, pp. 1–25 Submitted Oct. 12, 2017 https://lmcs.episciences.org/ Published Sep. 03, 2018 UNIVERSAL CONSTRUCTIONS FOR (CO)RELATIONS: CATEGORIES, MONOIDAL CATEGORIES, AND PROPS BRENDAN FONG AND FABIO ZANASI Massachusetts Institute of Technology, United States of America e-mail address: [email protected] University College London, United Kingdom e-mail address: [email protected] Abstract. Calculi of string diagrams are increasingly used to present the syntax and algebraic structure of various families of circuits, including signal flow graphs, electrical circuits and quantum processes. In many such approaches, the semantic interpretation for diagrams is given in terms of relations or corelations (generalised equivalence relations) of some kind. In this paper we show how semantic categories of both relations and corelations can be characterised as colimits of simpler categories. This modular perspective is important as it simplifies the task of giving a complete axiomatisation for semantic equivalence of string diagrams. Moreover, our general result unifies various theorems that are independently found in literature and are relevant for program semantics, quantum computation and control theory. 1. Introduction Network-style diagrammatic languages appear in diverse fields as a tool to reason about computational models of various kinds, including signal processing circuits, quantum pro- cesses, Bayesian networks and Petri nets, amongst many others. In the last few years, there have been more and more contributions towards a uniform, formal theory of these languages which borrows from the well-established methods of programming language semantics. A significant insight stemming from many such approaches is that a compositional analysis of network diagrams, enabling their reduction to elementary components, is more effective when system behaviour is thought of as a relation instead of a function. -
PULLBACK and BASE CHANGE Contents
PART II.2. THE !-PULLBACK AND BASE CHANGE Contents Introduction 1 1. Factorizations of morphisms of DG schemes 2 1.1. Colimits of closed embeddings 2 1.2. The closure 4 1.3. Transitivity of closure 5 2. IndCoh as a functor from the category of correspondences 6 2.1. The category of correspondences 6 2.2. Proof of Proposition 2.1.6 7 3. The functor of !-pullback 8 3.1. Definition of the functor 9 3.2. Some properties 10 3.3. h-descent 10 3.4. Extension to prestacks 11 4. The multiplicative structure and duality 13 4.1. IndCoh as a symmetric monoidal functor 13 4.2. Duality 14 5. Convolution monoidal categories and algebras 17 5.1. Convolution monoidal categories 17 5.2. Convolution algebras 18 5.3. The pull-push monad 19 5.4. Action on a map 20 References 21 Introduction Date: September 30, 2013. 1 2 THE !-PULLBACK AND BASE CHANGE 1. Factorizations of morphisms of DG schemes In this section we will study what happens to the notion of the closure of the image of a morphism between schemes in derived algebraic geometry. The upshot is that there is essentially \nothing new" as compared to the classical case. 1.1. Colimits of closed embeddings. In this subsection we will show that colimits exist and are well-behaved in the category of closed subschemes of a given ambient scheme. 1.1.1. Recall that a map X ! Y in Sch is called a closed embedding if the map clX ! clY is a closed embedding of classical schemes. -
DUAL TANGENT STRUCTURES for ∞-TOPOSES In
DUAL TANGENT STRUCTURES FOR 1-TOPOSES MICHAEL CHING Abstract. We describe dual notions of tangent bundle for an 1-topos, each underlying a tangent 1-category in the sense of Bauer, Burke and the author. One of those notions is Lurie's tangent bundle functor for presentable 1-categories, and the other is its adjoint. We calculate that adjoint for injective 1-toposes, where it is given by applying Lurie's tangent bundle on 1-categories of points. In [BBC21], Bauer, Burke, and the author introduce a notion of tangent 1- category which generalizes the Rosick´y[Ros84] and Cockett-Cruttwell [CC14] axiomatization of the tangent bundle functor on the category of smooth man- ifolds and smooth maps. We also constructed there a significant example: diff the Goodwillie tangent structure on the 1-category Cat1 of (differentiable) 1-categories, which is built on Lurie's tangent bundle functor. That tan- gent structure encodes the ideas of Goodwillie's calculus of functors [Goo03] and highlights the analogy between that theory and the ordinary differential calculus of smooth manifolds. The goal of this note is to introduce two further examples of tangent 1- categories: one on the 1-category of 1-toposes and geometric morphisms, op which we denote Topos1, and one on the opposite 1-category Topos1 which, following Anel and Joyal [AJ19], we denote Logos1. These two tangent structures each encode a notion of tangent bundle for 1- toposes, but from dual perspectives. As described in [AJ19, Sec. 4], one can view 1-toposes either from a `geometric' or `algebraic' point of view. -
Toward a Non-Commutative Gelfand Duality: Boolean Locally Separated Toposes and Monoidal Monotone Complete $ C^{*} $-Categories
