**1. Definition of operad**

An **operad** consists of:

* A set of 'objects'

* For objects \\(X_1,...,X_n,Y\\), a set of 'morphisms' \\(\phi: X_1,...,X_n \rightarrow Y\\)

* For each object, a designated identity morphism

* A rule for combining connected "trees" of morphisms into a unique composite morphism

An **operad** consists of:

* A set of 'objects'

* For objects \\(X_1,...,X_n,Y\\), a set of 'morphisms' \\(\phi: X_1,...,X_n \rightarrow Y\\)

* For each object, a designated identity morphism

* A rule for combining connected "trees" of morphisms into a unique composite morphism