<story><title>Infinity Category Theory Offers a Bird’s-Eye View of Mathematics</title><url>https://www.scientificamerican.com/article/infinity-category-theory-offers-a-birds-eye-view-of-mathematics/</url></story><parent_chain><item><author>OscarCunningham</author><text>&amp;#x27;The category of algebras of a monad on Set is complete and cocomplete, and the forgetful functor preserves limits.&amp;#x27; gets you pretty far in a lot of subjects.</text></item><item><author>red_trumpet</author><text>&amp;gt;  We&amp;#x27;d spend most of the semester proving some trivial facts about morphisms and functors, just specialized to an unfamiliar category&lt;p&gt;Even if you know category theory, you still need to prove facts about the different categories you are dealing with. Just phrasing it in the language of category theory seldom suffices to prove something.</text></item><item><author>benrbray</author><text>I&amp;#x27;ve recently been studying category theory, mostly from Riehl&amp;#x27;s book, a few years after finishing my undergraduate math degree at a well-regarded school.  I can&amp;#x27;t help but feel a little cheated that not one of my professors thought it was important to show us a little category theory.&lt;p&gt;Learning category theory has brought clarity that I never felt as a student.  I always had this strange feeling that many of my courses wasted time proving &amp;quot;the same&amp;quot; facts over and over again.  From group theory, to linear algebra, to topology, to measure theory.  We&amp;#x27;d spend most of the semester proving some trivial facts about morphisms and functors, just specialized to an unfamiliar category, and by the time we were done, we only had time to prove one or two interesting theorems before the semester ended.&lt;p&gt;Back then, I had stumbled upon some Wikipedia pages about category theory, but all the jargon seemed impenetrable at the time.  Now, I really wish it had part of the curriculum from the very beginning.  Simply giving students in an abstract algebra class the simple definitions for category, functor, and natural transformation would go a long way.  Math programs should probably also start teaching Haskell or Coq in the first or second year.</text></item></parent_chain><comment><author>ogogmad</author><text>Doesn&amp;#x27;t the preservation of limits follow from the fact that the forgetful functor is right adjoint to the free functor? A left adjoint functor is always cocontinuous and a right adjoint functor is always continuous.&lt;p&gt;Looking at examples of &amp;quot;algebras of a monad on set&amp;quot;, I see groups and monoids. Does this explain why:&lt;p&gt;- The trivial group corresponds to a singleton set?&lt;p&gt;- The direct product of two groups G and H corresponds to their Cartesian products as sets? And so on to direct products of infinitely many groups?&lt;p&gt;- An equaliser of G under homs f and g has as its underlying set {x \in G : f(x) = g(x)}?&lt;p&gt;And so on, I see that the same must be true for any other algebraic structure which is a &amp;quot;category of algebras of a monad on Set&amp;quot;. Or have I misunderstood?&lt;p&gt;[edit]&lt;p&gt;I also see that because the free functor is cocontinuous (because it&amp;#x27;s left adjoint to the forgetful functor), the free group generated by the union of some sets is the same as the free product (&lt;i&gt;not&lt;/i&gt; direct sum) of their individual free groups (because the free product is a coproduct but the direct sum is not). If we want a correspondence with direct sum instead, we need to generate free abelian groups instead of free groups.</text></comment>
<story><title>Infinity Category Theory Offers a Bird’s-Eye View of Mathematics</title><url>https://www.scientificamerican.com/article/infinity-category-theory-offers-a-birds-eye-view-of-mathematics/</url></story><parent_chain><item><author>OscarCunningham</author><text>&amp;#x27;The category of algebras of a monad on Set is complete and cocomplete, and the forgetful functor preserves limits.&amp;#x27; gets you pretty far in a lot of subjects.</text></item><item><author>red_trumpet</author><text>&amp;gt;  We&amp;#x27;d spend most of the semester proving some trivial facts about morphisms and functors, just specialized to an unfamiliar category&lt;p&gt;Even if you know category theory, you still need to prove facts about the different categories you are dealing with. Just phrasing it in the language of category theory seldom suffices to prove something.</text></item><item><author>benrbray</author><text>I&amp;#x27;ve recently been studying category theory, mostly from Riehl&amp;#x27;s book, a few years after finishing my undergraduate math degree at a well-regarded school.  I can&amp;#x27;t help but feel a little cheated that not one of my professors thought it was important to show us a little category theory.&lt;p&gt;Learning category theory has brought clarity that I never felt as a student.  I always had this strange feeling that many of my courses wasted time proving &amp;quot;the same&amp;quot; facts over and over again.  From group theory, to linear algebra, to topology, to measure theory.  We&amp;#x27;d spend most of the semester proving some trivial facts about morphisms and functors, just specialized to an unfamiliar category, and by the time we were done, we only had time to prove one or two interesting theorems before the semester ended.&lt;p&gt;Back then, I had stumbled upon some Wikipedia pages about category theory, but all the jargon seemed impenetrable at the time.  Now, I really wish it had part of the curriculum from the very beginning.  Simply giving students in an abstract algebra class the simple definitions for category, functor, and natural transformation would go a long way.  Math programs should probably also start teaching Haskell or Coq in the first or second year.</text></item></parent_chain><comment><author>orbifold</author><text>Yeah no. There are whole fields completely untouched by such reasoning, like analysis.</text></comment>