Toward a non-commutative Gelfand duality: Boolean locally separated toposes and Monoidal monotone complete C∗-categories Simon Henry February 23, 2018 Abstract ** Draft Version ** To any boolean topos one can associate its category of internal Hilbert spaces, and if the topos is locally separated one can consider a full subcategory of square integrable Hilbert spaces. In both case it is a symmetric monoidal monotone complete C∗-category. We will prove that any boolean locally separated topos can be reconstructed as the classifying topos of “non-degenerate” monoidal normal ∗-representations of both its category of internal Hilbert spaces and its category of square integrable Hilbert spaces. This suggest a possible extension of the usual Gelfand duality between a class of toposes (or more generally localic stacks or localic groupoids) and a class of symmetric monoidal C∗-categories yet to be discovered. Contents 1 Introduction 2 2 General preliminaries 3 3 Monotone complete C∗-categories and booleantoposes 6 4 Locally separated toposes and square integrable Hilbert spaces 7 5 Statementofthemaintheorems 9 arXiv:1501.07045v1 [math.CT] 28 Jan 2015 6 From representations of red togeometricmorphisms 13 6.1 Construction of the geometricH morphism on separating objects . 13 6.2 Functorialityonseparatingobject . 20 6.3 ConstructionoftheGeometricmorphism . 22 6.4 Proofofthe“reduced”theorem5.8 . 25 Keywords. Boolean locally separated toposes, monotone complete C*-categories, recon- struction theorem. 2010 Mathematics Subject Classification. 18B25, 03G30, 46L05, 46L10 . email: [email protected] 1 7 On the category ( ) and its representations 26 H T 7.1 The category ( /X ) ........................ 26 7.2 TensorisationH byT square integrable Hilbert space . -
Monoidal Functors, Equivalence of Monoidal Categories
14 1.4. Monoidal functors, equivalence of monoidal categories. As we have explained, monoidal categories are a categorification of monoids. Now we pass to categorification of morphisms between monoids, namely monoidal functors. 0 0 0 0 0 Definition 1.4.1. Let (C; ⊗; 1; a; ι) and (C ; ⊗ ; 1 ; a ; ι ) be two monoidal 0 categories. A monoidal functor from C to C is a pair (F; J) where 0 0 ∼ F : C ! C is a functor, and J = fJX;Y : F (X) ⊗ F (Y ) −! F (X ⊗ Y )jX; Y 2 Cg is a natural isomorphism, such that F (1) is isomorphic 0 to 1 . and the diagram (1.4.1) a0 (F (X) ⊗0 F (Y )) ⊗0 F (Z) −−F− (X−)−;F− (Y− )−;F− (Z!) F (X) ⊗0 (F (Y ) ⊗0 F (Z)) ? ? J ⊗0Id ? Id ⊗0J ? X;Y F (Z) y F (X) Y;Z y F (X ⊗ Y ) ⊗0 F (Z) F (X) ⊗0 F (Y ⊗ Z) ? ? J ? J ? X⊗Y;Z y X;Y ⊗Z y F (aX;Y;Z ) F ((X ⊗ Y ) ⊗ Z) −−−−−−! F (X ⊗ (Y ⊗ Z)) is commutative for all X; Y; Z 2 C (“the monoidal structure axiom”). A monoidal functor F is said to be an equivalence of monoidal cate gories if it is an equivalence of ordinary categories. Remark 1.4.2. It is important to stress that, as seen from this defini tion, a monoidal functor is not just a functor between monoidal cate gories, but a functor with an additional structure (the isomorphism J) satisfying a certain equation (the monoidal structure axiom). -
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Arxiv:1608.02901V1 [Math.AT] 9 Aug 2016 Functors
STABLE ∞-OPERADS AND THE MULTIPLICATIVE YONEDA LEMMA THOMAS NIKOLAUS Abstract. We construct for every ∞-operad O⊗ with certain finite limits new ∞-operads of spectrum objects and of commutative group objects in O. We show that these are the universal stable resp. additive ∞-operads obtained from O⊗. We deduce that for a stably (resp. additively) symmetric monoidal ∞-category C the Yoneda embedding factors through the ∞-category of exact, contravari- ant functors from C to the ∞-category of spectra (resp. connective spectra) and admits a certain multiplicative refinement. As an application we prove that the identity functor Sp → Sp is initial among exact, lax symmetric monoidal endo- functors of the symmetric monoidal ∞-category Sp of spectra with smash product. 1. Introduction Let C be an ∞-category that admits finite limits. Then there is a new ∞-category Sp(C) of spectrum objects in C that comes with a functor Ω∞ : Sp(C) → C. It is shown in [Lur14, Corollary 1.4.2.23] that this functors exhibits Sp(C) as the universal stable ∞-category obtained from C. This means that for every stable ∞-category D, post-composition with the functor Ω∞ induces an equivalence FunLex D, Sp(C) → FunLex D, C , where FunLex denotes the ∞-category of finite limits preserving (a.k.a. left exact) functors. The question that we want to address in this paper is the following. Suppose C admits a symmetric monoidal structure. Does Sp(C) then also inherits some sort of symmetric monoidal structure which satisfies a similar universal property in the world of symmetric monoidal ∞-categories? This is a question of high practical importance in applications in particular for the construction of algebra structures on mapping spectra and the answer that we give will be applied in future work by the author. -
Presentably Symmetric Monoidal Infinity-Categories Are Represented
