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@ -309,7 +309,10 @@ If we apply the first function $g$ to some element from set $Y$, we will get an
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![Applying one function after another](../01_set/functions_one_after_another.svg)
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We can define a function that is the equivalent to performing the operation described above. Let us call it $h: Y → G$. We may say that $h$ is the *composition* of $g$ and $f$, or $h = f \bullet g$ (notice that the first function is on the right, so it's similar to $b = f(g(a)$).
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We can define a function that is the equivalent to performing the operation described above.
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That would be a function such that, if you follow the arrow $h$ for any element of set $Y$ you will get to the same element of the set $G$ as the one you will get if you follow the $g$ and then follow $f$.
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Let us call it $h: Y → G$. We may say that $h$ is the *composition* of $g$ and $f$, or $h = f \bullet g$ (notice that the first function is on the right, so it's similar to $b = f(g(a)$).
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![Functional composition](../01_set/functions_compose.svg)
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@ -341,32 +344,32 @@ One of the main ways in which modern engineering differs from ancient craftsmans
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**Task:** Think about what would be those functions' sources and targets.
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The diagram
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By the way, diagrams that are "zoomed out" that show functions without showing set elements are called *external diagrams*, as opposed to the ones that we saw before, which are *internal*.
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Representing composition with commutative diagrams
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Composition and external diagrams
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---
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In functional composition diagrams, the equivalence between the successive application of the two composed functions ($f \bullet g$) and the new function ($h$) is expressed by the fact that if you follow the arrow $h$ for any element of set $Y$ you will get to the same element of the set $G$ as the one you will get if you follow the $g$ and then follow $f$. Diagrams that express such equivalence between sequences of function applications are called *commutative diagrams*.
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Let's look at the diagram that demonstrates functional composition in which we showed that successive application of the two composed functions ($f \bullet g$) and the new function ($h$) are equivalent.
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![Functional composition](../01_set/functions_compose.svg)
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If we "zoom-out" the view of the last diagram so it does not show the individual set elements, we get a more general view of functional composition.
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We showed this equivalence by drawing an *internal* diagram, and explicitly drawing the elements of the functions' sources and targets in such a way that the two paths are equivalent.
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Alternatively, we can just *say* that the arrow paths are all equivalent (all arrows starting from a given set element ultimately lead to the same corresponding element from the resulting set) and draw the equivalence as an external diagram.
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![Functional composition for sets](../01_set/functions_compose_sets.svg)
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In fact, because this diagram commutes (that is, all arrows starting from a given set element ultimately lead to the same corresponding element from the resulting set), this view is a more appropriate representation of the concept (as enumerating the elements is redundant).
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The external diagram is a more appropriate representation of the concept of composition, as it is more general. In fact, it is *so* general that it can actually serve as a *definition of functional composition*.
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Having this insight allows us to redefine functional composition in a more visual way.
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> The composition of two functions $f$ and $g$ is a third function $h$ defined in such a way that this diagram commutes.
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> The composition of two functions $f$ and $g$ is a third function $h$ defined in such a way that all the paths in this diagram are equivalent.
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![Functional composition - general definition](../01_set/functions_compose_general.svg)
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Diagrams that show functions without showing the elements of the sets are called *external diagrams*, as opposed to the ones that we saw before, which are *internal*.
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If you continue reading this book, you will hear more about diagrams in which all paths are equivalent (they are called *commuting diagrams*, by the way)
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At this point you might be worried that I had forgotten that I am supposed to talk about category theory and I am just presenting a bunch of irrelevant concepts. I really tend to do that, but not now — the fact that *functional composition* can be presented without even mentioning category theory doesn't stop it from being one of category theory's *most important concepts*.
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At this point you might be worried that I had forgotten that I am supposed to talk about category theory and I am just presenting a bunch of irrelevant concepts. I may indeed do that sometimes, but not right now - the fact that *functional composition* can be presented without even mentioning category theory doesn't stop it from being one of category theory's *most important concepts*.
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In fact, we can say (although this is not an official definition) that category theory is the study of things that are *function-like* (we call them *morphisms*) --- ones that have source and target, and that can be composed with one another in an associative way.
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In fact, we can say (although this is not an official definition) that category theory is the study of things that are *function-like* (we call them *morphisms*) --- ones that have source and target, that can be composed with one another in an associative way, that can be represented by external diagrams etc.
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And there is another way of defining category theory without defining category theory: it is what you get if you replace the concept of equality with the concept of *isomorphism*. We haven't talked about isomorphisms yet, but this is what we will be doing till the end of this chapter.
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@ -370,7 +370,6 @@ The diagrams above, use colors to illustrate the fact that the green morphism is
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Diagrams that are like that (ones in which any two paths between two objects are equivalent to one another) are called *commutative diagrams* (or diagrams that *commute*). All diagrams in this book (except the wrong ones) commute.
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A summary
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---
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