The theory developed by Gambino and Kock, of polynomials over a locally cartesian closed category E, is generalised for E just having pullbacks. The 2-categorical analogue of the theory of polynomials and polynomial functors is given, and its relationship with Street's theory of fibrations within 2-categories is explored. Johnstone's notion of "bagdomain data" is adapted to the present framework to make it easier to completely exhibit examples of polynomial monads.
Span categories provide an abstract framework for formalizing mathematical models of certain systems. The mathematical descriptions of some systems, such as classical mechanical systems, require categories that do not have pullbacks, and this limits the utility of span categories as a formal framework. Given categories $\mathscr{C}$ and $\mathscr{C}^\prime$ and a functor $\mathcal F$ from $\mathscr{C}$ to $\mathscr{C}^\prime$, we introduce the notion of an $\mathcal F$ pullback of a cospan in $\mathscr{C}$, as well as the notion of span tightness of $\mathcal F$. If $\mathcal F$ is span tight,…
We firstly prove the completeness of the category of crossed modules in a modified category of interest. Afterwards, we define pullback crossed modules and pullback cat$^1$-objects that are both obtained by pullback diagrams with extra structures on certain arrows. These constructions unify many corresponding results for the cases of groups, commutative algebras and can also be adapted to various algebraic structures.
The construction of a category of spans can be made in some categories $\CC$ which do not have pullbacks in the traditional sense. The PROP for monoids is a good example of such a $\CC$. The 2012 book concerning homological algebra by Marco Grandis gives the proof of associativity of relations in a Puppe-exact category based on a 1967 paper of M.Š. Calenko. The proof here is a restructuring of that proof in the spirit of the first sentence of this Abstract. We observe that these relations are spans of EM-spans and that EM-spans admit fake pullbacks so that spans of EM-spans compose. Our settin…
In this paper we provide purely categorical proofs of two important results of structural Ramsey theory: the result of M.\ Sokić that the free product of Ramsey classes is a Ramsey class and the result of M.\ Bodirsky, M.\Pinsker and T.\ Tsankov that adding constants to the language of a Ramsey class preserves the Ramsey property. The proofs that we present here ignore the model-theoretic background of these statements. Instead, they focus on categorical constructions by which the classes can be constructed, generalizing the original statements along the way. It turns out that the restriction …
We consider the ordinary category Span(C) of (isomorphism classes of) spans of morphisms in a category C with finite limits as needed, composed horizontally via pullback, and give a general criterion for a quotient of Span(C) to be an allegory. In particular, when C carries a pullback-stable, but not necessarily proper, (E, M)-factorization system, we establish a quotient category Span_E(C) that is isomorphic to the category Rel_M(C) of M-relations in C, and show that it is a (unitary and tabular) allegory precisely when M is a class of monomorphisms in C. Without this restriction, one can sti…
Generalising slightly the notions of a strict computability model and of a simulation between them, which were elaborated by Longley and Normann, we define canonical computability models over categories and appropriate Set-valued functors on them. We study the canonical total computability model over a category, and the partial one over a category with pullbacks. Our notions and results are generalised to categories with a base of computability, connecting Rosolini's theory of dominions with the theory of computability models.
The Biequivalence of Locally Cartesian Closed Categories and Martin-Löf Type Theories
Seely的论文“Locally Carnival closed categories and type theory“包含了范畴类型论中一个著名的结果：局部Carnival closed categories等价于Martin-Löf类型论中的Pi-类型、Sigma-类型和外延恒等式。然而，Seely的证明依赖于一个有问题的假设，即类型中的替换可以通过回调来解释。在这里，我们证明了西利定理的一个修正版本：马丁-洛夫类型论的Bénabou-Hofmann解释在局部范畴闭范畴中产生2-范畴的双等价。为了促进技术的发展，我们采用了具有族的范畴来代替句法的马丁-洛夫类型论。作为第二个结果，我们证明了，如果我们删除PI型所产生的类别与家庭是双等价的左正合范畴。
Given a locally cartesian closed category E, a polynomial (s,p,t) may be defined as a diagram consisting of three arrows in E of a certain shape. In this paper we define the homogeneous and monomial terms comprising a polynomial (s,p,t) and give sufficient conditions on E such that the homogeneous terms of polynomials exist. We use these homogeneous terms to exhibit an infinite family of coreflection adjunctions between polynomials and homogeneous polynomials of order n. We show that every locally cartesian closed category E with a strict initial object admits a negation operator and a (dense,…
University of Southampton（南安普顿大学）；University College London（伦敦大学学院）
Reverse derivative categories (RDCs) have recently been shown to be a suitable semantic framework for studying machine learning algorithms. Whereas emphasis has been put on training methodologies, less attention has been devoted to particular \emph{model classes}: the concrete categories whose morphisms represent machine learning models. In this paper we study presentations by generators and equations of classes of RDCs. In particular, we propose \emph{polynomial circuits} as a suitable machine learning model. We give an axiomatisation for these circuits and prove a functional completeness res…
We define indexed categories of (open) dynamical system and random dynamical system over polynomial interfaces, where time is given by an arbitrary monoid $\mathbb{T}$. We consider the case of open random dynamical systems over both open and closed noise sources, and the case where the interface of the random system is `nested' over the interface of its noise source. We show that, in discrete time, our categories of dynamical systems over polynomial interfaces $p$ are equivalent to Spivak's categories $p$-$\mathbf{Coalg}$ of $p$-coalgebras. We then define a notion of generalized $pT$-coalgebra…
We continue the theory of $\tT$-systems from the work of the second author, describing both ground systems and module systems over a ground system (paralleling the theory of modules over an algebra). The theory, summarized categorically at the end, encapsulates general algebraic structures lacking negation but possessing a map resembling negation, such as tropical algebras, hyperfields and fuzzy rings. We see explicitly how it encompasses tropical algebraic theory and hyperfields. Prime ground systems are introduced as a way of developing geometry. The polynomial system over a prime system is …