PRESENTABLY SYMMETRIC MONOIDAL 1-CATEGORIES ARE REPRESENTED BY SYMMETRIC MONOIDAL MODEL CATEGORIES THOMAS NIKOLAUS AND STEFFEN SAGAVE Abstract. We prove the theorem stated in the title. More precisely, we show the stronger statement that every symmetric monoidal left adjoint functor be- tween presentably symmetric monoidal 1-categories is represented by a strong symmetric monoidal left Quillen functor between simplicial, combinatorial and left proper symmetric monoidal model categories. 1. Introduction The theory of 1-categories has in recent years become a powerful tool for study- ing questions in homotopy theory and other branches of mathematics. It comple- ments the older theory of Quillen model categories, and in many application the interplay between the two concepts turns out to be crucial. In an important class of examples, the relation between 1-categories and model categories is by now com- pletely understood, thanks to work of Lurie [Lur09, Appendix A.3] and Joyal [Joy08] based on earlier results by Dugger [Dug01a]: On the one hand, every combinatorial simplicial model category M has an underlying 1-category M1. This 1-category M1 is presentable, i.e., it satisfies the set theoretic smallness condition of being accessible and has all 1-categorical colimits and limits. On the other hand, every presentable 1-category is equivalent to the 1-category associated with a combi- natorial simplicial model category [Lur09, Proposition A.3.7.6]. The presentability assumption is essential here since a sub 1-category of a presentable 1-category is in general not presentable, and does not come from a model category. In many applications one studies model categories M equipped with a symmet- ric monoidal product that is compatible with the model structure. -
Enhancing the Filtered Derived Category
Enhancing the filtered derived category Owen Gwilliam, https://people.mpim-bonn.mpg.de/gwilliam/, Max Planck Institute for Mathematics, Bonn Dmitri Pavlov, https://dmitripavlov.org/, Department of Mathematics and Statistics, Texas Tech University Contents 1. Introduction 2. Filtered objects in the language of stable ∞-categories 3. Filtered objects in the language of stable model categories 4. Comparison of the quasicategorical and the model categorical constructions 5. Filtered objects in the language of differential graded categories 6. Application: filtered operads and algebras over them 7. Application: filtered D-modules 8. References Abstract. The filtered derived category of an abelian category has played a useful role in subjects including geometric representation theory, mixed Hodge modules, and the theory of motives. We develop a natural generalization using current methods of homotopical algebra, in the formalisms of stable ∞-categories, sta- ble model categories, and pretriangulated, idempotent-complete dg categories. We characterize the filtered stable ∞-category Fil(C) of a stable ∞-category C as the left exact localization of sequences in C along the ∞-categorical version of completion (and prove analogous model and dg category statements). We also spell out how these constructions interact with spectral sequences and monoidal structures. As examples of this machinery, we construct a stable model category of filtered D-modules and develop the rudiments of a theory of filtered operads and filtered algebras over operads. 1 Introduction The filtered derived category plays an important role in the setting of constructible sheaves, D-modules, mixed Hodge modules, and elsewhere. Our goal is to revisit this construction using current technology for ho- motopical algebra, extending it beyond the usual setting of chain complexes (or sheaves of chain complexes). -
Enriched Monoidal Categories I: Centers
Enriched monoidal categories I: centers Liang Konga,b, Wei Yuanc,d, Zhi-Hao Zhange,a, Hao Zheng f,a,b 1 a Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen, 518055, China b Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen, 518055, China c Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China d School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China e Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China f Department of Mathematics, Peking University, Beijing 100871, China Abstract Thisworkisthe firstone inaseries,inwhich wedevelopamathematical theory of enriched (braided) monoidal categories and their representations. In this work, we introduce the notion of the E0-center (E1-center or E2-center) of an enriched (monoidal or braided monoidal) cate- gory, and compute the centers explicitly when the enriched (braided monoidal or monoidal) categories are obtained from the canonical constructions. These centers have important appli- cations in the mathematical theory of gapless boundaries of 2+1D topological orders and that of topological phase transitions in physics. They also play very important roles in the higher representation theory, which is the focus of the second work in the series. Contents 1 Introduction 2 2 2-categories 3 2.1 Examplesof2-categories . ....... 3 2.2 Centersin2-categories . ....... 5 arXiv:2104.03121v1 [math.CT] 7 Apr 2021 3 Enriched categories 8 3.1 Enriched categories, functors and natural transformations ............ -
