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Let M be a monoidal ∞-category with colimits. In this paper we study colimits of M-functors A --> B where B is left-tensored over M and A is an M-enriched category. We prove that the enriched Yoneda embedding Y: A --> P_M(A) yields a universal M-functor. In case when A has a certain monoidal structure, the category of enriched presheaves P_M(A) inherits the same monoidal structure and the enriched Yoneda embedding acquires the structure of universal monoidal M-functor. [ABSTRACT FROM AUTHOR]

The notion of Gray-category, a semi-strict 3-category in which the middle four interchange is weakened to an isomorphism, is central in the study of three-dimensional category theory. In this context it is common practice to use 2-dimensional pasting diagrams to express composites of 2-cells, however there is no thorough treatment in the literature justifying this procedure. We fill this gap by providing a formal approach to pasting in Gray-categories and by proving that such composites are uniquely defined up to a contractible groupoid of choices. [ABSTRACT FROM AUTHOR]

Lack described a Quillen model structure on the category GrayCat of Gray-categories and Gray-functors, for which the weak equivalences are the weak 3-equivalences. Restricted to Gray-groupoids, the resulting homotopy category is equivalent to the homotopy category of 3-types. In this note, we adapt the technique of Gurski, Johnson, and Osorno to show the localization of GrayCat at the weak equivalences is equivalent to the category of algebraic tricategories and pseudo-natural equivalence classes of weak 3-functors. This finishes establishing the homotopy hypothesis for algebraic trigroupoids. [ABSTRACT FROM AUTHOR]

We define and study the notion of a locally bounded enriched category over a (locally bounded) symmetric monoidal closed category, generalizing the locally bounded ordinary categories of Freyd and Kelly. In addition to proving several general results for constructing examples of locally bounded enriched categories and locally bounded closed categories, we demonstrate that locally bounded enriched categories admit fully enriched analogues of many of the convenient results enjoyed by locally bounded ordinary categories. In particular, we prove full enrichments of Freyd and Kelly's reflectivity and local boundedness results for orthogonal subcategories and categories of models for sketches and theories. We also provide characterization results for locally bounded enriched categories in terms of enriched presheaf categories, and we show that locally bounded enriched categories admit useful adjoint functor theorems and a representability theorem. We also define and study the notion of α-bounded-small weighted limit enriched in a locally α-bounded closed category, which parallels Kelly's notion of α-small weighted limit enriched in a locally α-presentable closed category, and we show that enriched categories of models of α-bounded-small weighted limit theories are locally α-bounded. [ABSTRACT FROM AUTHOR]

We introduce some general categorical machinery for studying Gysin functors (certain kinds of functors with transfers) and their associated categories of correspondences. These correspondence categories are closed, symmetric monoidal categories where all objects are self-dual. We also prove a limited reconstruction theorem: given such a closed, symmetric monoidal category (and some extra information) it is isomorphic to the correspondence category associated to a Gysin functor. Finally, if k is a field we use this technology to define and explore a new construction called the Grothendieck- Witt category of k. [ABSTRACT FROM AUTHOR]

We first take a whimsical look at a couple of questions relating to categories as generalized posets. Then we study the question of functorial choice of pullbacks. Finally, we consider a simple question in basic category theory, with an elementary solution which is surprisingly difficult to generalize to 2-categories. [ABSTRACT FROM AUTHOR]

M. Johnson, R. Rosebrugh and R. J. Wood, Entity-relationship-attribute designs and sketches, Theory and Applications of Categories 10(2002), 94-112. M. Johnson, R. Rosebrugh and R. J. Wood, Lenses, brations, and universal translations, Mathematical Structures in Computer Science, 22 (2012), 25{42. M. Johnson, R. Rosebrugh and R. J. Wood, Algebras and update strategies, Journal of Universal Computer Science 16(2010), 729{748. M. Johnson and R. Rosebrugh, Unifying set-based, delta-based and edit-based lenses, Proceedings of the 5th International Workshop on Bidirectional Transformations, Eindhoven, Netherlands, April, 2016, 1{13. [Extracted from the article]

We go back to the roots of enriched category theory and study categories enriched in chain complexes; that is, we deal with differential graded categories (DGcategories for short). In particular, we recall weighted colimits and provide examples. We solve the 50 year old question of how to characterize Cauchy complete DG-categories in terms of existence of some specific finite absolute colimits. As well as the interactions between absolute weighted colimits, we also examine the total complex of a chain complex in a DG-category as a non-absolute weighted colimit. [ABSTRACT FROM AUTHOR]

We introduce, for C a regular Cartesian Reedy category a model category whose fibrant objects are an analogue of quasicategories enriched in simplicial presheaves on C. We then develop a coherent realization and nerve for this model structure and demonstrate that these give a Quillen equivalence, in particular recovering the classical one in the process. We then demonstrate that this equivalence descends to any Cartesian closed left Bousfield localization in a natural way. As an application, we demonstrate a version of Yoneda's lemma for quasicategories enriched in any such Cartesian closed localization. [ABSTRACT FROM AUTHOR]

We prove that the homotopy theory of parsummable categories (as defined by Schwede) with respect to the underlying equivalences of categories is equivalent to the usual homotopy theory of symmetric monoidal categories. In particular, this yields a model of symmetric monoidal categories in terms of categories equipped with a strictly commutative, associative, and unital (but only partially defined) operation. [ABSTRACT FROM AUTHOR]

We prove that every elementary (∞, 1)-topos has a natural number object. We achieve this by defining the loop space of the circle and showing that we can construct a natural number object out of it. Part of the proof involves showing that various definitions of natural number objects (Lawvere, Freyd and Peano) agree with each other in an elementary (∞, 1)-topos. As part of this effort we also study the internal object of contractibility in (∞, 1)-categories, which is of independent interest. Finally, we discuss various applications of natural number objects. In particular, we use it to define internal sequential colimits in an elementary (∞, 1)-topos. [ABSTRACT FROM AUTHOR]

We generalize the notion of ends and coends in category theory to the realm of module categories over finite tensor categories. We call this new concept module (co)end. This tool allows us to give different proofs to several known results in the theory of representations of finite tensor categories. As a new application, we present a description of the relative Serre functor for module categories in terms of a module coend, in a analogous way as a Morita invariant description of the Nakayama functor of abelian categories presented in [4]. [ABSTRACT FROM AUTHOR]

It is shown that any compact metric space of diameter at most 2 embeds isometrically as a linearly independent set of extreme points of the unit ball of a separable Banach space. The proof illustrates how category theory can play a useful role in a problem of functional analysis. [ABSTRACT FROM AUTHOR]

One goal of applied category theory is to better understand networks appearing throughout science and engineering. Here we introduce 'structured cospans' as a way to study networks with inputs and outputs. Given a functor L: A → X, a structured cospan is a diagram in X of the form L(a) → x ← L(b). If A and X have finite colimits and L is a left adjoint, we obtain a symmetric monoidal category whose objects are those of A and whose morphisms are isomorphism classes of structured cospans. This is a hypergraph category. However, it arises from a more fundamental structure: a symmetric monoidal double category where the horizontal 1-cells are structured cospans. We show how structured cospans solve certain problems in the closely related formalism of 'decorated cospans', and explain how they work in some examples: electrical circuits, Petri nets, and chemical reaction networks. [ABSTRACT FROM AUTHOR]

We develop a compositional framework for formal synthesis of hybrid systems using the language of category theory. More specifically, we provide mutually compatible tools for hierarchical, sequential, and independent parallel composition. In our framework, hierarchies of hybrid systems correspond to template-anchor pairs, which we model as spans of subdividing and embedding semiconjugacies. Hierarchical composition of template-anchor pairs corresponds to the composition of spans via fiber product. To model sequential composition, we introduce "directed hybrid systems," each of which ows from an initial subsystem to a final subsystem in a Conley-theoretic sense. Sequential composition of directed systems is given by a pushout of graph embeddings, rewriting the continuous dynamics of the overlapping subsystem to prioritize the second directed system. Independent parallel composition corresponds to a categorical product with respect to semiconjugacy. To formalize the compatibility of these three types of composition, we construct a vertically cartesian double category of hybrid systems where the vertical morphisms are semiconjugacies, and the horizontal morphisms are directed hybrid systems. [ABSTRACT FROM AUTHOR]

