In this paper we want to investigate two notions of the cardinality of relations in the context of allegories. The different axiom systems are motivated on the existence of injective and surjective functions, respectively. In both cases we provide a canonical cardinality function and show that it is initial in the category of all cardinality functions over the given allegory.

This paper presents three paradigms for circuit design, and investigates the relationships between them. These paradigms are syntactic (based on Freyd and Scedrov's unitary pre-tabular allegories (upas)), pictorial (based on the net list model of circuit connectivity), and relational (based on Sheeran's relational model of circuit design Ruby). We show that net lists over a given signature cons...

I was writing and thinking about the algorithmic and the allegoric. I’ve been struck by how so much contemporary art practice has been informed by the algorithmic. Having done work in information engines and various net art pieces using algorithms for the display and organization of information, I was equally intrigued by what we get after we’ve deployed all these little engines and are outside...

We develop a new compilation scheme and categorical abstract machine for execution of logic programs based on allegories, the categorical version of the calculus of relations. In our case, both operational and denotational semantics are built upon the same theory, thus execution of a query is performed using algebraic reasoning, giving immediate proof of correctness properties. Our work serves ...

Relational languages such as Ruby are used to derive hardware circuits from abstract speci cations of their behaviour. Much reasoning is done informally in Ruby using pictorial representations of relational terms. We formalise this use of pictures in circuit design. We show that pictures naturally form a unitary pretabular allegory. Homomorphisms of pictures correspond to adding new wires or ci...