Symmetric Monoidal Structures and the Frobenius
Symmetric monoidal structures and the Frobenius Dylan Wilson February 25, 2020 If R is a commutative ring then we can write down a sort of diagonal, which is multi- plicative: r ÞÑ rbp P Rbp This lands in the symmetric tensors, pRbpqΣp . This map is not usually additive. For example, we have px ` yqb2 “ x b x ` y b y ` px b y ` y b xq Here, the error term x b y ` y b x is in the image of the transfer map R b R Ñ pR b RqΣ2 . And in fact, in general, the composite R Ñ pRbpqΣp Ñ pRbpqΣp {tr is additive. The goal of this lecture and part of the next is to set up a derived or homotopy coherent analogue of this story, and then see how it plays out in the case R “ F2 ^ F2. (1) First we need to make sense of commutative rings. It's easier to do commutative monoids (i.e. commutative algebras in a cartesian monoidal category) first, so let's do that. 8 8 Example 1.1. Let Gr “ n¥0 GrnpR q be the space of finite dimensional subspaces of R . This kind of has a binary operation. If I have V; W Ď R8 then if I choose an embedding 8 8 8 ² R ` R ãÑ R , we may view V ` W as a new element of Gr. But this depends on a choice. More generally, the natural thing to write down for any finite set J is a zig-zag ˆJ J ˆJ Gr Ð IsompR8 ; R8q ˆ Gr Ñ Gr Here, Isom denotes the space of isometric embeddings, the left-hand arrow is projection, and the right hand arrow is the operation where we take direct sums and then use our embedding. -
Symmetric Monoidal G-Categories and Their Strictification
SYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION B. GUILLOU, J.P. MAY, M. MERLING, AND A.M. OSORNO Abstract. We give an operadic definition of a genuine symmetric monoidal G-category, and we prove that its classifying space is a genuine E1 G-space. We do this by developing some very general categorical coherence theory. We combine results of Corner and Gurski, Power, and Lack, to develop a stric- tification theory for pseudoalgebras over operads and monads. It specializes to strictify genuine symmetric monoidal G-categories to genuine permutative G-categories. All of our work takes place in a general internal categorical framework that has many quite different specializations. When G is a finite group, the theory here combines with previous work to generalize equivariant infinite loop space theory from strict space level input to considerably more general category level input. It takes genuine symmetric monoidal G-categories as input to an equivariant infinite loop space machine that gives genuine Ω-G-spectra as output. Contents Introduction and statements of results2 Acknowledgements5 1. Categorical preliminaries6 1.1. Internal categories6 1.2. Chaotic categories8 1.3. The embedding of Set in V 9 2. Pseudoalgebras over operads and 2-monads 11 2.1. Pseudoalgebras over 2-Monads 11 2.2. The 2-monads associated to operads 12 2.3. Pseudoalgebras over operads 14 3. Operadic specification of symmetric monoidal G-categories 18 4. Enhanced factorization systems 21 4.1. Enhanced factorization systems 21 4.2. The enhanced factorization system on Cat(V ) 23 4.3. Proof of the properties of the EFS on Cat(V ) 24 5. -
Props in Network Theory
Props in Network Theory John C. Baez∗y Brandon Coya∗ Franciscus Rebro∗ [email protected] [email protected] [email protected] January 23, 2018 Abstract Long before the invention of Feynman diagrams, engineers were using similar diagrams to reason about electrical circuits and more general networks containing mechanical, hydraulic, thermo- dynamic and chemical components. We can formalize this reasoning using props: that is, strict symmetric monoidal categories where the objects are natural numbers, with the tensor product of objects given by addition. In this approach, each kind of network corresponds to a prop, and each network of this kind is a morphism in that prop. A network with m inputs and n outputs is a morphism from m to n, putting networks together in series is composition, and setting them side by side is tensoring. Here we work out the details of this approach for various kinds of elec- trical circuits, starting with circuits made solely of ideal perfectly conductive wires, then circuits with passive linear components, and then circuits that also have voltage and current sources. Each kind of circuit corresponds to a mathematically natural prop. We describe the `behavior' of these circuits using morphisms between props. In particular, we give a new construction of the black-boxing functor of Fong and the first author; unlike the original construction, this new one easily generalizes to circuits with nonlinear components. We also use a morphism of props to clarify the relation between circuit diagrams and the signal-flow diagrams in control theory. Technically, the key tools are the Rosebrugh{Sabadini{Walters result relating circuits to special commutative Frobenius monoids, the monadic adjunction between props and signatures, and a result saying which symmetric monoidal categories are equivalent to props.