Algebras for the partial map classifier monad. (English) Zbl 0747.18005
Category theory, Proc. Int. Conf., Como/Italy 1990, Lect. Notes Math. 1488, 262-278 (1991).
[For the entire collection see Zbl 0733.00009.]
An elementary topos \(E\) is a finitely complete Cartesian closed category for which the partial maps into each object \(x\) are represented by actual maps into an object \(tx\). This construction \(t\) enriches to a monad on the category \(E\). When \(E\) is the category of sets, \(tx\) is the set of subsets of \(x\) of cardinality \(\leq 1\), and an Eilenberg-Moore \(t\)-algebra is a pointed set. The present paper identifies the \(t\)-algebras for any \(E\) using actions of the subobject classifier \(t1\).
An elementary topos \(E\) is a finitely complete Cartesian closed category for which the partial maps into each object \(x\) are represented by actual maps into an object \(tx\). This construction \(t\) enriches to a monad on the category \(E\). When \(E\) is the category of sets, \(tx\) is the set of subsets of \(x\) of cardinality \(\leq 1\), and an Eilenberg-Moore \(t\)-algebra is a pointed set. The present paper identifies the \(t\)-algebras for any \(E\) using actions of the subobject classifier \(t1\).
Reviewer: R.H.Street (North Ryde)
MSC:
| 18B25 | Topoi |
| 18C15 | Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads |
