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stuff
This commit is contained in:
Boris Marinov
parent
1bfe94d307
commit
ecb567bc4a
191
Gemfile.lock
191
Gemfile.lock
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@ -1,13 +1,13 @@
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GEM
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remote: https://rubygems.org/
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specs:
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||||
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@ -16,26 +16,44 @@ GEM
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@ -44,50 +62,51 @@ GEM
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jekyll-titles-from-headings (= 0.5.3)
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jemoji (= 0.12.0)
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kramdown (= 2.3.0)
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kramdown (= 2.3.1)
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kramdown-parser-gfm (= 1.1.0)
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liquid (= 4.0.3)
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mercenary (~> 0.3)
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minima (= 2.5.1)
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nokogiri (>= 1.10.4, < 2.0)
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rouge (= 3.23.0)
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rouge (= 3.26.0)
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terminal-table (~> 1.4)
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addressable (~> 2.3)
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typhoeus (~> 1.3)
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activesupport (>= 2)
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nokogiri (>= 1.4)
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i18n (0.9.5)
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concurrent-ruby (~> 1.0)
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jekyll (3.9.0)
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@ -124,6 +143,8 @@ GEM
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jekyll-github-metadata (2.13.0)
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@ -136,57 +157,57 @@ GEM
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jekyll (>= 3.3, < 5.0)
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@ -196,47 +217,43 @@ GEM
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gemoji (~> 3.0)
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html-pipeline (~> 2.2)
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jekyll (>= 3.0, < 5.0)
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rexml
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racc (1.6.0)
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rb-inotify (0.10.1)
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sawyer (0.8.2)
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addressable (>= 2.3.5)
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faraday (> 0.8, < 2.0)
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@ -251,9 +268,9 @@ GEM
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|||
thread_safe (~> 0.1)
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unf (0.1.4)
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unf_ext
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||||
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zeitwerk (2.4.2)
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unicode-display_width (1.8.0)
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zeitwerk (2.5.4)
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PLATFORMS
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ruby
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@ -312,6 +312,8 @@ Those two operations and their composite results in a group called $Dih3$ that i
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**Question:** Besides having two main actions, what is the defining factor that makes this and any other group non-abelian?
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Groups that represent the set of rotations and reflections of any 2D shape are called *dihedral groups*
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<!--
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TODO: FSM as monoids
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https://faculty.uml.edu/klevasseur/ads/s-monoid-of-fsm.html
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@ -433,10 +435,16 @@ And with $\mathrm{S}_3$ we are already feeling the power of exponential (and eve
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Each symmetric group $\mathrm{S}_n$ contains all groups of order $n$ - this is so, because (as we saw in the prev section) every group with $n$ elements is isomorphic to a set of permutations on the set of $n$ elements and the group $\mathrm{S}_n$ contains all such permutations. In particular, $\mathrm{S}_1$ is isomorphic to $Z_1$ (the *trivial group*), $\mathrm{S}_2$ is isomorphic to $Z_2$ (the *boolean group*). And the top three permutations of $\mathrm{S}_3$ are isomorphic to the group $Z_3$.
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Each symmetric group $\mathrm{S}_n$ contains all groups of order $n$ - this is so, because (as we saw in the prev section) every group with $n$ elements is isomorphic to a set of permutations on the set of $n$ elements and the group $\mathrm{S}_n$ contains *all such* permutations that exist.
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Here are some examples:
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- $\mathrm{S}_1$ is isomorphic to $Z_1$, the *trivial group*, and $\mathrm{S}_2$ is isomorphic to $Z_2$ , the *boolean group*, (but no other symmetric grops are isomorphic to cycle groups)
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- The top three permutations of $\mathrm{S}_3$ are isomorphic to the group $Z_3$.
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- $\mathrm{S}_3$ is also isomorphic to $Dih3$ (but no other symmetric group is isomorphic to a dihedral group)
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Based on this insight, can state Cayley's theorem in terms of symmetric groups in the following way:
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> All groups are isomorphic to subgroups of symmetric groups.
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@ -523,7 +531,9 @@ And then, if we start applying the two generators and follow the laws, we get th
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The set of generators and laws that defines a given group is called the *presentation of a group*. Every group has a presentation.
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Free monoids
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{% if site.distribution == 'print' %}
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Interlude: Free monoids
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---
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We saw how picking a different selection of laws gives rise to different types of monoid. But what monoids would we get if we pick no laws at all? These monoids (we get a different one depending on the set we picked) are called a *free monoids* (the word "free" is used in the sense that once you have the set, you can upgrade it to a monoid for free (i.e. without having to define anything else.)