In this paper we develop a theory of Segal enriched categories. Our motivation was to generalize the notion of up-to-homotopy monoid in a monoidal category, introduced by Leinster. Our formalism generalizes the classical theory of Segal categories and extends the theory of categories enriched over a 2-category. We introduce Segal dg-categories which did not exist so far. We show that the homotopy transfer problem for algebras leads directly to a Leinster{Segal algebra. [ABSTRACT FROM AUTHOR]

The goal of this article is to emphasize the role of cubical sets in enriched category theory and infinity-category theory. We show in particular that categories enriched in cubical sets provide a convenient way to describe many infinity-categories appearing in the context of homological algebra. [ABSTRACT FROM AUTHOR]

We introduce the notion of a braiding on a skew monoidal category, whose curious feature is that the defining isomorphisms involve three objects rather than two. Examples are shown to arise from 2-category theory and from bialgebras. In order to describe the 2-categorical examples, we take a multicategorical approach. We explain how certain braided skew monoidal structures in the 2-categorical setting give rise to braided monoidal bicategories. For the bialgebraic examples, we show that, for a skew monoidal category arising from a bialgebra, braidings on the skew monoidal category are in bijection with cobraidings (also known as coquasitriangular structures) on the bialgebra. [ABSTRACT FROM AUTHOR]

We relate the relative nerve Nf (D) of a diagram of simplicial sets f : D → sSet with the Grothendieck construction GrF of a simplicial functor F : D → sCat in the case where f = NF. We further show that any strict monoidal simplicial category C gives rise to a functor C• : Δop → sCat, and that the relative nerve of NC• is the operadic nerve N⊕(C). Finally, we show that all the above constructions commute with appropriately defined opposite functors. [ABSTRACT FROM AUTHOR]

We consider Toeplitz and Cuntz-Krieger C*-algebras associated with finitely aligned left cancellative small categories. We pay special attention to the case where such a category arises as the Zappa-Szép product of a category and a group linked by a one-cocycle. As our main application, we obtain a new approach to Exel-Pardo algebras in the case of row-finite graphs. We also present some other ways of constructing C*-algebras from left cancellative small categories and discuss their relationship. [ABSTRACT FROM AUTHOR]

Passive linear networks are used in a wide variety of engineering applications, but the best studied are electrical circuits made of resistors, inductors and capacitors. We describe a category where a morphism is a circuit of this sort with marked input and output terminals. In this category, composition describes the process of attaching the outputs of one circuit to the inputs of another. We construct a functor, dubbed the 'black box functor', that takes a circuit, forgets its internal structure, and remembers only its external behavior. Two circuits have the same external behavior if and only if they impose same relation between currents and potentials at their terminals. The space of these currents and potentials naturally has the structure of a symplectic vector space, and the relation imposed by a circuit is a Lagrangian linear relation. Thus, the black box functor goes from our category of circuits to a category with Lagrangian linear relations as morphisms. We prove that this functor is symmetric monoidal and indeed a hypergraph functor. We assume the reader is familiar with category theory, but not with circuit theory or symplectic linear algebra. [ABSTRACT FROM AUTHOR]

In this paper, we prove that there is a canonical homotopy (n+1)-algebra structure on the shifted operadic deformation complex Def(en→P)[-n] for any operad P and a map of operads f:en→P. This result generalizes the result of [T2], where the case ... was considered. Another more computational proof of the same statement was recently sketched in [CW]. Our method combines the one of Tamarkin with the categorical algebra on the category of symmetric sequences, introduced in [R] and further developed in Kapranov-Manin and Fresse. We define suitable deformation functors on n-coalgebras, which are considered as the "non-commutative" base of deformation, prove their representability, and translate properties of the functors to the corresponding properties of the representing objects. A new point, which makes the method more powerful, is to consider the argument of our deformation theory as an object of the category of symmetric sequences of dg vector spaces, not as just a single dg vector space. [ABSTRACT FROM AUTHOR]

