By Garrett P.

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Universal properties. The most natural context in which to introduce universal properties requires a good familiarity with the language of functors, which we will only introduce at a later stage (cf. 1). For the purpose of the examples we will run across in (most of) this course, the following ‘working deﬁnition’ should suﬃce. We say that a construction satisﬁes a universal property (or: ‘is the solution to a universal problem’) when it may be viewed as a terminal object of a category. The category depends on the context, and is usually explained ‘in words’ (and often without even mentioning the word category).

11This is a particularly important exercise, and I recommend that the reader writes out all the gory details carefully. 2. 7. Basic examples. The basic operations on sets provided us with several important examples of injective and surjective functions. 4. Let A, B be sets. Then there are natural projections πA , πB : A×B ` ÒÒ ```πB Ò `` ÒÒ `0 0 ÑÒÑ Ò πA A B deﬁned by πA ((a, b)) := a , πB ((a, b)) := b for all (a, b) ∈ A × B. Both these maps are (clearly) surjective. 5. , B of B) in A B. 6. If ∼ is an equivalence relation on a set A, there is a (clearly surjective) canonical projection G G A/∼ A obtained by sending every a ∈ A to its equivalence class [a]∼ .

Is not part of the deﬁnition of category: these operations highlight interesting features of Set, which may or may not be shared by other categories. We will soon come back to some of these operations and understand more precisely what they say about Set. 3. Here is a completely diﬀerent example. Suppose S is a set, and ∼ is a relation on S satisfying the reﬂexive and transitive property. Then we can encode this data into a category: • Objects: the elements of S; • Morphisms:, if a, b are objects (that is: if a, b ∈ S) then let Hom(a, b) be the set consisting of the element (a, b) ∈ S × S if a ∼ b, and Hom(a, b) = ∅ otherwise.