<pre>
for all s1, s2, s3 in S:
- (i) s1⋆s2 etc are also in S
- (ii) (s1⋆s2)⋆s3 = s1⋆(s2⋆s3)
+ (i) s1⋆s2 etc are also in S
+ (ii) (s1⋆s2)⋆s3 = s1⋆(s2⋆s3)
(iii) z⋆s1 = s1 = s1⋆z
</pre>
To have a category, the elements and morphisms have to satisfy some constraints:
<pre>
- (i) the class of morphisms has to be closed under composition:
- where f:C1→C2 and g:C2→C3, g ∘ f is also a
- morphism of the category, which maps C1→C3.
- (ii) composition of morphisms has to be associative
+ (i) the class of morphisms has to be closed under composition:
+ where f:C1→C2 and g:C2→C3, g ∘ f is also a
+ morphism of the category, which maps C1→C3.
+
+ (ii) composition of morphisms has to be associative
+
(iii) every element E of the category has to have an identity
- morphism 1<sub>E</sub>, which is such that for every morphism
- f:C1→C2: 1<sub>C2</sub> ∘ f = f = f ∘ 1<sub>C1</sub>
+ morphism 1<sub>E</sub>, which is such that for every morphism f:C1→C2:
+ 1<sub>C2</sub> ∘ f = f = f ∘ 1<sub>C1</sub>
</pre>
These parallel the constraints for monoids. Note that there can be multiple distinct morphisms between an element `E` and itself; they need not all be identity morphisms. Indeed from (iii) it follows that each element can have only a single identity morphism.
Some examples of categories are:
-* Categories whose elements are sets and whose morphisms are functions between those sets. Here the source and target of a function are its domain and range, so distinct functions sharing a domain and range (e.g., sin and cos) are distinct morphisms between the same source and target elements. The identity morphism for any element/set is just the identity function for that set.
+* Categories whose elements are sets and whose morphisms are functions between those sets. Here the source and target of a function are its domain and range, so distinct functions sharing a domain and range (e.g., `sin` and `cos`) are distinct morphisms between the same source and target elements. The identity morphism for any element/set is just the identity function for that set.
* any monoid <code>(S,⋆,z)</code> generates a category with a single element `x`; this `x` need not have any relation to `S`. The members of `S` play the role of *morphisms* of this category, rather than its elements. All of these morphisms are understood to map `x` to itself. The result of composing the morphism consisting of `s1` with the morphism `s2` is the morphism `s3`, where <code>s3=s1⋆s2</code>. The identity morphism for the (single) category element `x` is the monoid's identity `z`.
-* a **preorder** is a structure `(S, ≤)` consisting of a reflexive, transitive, binary relation on a set `S`. It need not be connected (that is, there may be members `x`,`y` of `S` such that neither `x≤y` nor `y≤x`). It need not be anti-symmetric (that is, there may be members `s1`,`s2` of `S` such that `s1≤s2` and `s2≤s1` but `s1` and `s2` are not identical). Some examples:
+* a **preorder** is a structure <code>(S, ≤)</code> consisting of a reflexive, transitive, binary relation on a set `S`. It need not be connected (that is, there may be members `x`,`y` of `S` such that neither <code>x≤y</code> nor <code>y≤x</code>). It need not be anti-symmetric (that is, there may be members `s1`,`s2` of `S` such that <code>s1≤s2</code> and <code>s2≤s1</code> but `s1` and `s2` are not identical). Some examples:
* sentences ordered by logical implication ("p and p" implies and is implied by "p", but these sentences are not identical; so this illustrates a pre-order without anti-symmetry)
* sets ordered by size (this illustrates it too)
A **functor** is a "homomorphism", that is, a structure-preserving mapping, between categories. In particular, a functor `F` from category <b>C</b> to category <b>D</b> must:
<pre>
- (i) associate with every element C1 of <b>C</b> an element F(C1) of <b>D</b>
- (ii) associate with every morphism f:C1→C2 of <b>C</b> a morphism F(f):F(C1)→F(C2) of <b>D</b>
- (iii) "preserve identity", that is, for every element C1 of <b>C</b>: F of C1's identity morphism in <b>C</b> must be the identity morphism of F(C1) in <b>D</b>: F(1<sub>C1</sub>) = 1<sub>F(C1)</sub>.
- (iv) "distribute over composition", that is for any morphisms f and g in <b>C</b>: F(g ∘ f) = F(g) ∘ F(f)
+ (i) associate with every element C1 of <b>C</b> an element F(C1) of <b>D</b>
+
+ (ii) associate with every morphism f:C1→C2 of <b>C</b> a morphism
+ F(f):F(C1)→F(C2) of <b>D</b>
+
+ (iii) "preserve identity", that is, for every element C1 of <b>C</b>:
+ F of C1's identity morphism in <b>C</b> must be the identity morphism
+ of F(C1) in <b>D</b>: F(1<sub>C1</sub>) = 1<sub>F(C1)</sub>.
+
+ (iv) "distribute over composition", that is for any morphisms f and g in <b>C</b>:
+ F(g ∘ f) = F(g) ∘ F(f)
</pre>
A functor that maps a category to itself is called an **endofunctor**. The (endo)functor that maps every element and morphism of <b>C</b> to itself is denoted `1C`.