The present article is the first of a series whose goal is to define a logical formalism in which it is possible to reason about genetics. In this paper, we introduce the main concepts of our language whose domain of discourse consists of a class of limit-sketches and their associated models. While our program will aim to show that different phenomena of genetics can be modeled by changing the category in which the models take their values, in this paper, we study models in the category of sets to capture mutation mechanisms such as insertions, deletions, substitutions, duplications and inversions. We show how the proposed formalism can be used for constructing multiple sequence alignments with an emphasis on mutation mechanisms. [ABSTRACT FROM AUTHOR]

We study the structure of the category of polynomials in a locally cartesian closed category. Formalizing the conceptual view that polynomials are constructed from sums and products, we characterize this category in terms of the composite of the pseudomonads which freely add fibred sums and products to fibrations. The composite pseudomonad structure corresponds to a pseudo-distributive law between these two pseudomonads, which exists if and only if the base category is locally cartesian closed. [ABSTRACT FROM AUTHOR]

We show that a Hopf monad on a *-autonomous category lifts *-autonom- ous structure to the category of algebras precisely when there is an algebra structure on the dualizing object. Our proof is based on Pastro's characterization of *-autonomous (co)monads as linearly distributive (co)monads with negation. [ABSTRACT FROM AUTHOR]

We introduce and compare several new exactness conditions involving what we call split cubes. These conditions are motivated by their special cases, some which become familiar, in the pointed context, once we reformulate them with split cubes replaced with split extensions. [ABSTRACT FROM AUTHOR]

It has been shown by J.Funk, P.Hofstra and B.Steinberg that any Grothendieck topos T is endowed with a canonical group object, called its isotropy group, which acts functorially on every object of T. We show that this group is in fact the group of points of a localic group object, called the localic isotropy group, which also acts on every object, and in fact also on every internal locales and on every T-topos. This new localic isotropy group has better functoriality and stability property than the original version and shed some lights on the phenomenon of higher isotropy observed for the ordinary isotropy group. We prove in particular using a localic version of the isotropy quotient that any geometric morphism can be factored uniquely as a connected atomic geometric morphism followed by a so called "essentially anisotropic" geometric morphism, and that connected atomic morphism are exactly the quotient by an open isotropy action. [ABSTRACT FROM AUTHOR]

Coarse-graining is a standard method of extracting a simple Markov process from a more complicated one by identifying states. Here we extend coarse-graining to open Markov processes. An "open" Markov process is one where probability can flow in or out of certain states called "inputs" and "outputs". One can build up an ordinary Markov process from smaller open pieces in two basic ways: composition, where we identify the outputs of one open Markov process with the inputs of another, and tensoring, where we set two open Markov processes side by side. In previous work, Fong, Pollard and the first author showed that these constructions make open Markov processes into the morphisms of a symmetric monoidal category. Here we go further by constructing a symmetric monoidal double category where the 2-morphisms are ways of coarse-graining open Markov processes. We also extend the already known "black-boxing" functor from the category of open Markov processes to our double category. Black-boxing sends any open Markov process to the linear relation between input and output data that holds in steady states, including nonequilibrium steady states where there is a nonzero flow of probability through the process. To extend black-boxing to a functor between double categories, we need to prove that black-boxing is compatible with coarse-graining. [ABSTRACT FROM AUTHOR]

The Faa di Bruno construction, introduced by Cockett and Seely, constructs a comonad Faa whose coalgebras are precisely Cartesian differential categories. In other words, for a Cartesian left additive category X, Faa(X) is the cofree Cartesian differential category over X. Composition in these cofree Cartesian differential categories is based on the Faa di Bruno formula, and corresponds to composition of differential forms. This composition, however, is somewhat complex and difficult to work with. In this paper we provide an alternative construction of cofree Cartesian differential categories inspired by tangent categories. In particular, composition defined here is based on the fact that the chain rule for Cartesian differential categories can be expressed using the tangent functor, which simplifies the formulation of the higher order chain rule. [ABSTRACT FROM AUTHOR]

