algebraic theory
- informal definition from Categorical System Theory (1.3.4.1)
- A algebraic theory \(\mathbb{T}\) consists of:
- a set \(\mathbb{T}_{n}\) of \(n\)-ary operations for each \(n \in \mathbb{N}\)
- a set of laws setting some composites of operations equal to others
- A algebraic theory \(\mathbb{T}\) consists of:
- we can use it to organize the operations we can perform in wiring diagram
- informal: A wiring diagram with operations from an algebraic theory \(\mathbb{T}\) is a wiring diagram where operations from the theory \(\mathbb{T}\) can be drawn in little green beads on the wires.
- this is in contrast to a wiring diagram without any operation (pure wiring diagram)
- ultimately, we want to say "wriring diagrams with operations from \(\mathbb{T}\) are lenses in ... cartesian category"
- Lawvere theories is what we need
- informal: A wiring diagram with operations from an algebraic theory \(\mathbb{T}\) is a wiring diagram where operations from the theory \(\mathbb{T}\) can be drawn in little green beads on the wires.
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Lawvere Theories
If this is the case, we say that the Lawvere theory is presented by the algebraic theory.