Related papers: Uniform Elgot Iteration in Foundations
Classic Topic Models are built under the Bag Of Words assumption, in which word position is ignored for simplicity. Besides, symmetric priors are typically used in most applications. In order to easily learn topics with different properties…
Many of the properties of sectional category, topological complexity and homotopic distance are in fact derived from a small number of basic properties, which, once established, lead to all the others without further recourse to topology.…
We illustrate the generative power of the lifting property (orthogonality of morphisms in a category) as means of defining natural elementary mathematical concepts by giving a number of examples in various categories, in particular showing…
Category theory is the language of homological algebra, allowing us to state broadly applicable theorems and results without needing to specify the details for every instance of analogous objects. However, authors often stray from the realm…
The free algebra adjunction, between the category of algebras of a monad and the underlying category, induces a comonad on the category of algebras. The coalgebras of this comonad are the topic of study in this paper (following earlier…
Starting from the observation that distinct notions of copying have arisen in different categorical fields (logic and computation, contrasted with quantum mechanics) this paper addresses the question of when, or whether, they may coincide.…
Capretta's delay monad can be used to model partial computations, but it has the "wrong" notion of built-in equality, strong bisimilarity. An alternative is to quotient the delay monad by the "right" notion of equality, weak bisimilarity.…
In various subjects including mathematics, one can hope to use mathematical thinking well when the right kinds of algebraic structure to consider can be discovered or spotted. Therefore, it would help to understand kinds of algebraic…
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,…
The category of models of any theory $T$ in any first-order language $L$ has the surprising property that any small category that is elementarily equivalent with it, already embeds in it. The proof uses an abstract argument via ultrapowers,…
Categorical universal algebra can be developed either using Lawvere theories (single-sorted finite product theories) or using monads, and the category of Lawvere theories is equivalent to the category of finitary monads on Set. We show how…
Monads are a useful tool for structuring effectful features of computation such as state, non-determinism, and continuations. In the last decade, several generalisations of monads have been suggested which provide a more fine-grained model…
We develop a representation theory of categories as a means to explore characteristic structures in algebra. Characteristic structures play a critical role in isomorphism testing of groups and algebras, and their construction and…
In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of…
The purpose of this paper is to show that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, and universal constructions in the Sets, the category of sets and…
Like the notion of computation via (strong) monads serves to classify various flavours of impurity, including exceptions, non-determinism, probability, local and global store, the notion of guardedness classifies well-behavedness of cycles…
A concept of "evolving categories" is suggested to build a simple, scalable, mathematically consistent framework for representing in uniform way both data and algorithms. A state machine for executing algorithms becomes clear, rich and…
The ordinary Structure Identity Principle states that any property of set-level structures (e.g., posets, groups, rings, fields) definable in Univalent Foundations is invariant under isomorphism: more specifically, identifications of…
This paper investigates some issues arising in categorical models of reversible logic and computation. Our claim is that the structural (coherence) isomorphisms of these categorical models, although generally overlooked, have decidedly…
In this article we show how to build main aspects of our paper on globular weak $(\infty,n)$-categories, but now for the cubical geometry. Thus we define a monad on the category $\mathbb{C}\mathbb{S}ets$ of cubical sets which algebras are…