Quantum categories have been recently studied because of their relation to bialgebroids, small categories, and skew monoidales. This is the first of a series of papers based on the author's PhD thesis in which we examine the theory of quantum categories developed by Day, Lack, and Street. A quantum category is an opmonoidal monad on the monoidale associated to a biduality R a R°, or enveloping monoidale, in a monoidal bicategory of modules Mod(ν) for a monoidal category ν. Lack and Street proved that quantum categories are in equivalence with right skew monoidales whose unit has a right adjoint in Mod(ν). Our first important result is similar to that of Lack and Street. It is a characterisation of opmonoidal arrows on enveloping monoidales in terms of a new structure named oplax action. We then provide three different notions of comodule for an opmonoidal arrow, and using a similar technique we prove that they are equivalent. Finally, when the opmonoidal arrow is an opmonoidal monad, we are able to provide the category of comodules for a quantum category with a monoidal structure such that the forgetful functor is monoidal. [ABSTRACT FROM AUTHOR]

Good atlases are defined for effective orbifolds, and a spark complex is constructed on each good atlas. It is proved that this process is 2-functorial with compatible systems playing as morphisms between good atlases, and that the spark character 2-functor factors through this 2-functor. [ABSTRACT FROM AUTHOR]

Tangent categories provide an axiomatic approach to key structural aspects of differential geometry that exist not only in the classical category of smooth manifolds but also in algebraic geometry, homological algebra, computer science, and combinatorics. Generalizing the notion of (linear) connection on a smooth vector bundle, Cockett and Cruttwell have defined a notion of connection on a differential bundle in an arbitrary tangent category. Herein, we establish equivalent formulations of this notion of connection that reduce the amount of specified structure required. Further, one of our equivalent formulations substantially reduces the number of axioms imposed, and others provide useful abstract conceptualizations of connections. In particular, we show that a connection on a differential bundle E over M is equivalently given by a single morphism K that induces a suitable decomposition of T E as a biproduct. We also show that a connection is equivalently given by a vertical connection K for which a certain associated diagram is a limit diagram. [ABSTRACT FROM AUTHOR]

The Ehresmann-Schein-Nambooripad (ESN) Theorem asserts an equivalence between the category of inverse semigroups and the category of inductive groupoids. In this paper, we consider the category of inverse categories and functors - a natural generalization of inverse semigroups and semigroup homomorphisms - and extend the ESN Theorem to an equivalence between this category and the category of locally complete inductive groupoids and locally inductive functors. From the proof of this extension, we also generalize the ESN Theorem to an equivalence between the category of inverse semicategories and the category of locally inductive groupoids and to an equivalence between the category of inverse categories with oplax functors and the category of locally complete inductive groupoids and ordered functors. [ABSTRACT FROM AUTHOR]

Long before the invention of Feynman diagrams, engineers were using similar diagrams to reason about electrical circuits and more general networks containing mechanical, hydraulic, thermodynamic 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 electrical 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- ow 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. [ABSTRACT FROM AUTHOR]

Following the pattern of the Frobenius structure usually assigned to the 1-dimensional sphere, we investigate the Frobenius structures of spheres in all other dimensions. Starting from dimension d = 1, all the spheres are commutative Frobenius objects in categories whose arrows are (d + 1)-dimensional cobordisms. With respect to the language of Frobenius objects, there is no distinction between these spheres-they are all free of additional equations formulated in this language. The corresponding structure makes out of the 0-dimensional sphere not a commutative but a symmetric Frobenius object. This sphere is mapped to a matrix Frobenius algebra by a 1-dimensional topological quantum field theory, which corresponds to the representation of a class of diagrammatic algebras given by Richard Brauer [ABSTRACT FROM AUTHOR]

