Niles Johnson<p>New paper 🎉🎉<br />About coherence 🥱💤⁉️</p><p>joint work with Nick Gurski<br /><a href="https://arxiv.org/abs/2312.11261" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="">arxiv.org/abs/2312.11261</span><span class="invisible"></span></a></p><p>The title is:<br />Universal pseudomorphisms, [*deep breath*]<br />with applications to diagrammatic coherence for braided and symmetric monoidal functors 🙃😸</p><p>I've always thought coherence theorems sound boring, but actually they're good! In this paper we take a problem that is hard (coherence for structured functors), do a *bunch* of really abstract stuff (2-monad theory), and come out with a solution that makes your life* significantly better.</p><p>[*Here, "your life" means the part of your life you spend checking diagrams of braided monoidal functors. Or, more generally, pseudomorphisms for algebras over a 2-monad.]</p><p>Almost 1/5 of this paper is dedicated to real, genuine examples, and that's what I want to focus on below. I'll say just a bit about the more abstract machinery on which the examples are based. If you've been following along, this is the culmination of my series "weird facts about monoidal functors and coherence" [1,2,3,4].</p><p>[1] <a href="https://mathstodon.xyz/@nilesjohnson/110741323263984146" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">mathstodon.xyz/@nilesjohnson/1</span><span class="invisible">10741323263984146</span></a><br />[2] <a href="https://mathstodon.xyz/@nilesjohnson/110876487813747736" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">mathstodon.xyz/@nilesjohnson/1</span><span class="invisible">10876487813747736</span></a><br />[3] <a href="https://mathstodon.xyz/@nilesjohnson/110979458364785667" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">mathstodon.xyz/@nilesjohnson/1</span><span class="invisible">10979458364785667</span></a><br />[4] <a href="https://mathstodon.xyz/@nilesjohnson/111070640771166081" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">mathstodon.xyz/@nilesjohnson/1</span><span class="invisible">11070640771166081</span></a></p><p><a href="https://mathstodon.xyz/tags/CategoryTheory" class="mention hashtag" rel="tag">#<span>CategoryTheory</span></a> <a href="https://mathstodon.xyz/tags/MonoidalFunctor" class="mention hashtag" rel="tag">#<span>MonoidalFunctor</span></a> <a href="https://mathstodon.xyz/tags/Coherence" class="mention hashtag" rel="tag">#<span>Coherence</span></a> <a href="https://mathstodon.xyz/tags/Braided" class="mention hashtag" rel="tag">#<span>Braided</span></a> <a href="https://mathstodon.xyz/tags/Symmetric" class="mention hashtag" rel="tag">#<span>Symmetric</span></a> <a href="https://mathstodon.xyz/tags/PseudomorphismClassifier" class="mention hashtag" rel="tag">#<span>PseudomorphismClassifier</span></a></p><p>(1/14)</p>
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Niles Johnson<p>If you've been following my weird monoidal functor / coherence posts [1,2,3,4]... we are really getting close to finishing this project! It could be a matter of weeks. I'm excited because this project contains a cool blend of some mind-wringing 2-monad theory, followed by some (imo) genuinely useful applications to symmetric/braided monoidal functors, and then some real, detailed, actual examples. The doubling functor I mentioned a while back makes an appearance, along with (if we finally have it figured out, and we don't have to cut it) their even weirder friend, quadrupling!</p><p>We're working on getting the introduction and examples to be as clear as possible for readers who want to skip all of the not-entirely-easy middle part. I'm looking forward to saying more about it :)</p><p>[1] <a href="https://mathstodon.xyz/@nilesjohnson/110741323263984146" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">mathstodon.xyz/@nilesjohnson/1</span><span class="invisible">10741323263984146</span></a><br />[2] <a href="https://mathstodon.xyz/@nilesjohnson/110876487813747736" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">mathstodon.xyz/@nilesjohnson/1</span><span class="invisible">10876487813747736</span></a><br />[3] <a href="https://mathstodon.xyz/@nilesjohnson/110979458364785667" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">mathstodon.xyz/@nilesjohnson/1</span><span class="invisible">10979458364785667</span></a><br />[4] <a href="https://mathstodon.xyz/@nilesjohnson/111070640771166081" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">mathstodon.xyz/@nilesjohnson/1</span><span class="invisible">11070640771166081</span></a></p><p><a href="https://mathstodon.xyz/tags/CategoryTheory" class="mention hashtag" rel="tag">#<span>CategoryTheory</span></a> <br /><a href="https://mathstodon.xyz/tags/Monad" class="mention hashtag" rel="tag">#<span>Monad</span></a> <a href="https://mathstodon.xyz/tags/MonoidalFunctor" class="mention hashtag" rel="tag">#<span>MonoidalFunctor</span></a> <a href="https://mathstodon.xyz/tags/PseudomorphismClassifier" class="mention hashtag" rel="tag">#<span>PseudomorphismClassifier</span></a></p>
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