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@ -551,3 +561,5 @@ We understand that, being the most general of all monoids for a given set of gen
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Plus, what about other monoids that also can have this property (that they are connected to any other monoids) Simple - they are the free monoid with some extra structure, so we can filter them out by using an universal property.
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But, what would this property be, (and what the hell are universal properties anyways?), tune in after a few chapters to find out.
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{%endif%}
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@ -352,6 +352,7 @@ $gᵃ gᵇ= gᵃ⁺ᵇ$
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$Task:$ Although it's trivial, we didn't prove that the first functor law (the one about the preservation of identities always holds. Interestingly enough, for groups/monoids it actually follows from the second law. Try to prove that. Start with the definition of the identity function.
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<!-- TODO show isomorphism theorems -->
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Functors in orders
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@ -418,6 +419,8 @@ There may be many more levels of categories from categories. However that does n
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We can use the same tactict with our little journey in category theory - stick on the level that make sense for us and not be obsessed with forming picture of the whole thing Because there is no *whole thing* - category theory is an *abstract* theory. That is, it does not seek to represent an actual state of affairs, but to provide a language that you can use to express many different ideas, actual or purely imaginary. So view the category of categories not as a structure, but as a space, where all these concepts live.
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{% if site.distribution == 'print' %}
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Free and forgetful functors
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===
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@ -524,4 +527,5 @@ Let's take another example. The free monoid with just one generator can be mappe
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There definitely is more to be said about free and forgetful functors, as well as for adjunctions (which are not only of free/forgetful type), but as always, I will leave things at the most interesting moment in order to revisit them from a different perspective later.
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{%endif%}
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@ -43,7 +43,7 @@ highlighter: rouge
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permalink: /:title/
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# The release of Jekyll Now that you're using
|
||||
version: v1.1.0
|
||||
version: v4.2.0
|
||||
|
||||
# Use the following plug-ins
|
||||
gems:
|
||||
|
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|||
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@ -77,8 +77,7 @@ window.MathJax = {
|
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|
||||
<h1>Support this project</h1>
|
||||
<p><a target="_blank" href="https://ko-fi.com/borismarinov">Ko-fi</a></p>
|
||||
|
||||
|
||||
<p><a target="_blank" href="https://patreon.com/borismarinov">Patreon</a></p>
|
||||
</div>
|
||||
</div>
|
||||
<div class="footer">
|
||||
|
|
|
|||
18
index.md
18
index.md
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@ -4,3 +4,21 @@ title: index
|
|||
---
|
||||
|
||||
[](00_about)
|
||||
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||||
|
||||
Category Theory Illustrated is a primer in category theory and various related concepts in "higher" mathematics that is *really* accessible to people with no prior exposure to the subject without being dumbed down, by utilizing visual explanations.
|
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|
||||
The book serves as chapter 0 going through the gist of the material covered by other similar introductory books, but doing so in a way that would enable non-mathematicians to swift through with ease.
|
||||
|
||||
Reading it would enable my readers to effortlessly go through any academic introduction to category theory, as well as to start tackling resources that use category theory as a tool to treat other subjects.
|
||||
|
||||
> "The range of applications for category theory is immense, and visually conveying meaning through illustration is an indispensable skill for organizational and technical work. Unfortunately, the foundations of category theory, despite much of their utility and simplicity being on par with Venn Diagrams, are locked behind resources that assume far too much academic background.
|
||||
>
|
||||
>Should category theory be considered for this academic purpose or any work wherein clear thinking and explanations are valued, beginner-appropriate resources are essential. There is no book on category theory that makes its abstractions so tangible as "Category Theory Illustrated" does. I recommend it for programmers, managers, organizers, designers, or anyone else who values the structure and clarity of information, processes, and relationships."
|
||||
|
||||
[Evan Burchard](https://www.oreilly.com/pub/au/7124), Author of "The Web Game Developer's Cookbook" and "Refactoring JavaScript"
|
||||
|
||||
|
||||
> "The clarity, consistency and elegance of diagrams in 'Category Theory Illustrated' has helped us demystify and explain in simple terms a topic often feared."
|
||||
|
||||
[Gonzalo Casas](https://gnz.io/), Software developer and lecturer for Introduction to Computational Methods for Digital Fabrication in Architecture at ETH Zurich
|
||||
|
|
|
|||
Loading…
Reference in New Issue
Block a user