The formation of the "strict" span category Span(C) of a category C with pullbacks is a standard organizational tool of category theory. Unfortunately, limits or colimits in Span(C) are not easily computed in terms of constructions in C. This paper shows how to form the pullback in Span(C) for many, but not all, pairs of spans, given the existence of some speci-c so-called lax pullback complements in C of the "left legs" of at least one of the two given spans. For some types of spans we require the ambient category to be adhesive to be able to form at least a weak pullback in Span(C). The existence of all lax pullback complements in C along a given morphism is equivalent to the exponentiability of that morphism. Since exponentiability is a rather restrictive property of a morphism, the paper first develops a comprehensive framework of rules for individual lax pullback complement diagrams, which resembles the set of pasting and cancellation rules for pullback diagrams, including their behaviour under pullback. We also present examples of lax pullback complements along non-exponentiable morphisms, obtained via lifting along a fibration. [ABSTRACT FROM AUTHOR]

We introduce a new class of categories generalizing locally presentable ones. The distinction does not manifest in the abelian case and, assuming Vopěenka's principle, the same happens in the regular case. The category of complete partial orders is the natural example of a nearly locally finitely presentable category which is not locally presentable. [ABSTRACT FROM AUTHOR]

We show that the regular patterns of Getzler (2009) form a 2-category biequivalent to the 2-category of substitudes of Day and Street (2003), and that the Feynman categories of Kaufmann and Ward (2013) form a 2-category biequivalent to the 2-category of coloured operads (with invertible 2-cells). These biequivalences induce equivalences between the corresponding categories of algebras. There are three main ingredients in establishing these biequivalences. The first is a strictification theorem (exploiting Power's General Coherence Result) which allows to reduce to the case where the structure maps are identity-on-objects functors and strict monoidal. Second, we subsume the Getzler and Kaufmann{Ward hereditary axioms into the notion of Guitart exactness, a general condition ensuring compatibility between certain left Kan exten- sions and a given monad, in this case the free-symmetric-monoidal-category monad. Finally we set up a biadjunction between substitudes and what we call pinned symmetric monoidal categories, from which the results follow as a consequence of the fact that the hereditary map is precisely the counit of this biadjunction. [ABSTRACT FROM AUTHOR]

We define strict and weak duality involutions on 2-categories, and prove a coherence theorem that every bicategory with a weak duality involution is biequivalent to a 2-category with a strict duality involution. For this purpose we introduce \2-categories with contravariance", a sort of enhanced 2-category with a basic notion of \contravariant morphism", which can be regarded either as generalized multicategories or as enriched categories. This enables a universal characterization of duality involutions using absolute weighted colimits, leading to a conceptual proof of the coherence theorem. [ABSTRACT FROM AUTHOR]

We explicitly show that symmetric Frobenius structures on a finite-dimensional, semi-simple algebra stand in bijection to homotopy fixed points of the trivial SO(2)-action on the bicategory of finite-dimensional, semi-simple algebras, bimodules and intertwiners. The results are motivated by the 2-dimensional Cobordism Hypothesis for oriented manifolds, and can hence be interpreted in the realm of Topological Quantum Field Theory. [ABSTRACT FROM AUTHOR]

Databases have been studied category-theoretically for decades. While mathematically elegant, previous categorical models have typically struggled with representing concrete data such as integers or strings. In the present work, we propose an extension of the earlier set-valued functor model, making use of multi-sorted algebraic theories (a.k.a. Lawvere theories) to incorporate concrete data in a principled way. This approach easily handles missing information (null values), and also allows constraints and queries to make use of operations on data, such as multiplication or comparison of numbers, helping to bridge the gap between traditional databases and programming languages. We also show how all of the components of our model-including schemas, instances, change-of-schema functors, and queries--t into a single double categorical structure called a proarrow equipment (a.k.a. framed bicategory). [ABSTRACT FROM AUTHOR]

In [Barnea, Schlank, 2015(2)], the author and Tomer Schlank studied a much weaker homotopical structure than a model category, which we called a \weak cofibration category". We showed that a small weak cofibration category induces in a natural way a model category structure on its ind-category, provided the ind-category satisfies a certain two out of three property. The main purpose of this paper is to give sufficient intrinsic conditions on a weak cofibration category for this two out of three property to hold. We consider an application to the category of compact metrizable spaces, and thus obtain a model structure on its ind-category. This model structure is defined on a category that is closely related to a category of topological spaces and has many convenient formal properties. A more general application of these results, to the (opposite) category of separable C*-algebras, appears in a paper by the author, Michael Joachim and Snigdhayan Mahanta [Barnea, Joachim, Mahanta, 2017]. [ABSTRACT FROM AUTHOR]

