By Barr M.

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**Sample text**

It does not preserve initial objects, but it does reﬂect initial objects although vacuously: the empty set is the only initial object in Set and the underlying set of a monoid cannot be empty since it must have an identity element. We leave the details to you. 4 Equivalences In this section we deﬁne what it means for two categories to be equivalent. The correct concept turns out to be weaker than requiring that they be isomorphic – that is, that there is a functor from one to the other which has an inverse in Cat.

Natural transformations are maps between functors; one way to think of them is as a deformation of one construction (construed as a functor) into another. 1 Diagrams We begin with diagrams in a graph and discuss commutativity later. 1 Deﬁnition Let I and G be graphs. A diagram in G of shape I is a homomorphism D : I − → G of graphs. I is called the shape graph of the diagram D. We have thus given a new name to a concept which was already deﬁned (not uncommon in mathematics). A diagram is a graph homomorphism from a diﬀerent point of view.

M (u) has domain M (0) and codomain M (1). But the domain and codomain of an arrow in a category are uniquely determined by the arrow. So that the only necessary information is which arrow M (u) is.

### Category theory by Barr M.

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