cartesian category
- definition
- a category \(\mathcal{C}\) is cartesian if
- for any two objects \(A\) and \(B\) they have a product
- and \(\mathcal{C}\) has a terminal object \(1\)
- a category \(\mathcal{C}\) is cartesian if
- equivalently, for any finite set \(I\) and and \(I\)-indexed family \(A_{(-)}: I \to \mathcal{C}\) of objects, there's a product \(\Pi_{i \in I} A_{i}\) in \(\mathcal{C}\)
- cartesian functor
- a functor between two cartesian categories is said to be cartesian if it preserve the product and terminal object
- which means, for two objects in the domain category, their product in the domain category is sent to codomain category and behave as the product of ...
Backlinks
Lawvere Theories
Definition of a model of a Lawere theory \(\mathcal{L}\) in a cartesian category \(\mathcal{C}\)
algebraic theory
ultimately, we want to say "wriring diagrams with operations from \(\mathbb{T}\) are lenses in ... cartesian category"
category Arity
wiring diagram are interpreted as lenses in the category of arities, which are the free cartesian categories
category Euc
It's a cartesian category with \(\mathbb{R}^{n} \times \mathbb{R}^{m} = \mathbb{R}^{n + m}\) and \(1 = \mathbb{R}^{0}\)