Finitary monads on a locally finitely presentable category A are well-known to possess a presentation by signatures and equations. Here we prove that, analogously, bases on A, i.e., finitary functors from A to the category of finitary monads on A, possess a presentation by parametrized signatures and equations. [ABSTRACT FROM AUTHOR]

At the heart of differential geometry is the construction of the tangent bundle of a manifold. There are various abstractions of this construction, and of particular interest here is that of Tangent Structures. Tangent Structure is defined via giving an underlying category M and a tangent functor T along with a list of natural transformations satisfying a set of axioms, then detailing the behaviour of T in the category End(M). However, this axiomatic definition at first seems somewhat disjoint from other approaches in differential geometry. The aim of this paper is to present a perspective that addresses this issue. More specifically, this paper highlights a very explicit relationship between the axiomatic definition of Tangent Structure and the Weil algebras (which have a well established place in differential geometry). [ABSTRACT FROM AUTHOR]

The main result of this paper may be stated as a construction of "almost representations" μn and μn for the presheaves Obn and Obn on the C-systems defined by locally cartesian closed universe categories with binary product structures and the study of the behavior of these "almost representations" with respect to the universe category functors. In addition, we study a number of constructions on presheaves on C-systems and on universe categories that are used in the proofs of our main results, but are expected to have other applications as well. [ABSTRACT FROM AUTHOR]

In this note, we prove the existence of E≤-injective hulls in the category PoSgr≤ of posemigroups and their submultiplicative order-preserving maps; here E≤ denotes the class of those morphisms h: A → B for which h(a1) ... h(an) ≤ h(a) always implies a1 ... an ≤ a. The result of this note subsumes the results given by Lambek et al. (2012) and by Zhang and Laan (2014). [ABSTRACT FROM AUTHOR]

Normal monomorphisms in the sense of Bourn describe the equivalence classes of an internal equivalence relation. Although the definition is given in the fairly general setting of a category with finite limits, later investigations on this subject often focus on protomodular settings, where normality becomes a property. This paper clarifies the connections between internal equivalence relations and Bourn-normal monomorphisms in regular Mal'tesv categories with pushouts of split monomorphisms along arbitrary morphisms, whereas a full description is achieved for quasi-pointed regular Mal'tsev categories with pushouts of split monomorphisms along arbitrary morphisms. [ABSTRACT FROM AUTHOR]

We define the notion of a (P, P)-structure on a universe p in a locally cartesian closed category category with a binary product structure and construct a (∏, λ)-structure on the C-systems CC(C, p) from a (P, P)-structure on p. We then define homomorphisms of C-systems with (∏, &#955,)-structures and functors of universe categories with (P, P)-structures and show that our construction is functorial relative to these definitions. [ABSTRACT FROM AUTHOR]

Sheaves are objects of a local nature: a global section is determined by how it looks locally. Hence, a sheaf cannot describe mathematical structures which contain global or nonlocal geometric information. To fill this gap, we introduce the theory of "gleaves", which are presheaves equipped with an additional "gluing operation" of compatible pairs of local sections. This generalizes the conditional product structures of Dawid and Studeny, which correspond to gleaves on distributive lattices. Our examples include the gleaf of metric spaces and the gleaf of joint probability distributions. A result of Johnstone shows that a category of gleaves can have a subobject classifier despite not being cartesian closed. Gleaves over the simplex category Δ, which we call compositories, can be interpreted as a new kind of higher category in which the composition of an m-morphism and an n-morphism along a common κ-morphism face results in an (m + n -- k)-morphism. The distinctive feature of this composition operation is that the original morphisms can be recovered from the composite morphism as initial and final faces. Examples of compositories include nerves of categories and compositories of higher spans. [ABSTRACT FROM AUTHOR]