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diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn
index 53a66112..7b52c3a6 100644
--- a/advanced_topics/monads_in_category_theory.mdwn
+++ b/advanced_topics/monads_in_category_theory.mdwn
@@ -19,35 +19,41 @@ corrections.
Monoids
-------
-A **monoid** is a structure `(S, *, z)` consisting of an associative binary operation `*` over some set `S`, which is closed under `*`, and which contains an identity element `z` for `*`. That is:
+A **monoid** is a structure (S,⋆,z)
consisting of an associative binary operation ⋆
over some set `S`, which is closed under ⋆
, and which contains an identity element `z` for ⋆
. That is:
for all s1, s2, s3 in S: - (i) s1*s2 etc are also in S - (ii) (s1*s2)*s3 = s1*(s2*s3) - (iii) z*s1 = s1 = s1*z + (i) s1⋆s2 etc are also in S + (ii) (s1⋆s2)⋆s3 = s1⋆(s2⋆s3) + (iii) z⋆s1 = s1 = s1⋆zSome examples of monoids are: -* finite strings of an alphabet `A`, with `*` being concatenation and `z` being the empty string -* all functions `X→X` over a set `X`, with `*` being composition and `z` being the identity function over `X` -* the natural numbers with `*` being plus and `z` being `0` (in particular, this is a **commutative monoid**). If we use the integers, or the naturals mod n, instead of the naturals, then every element will have an inverse and so we have not merely a monoid but a **group**.) -* if we let `*` be multiplication and `z` be `1`, we get different monoids over the same sets as in the previous item. +* finite strings of an alphabet `A`, with
⋆
being concatenation and `z` being the empty string
+* all functions X→X
over a set `X`, with ⋆
being composition and `z` being the identity function over `X`
+* the natural numbers with ⋆
being plus and `z` being `0` (in particular, this is a **commutative monoid**). If we use the integers, or the naturals mod n, instead of the naturals, then every element will have an inverse and so we have not merely a monoid but a **group**.)
+* if we let ⋆
be multiplication and `z` be `1`, we get different monoids over the same sets as in the previous item.
Categories
----------
A **category** is a generalization of a monoid. A category consists of a class of **elements**, and a class of **morphisms** between those elements. Morphisms are sometimes also called maps or arrows. They are something like functions (and as we'll see below, given a set of functions they'll determine a category). However, a single morphism only maps between a single source element and a single target element. Also, there can be multiple distinct morphisms between the same source and target, so the identity of a morphism goes beyond its "extension."
-When a morphism `f` in category C has source `C1` and target `C2`, we'll write `f:C1→C2`.
+When a morphism `f` in category C has source `C1` and target `C2`, we'll write f:C1→C2
.
To have a category, the elements and morphisms have to satisfy some constraints:
- (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 1E, which is such that for every morphism f:C1→C2: 1C2 ∘ f = f = f ∘ 1C1 + (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 1E, which is such that for every morphism f:C1→C2: + 1C2 ∘ f = f = f ∘ 1C1These 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. @@ -59,7 +65,7 @@ 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. -* any monoid `(S,*,z)` 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 `s3=s1*s2`. The identity morphism for the (single) category element `x` is the monoid's identity `z`. +* any monoid
(S,⋆,z)
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 s3=s1⋆s2
. 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: