Jekyll2019-05-04T20:30:57+00:00https://thosgood.github.io/feed.xmlTim Hosgoodmy websiteSkomer island2019-04-15T00:00:00+00:002019-04-15T00:00:00+00:00https://thosgood.github.io/life/2019/04/15/skomer-island<p>I haven’t posted anything in a while, and rather than trying to write about maths, I wanted to just share some lovely photos of Skomer island (which I recently visited).
I am even less knowledgeable about birds than I am about maths, but I do love them, and this was the first time in my life that I’d actually seen a puffin in the flesh!</p>
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<p>I am from North Devon, and just off the coast there is a small island called <a href="https://en.wikipedia.org/wiki/Lundy">Lundy</a>, which has a very interesting history, but is also famous for being the home to one of the few Atlantic puffin colonies in the British Isles.<sup id="fnref:1"><a href="#fn:1" class="footnote">1</a></sup>
In the last few decades, sadly, the number of breeding pairs of puffins has been in sharp decline, and when I went over there about 10 years ago I didn’t see a single one.
Thanks to lots of work by dedicated volunteers, it seems like the puffin population of Lundy is slowly on the rise again, and so hopefully the future will see the return of Lundy as a puffin sanctuary.</p>
<p>Just off the west coast of Wales, in Pembrokeshire, there is another island famous for its fauna: <a href="https://en.wikipedia.org/wiki/Skomer">Skomer</a>.
As well as housing about half of the world’s population of <a href="https://en.wikipedia.org/wiki/Manx_shearwater">Manx shearwaters</a><sup id="fnref:2"><a href="#fn:2" class="footnote">2</a></sup>, being the unique home of the <a href="https://en.wikipedia.org/wiki/Skomer_vole">Skomer vole</a>, and having numerous other species of birds, seals, and rabbits, it also has the largest puffin colony in southern Britain.</p>
<p>Puffins are extremely cute<sup id="fnref:3"><a href="#fn:3" class="footnote">3</a></sup> birds who mate for life, returning to the same burrows each year in order to find their partner, mate, and then leave (flying solo to the northern seas) for six or seven months, before coming back the next April.
To quote from the <a href="https://en.wikipedia.org/wiki/Atlantic_puffin">Wikipedia article</a>,</p>
<blockquote>
<p>Spending the autumn and winter in the open ocean of the cold northern seas, the Atlantic puffin returns to coastal areas at the start of the breeding season in late spring.
It nests in clifftop colonies, digging a burrow in which a single white egg is laid.
The chick mostly feeds on whole fish and grows rapidly.
After about 6 weeks, it is fully fledged and makes its way at night to the sea.
It swims away from the shore and does not return to land for several years.</p>
</blockquote>
<p>As of 2015, the Atlantic puffin is rated ‘vulnerable’ by the International Union for the Conservation of Nature, and was reported as being ‘threatened with extinction’ by BirdLife International in 2018.</p>
<p><a data-flickr-embed="true" href="https://www.flickr.com/photos/timhosgood/albums/72157677732207497" title="Skomer island"><img src="https://live.staticflickr.com/7852/40648359233_c37c6a3618_z.jpg" width="640" height="427" alt="Skomer island" /></a><script async="" src="//embedr.flickr.com/assets/client-code.js" charset="utf-8"></script></p>
<h3 id="footnotes">Footnotes</h3>
<div class="footnotes">
<ol>
<li id="fn:1">
<p>Indeed, it seems to be the case that the name <em>Lundy</em> comes from the Old Norse word for puffin (c.f. <a href="https://en.wikipedia.org/wiki/Lundey">Lundey</a>, off the coast of Reykjavík). <a href="#fnref:1" class="reversefootnote">↩</a></p>
</li>
<li id="fn:2">
<p>Interestingly, the Latin name for which is <em>Puffinus puffinus</em>. <a href="#fnref:2" class="reversefootnote">↩</a></p>
</li>
<li id="fn:3">
<p>No citation needed. <a href="#fnref:3" class="reversefootnote">↩</a></p>
</li>
</ol>
</div>I haven’t posted anything in a while, and rather than trying to write about maths, I wanted to just share some lovely photos of Skomer island (which I recently visited). I am even less knowledgeable about birds than I am about maths, but I do love them, and this was the first time in my life that I’d actually seen a puffin in the flesh!Twisting cochains and arbitrary dg-categories2018-12-12T00:00:00+00:002018-12-12T00:00:00+00:00https://thosgood.github.io/maths/2018/12/12/twisting-cochains-and-arbitrary-dg-categories<p>Having recently been thinking about twisting cochains (a major part of my thesis) a bit more, I think I better understand one reason why they are very useful (and why they were first introduced by Bondal and Kapranov), and that’s in ‘fixing’ a small quirk about dg-categories that I didn’t quite understand way back when I wrote <a href="/maths/2018/04/26/derived-dg-triangulated-and-infinity-categories.html">this post</a> about derived, dg-, and $A_\infty$-categories and their role in ‘homotopy things’.</p>
<p>This isn’t a long post and could probably instead be a tweet but then this blog would be a veritable ghost town.</p>
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<p>First of all, for the actual definitions of twisting/twisted cochains/complexes (the nomenclature varies wildly and seemingly inconsistently),<sup id="fnref:7"><a href="#fn:7" class="footnote">1</a></sup> I will shamelessly refer the interested reader to <a href="https://github.com/thosgood/papers/blob/master/twisted-complexes-summary/twisted-complexes-summary.pdf">some notes I wrote a while back</a>.</p>
<p>Secondly, the ‘quirk’ of dg-categories about which I’m talking<sup id="fnref:1"><a href="#fn:1" class="footnote">2</a></sup> is that, for a lot of people<sup id="fnref:2"><a href="#fn:2" class="footnote">3</a></sup>, it is the (pre-)triangulated structure that is interesting.
This means that (as far as I am aware)<sup id="fnref:3"><a href="#fn:3" class="footnote">4</a></sup> an arbitrary dg-category lacks some sort of homotopic interpretation because it has no structure corresponding to <em>stability</em> ‘upstairs’.
Twisting cochains then, as they were introduced by Bondal and Kapranov<sup id="fnref:4"><a href="#fn:4" class="footnote">5</a></sup>, are a sort of solution to this problem, in that (to quote from where else but the nLab) <em>“passing from a dg-category to its category of twisted complexes is a step towards enhancing it to a pretriangulated dg-category”</em>.<sup id="fnref:5"><a href="#fn:5" class="footnote">6</a></sup>
In essence, they give us the ‘smallest’ ‘bigger’ dg-category in which we have shifts and functorial cones.</p>
<p>Really I am just parroting back the reasons why these things were initially invented, but it’s something that I hadn’t fully appreciated, since I’ve been working with specific types of twisted complexes (ones that somehow correspond to projective/free things and concentrated in a single degree) that really arise in what appears (to me) to be a completely different manner: namely in the setting of (O’Brian), Toledo, and Tong<sup id="fnref:6"><a href="#fn:6" class="footnote">7</a></sup> where they are (to be vague) thought of as resolutions of coherent sheaves, or first-order perturbations of certain bicomplexes by flat connections.</p>
<p>I really have no geometric/homotopic intuition as to why this specific case of twisted complexes corresponds thusly, and haven’t been able to find any references at all.
Any ideas?</p>
<h3 id="footnotes">Footnotes</h3>
<div class="footnotes">
<ol>
<li id="fn:7">
<p>Although for me, at least, I (tend to) use <em>twisted complex</em> to refer to the concept of Bondal and Kapranov, and <em>twisting cochain</em> to refer to the concept of (O’Brian), Toledo, and Tong. <a href="#fnref:7" class="reversefootnote">↩</a></p>
</li>
<li id="fn:1">
<p>Having hidden this part in the main post and not the excerpt makes me feel like I’m writing the mathematical equivalent of click-bait journalism. Next will come posts with titles such as <em>“Nine functors that you wouldn’t believe have derived counterparts — number six will shock you!”</em> and <em>“You Will Laugh And Then Cry When You See What This Child Did With The Grothendieck Construction”</em>. I apologise in advance. <a href="#fnref:1" class="reversefootnote">↩</a></p>
</li>
<li id="fn:2">
<p>[weasel words] [citation needed] <a href="#fnref:2" class="reversefootnote">↩</a></p>
</li>
<li id="fn:3">
<p>which is, admittedly, best measured on the Planck scale. <a href="#fnref:3" class="reversefootnote">↩</a></p>
</li>
<li id="fn:4">
<p>A. I. Bondal, M. M. Kapranov, “Enhanced triangulated categories”, Mat. Sb., 181:5 (1990), 669–683; Math. USSR-Sb., 70:1 (1991), 93–107. <a href="#fnref:4" class="reversefootnote">↩</a></p>
</li>
<li id="fn:5">
<p><a href="https://ncatlab.org/nlab/show/twisted+complex">https://ncatlab.org/nlab/show/twisted+complex</a> <a href="#fnref:5" class="reversefootnote">↩</a></p>
</li>
<li id="fn:6">
<p>A bunch of papers, but in particular e.g. D. Toledo and Y. L. L. Tong, “Duality and Intersection Theory in Complex Manifolds. I.”, Math. Ann., 237 (1978), 41—77. <a href="#fnref:6" class="reversefootnote">↩</a></p>
</li>
</ol>
</div>Having recently been thinking about twisting cochains (a major part of my thesis) a bit more, I think I better understand one reason why they are very useful (and why they were first introduced by Bondal and Kapranov), and that’s in ‘fixing’ a small quirk about dg-categories that I didn’t quite understand way back when I wrote this post about derived, dg-, and $A_\infty$-categories and their role in ‘homotopy things’. This isn’t a long post and could probably instead be a tweet but then this blog would be a veritable ghost town.Torsors and principal bundles2018-10-31T00:00:00+00:002018-10-31T00:00:00+00:00https://thosgood.github.io/maths/2018/10/31/torsors-and-principal-bundles<p>In my thesis, switching between vector bundles and principal $\mathrm{GL}_r$-bundles has often made certain problems easier (or harder) to understand.
Due to my innate fear of all things differentially geometric, I often prefer working with principal bundles, and since reading Stephen Sontz’s (absolutely fantastic) book <a href="https://www.springer.com/fr/book/9783319147642">Principal Bundles — The Classical Case</a>, I’ve really grown quite fond of bundles, especially when you start talking about all the lovely $\mathbb{B}G$ and $\mathbb{E}G$ things therein<sup id="fnref:1"><a href="#fn:1" class="footnote">1</a></sup><sup id="fnref:3"><a href="#fn:3" class="footnote">2</a></sup>.
Point is, I haven’t posted anything in forever, and one of my supervisor’s strong pedagogical beliefs is that <em>‘affine vector spaces should be understood as $G$-torsors, where $G$ is the underlying vector space acting via translation’</em>,<sup id="fnref:4"><a href="#fn:4" class="footnote">3</a></sup> which makes a nice short topic of discussion, whence this post.</p>
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<p>We briefly<sup id="fnref:2"><a href="#fn:2" class="footnote">4</a></sup> recall the definition of a principal $G$-bundle over a space $X$, where $G$ is some <em>topological</em> group.</p>
<p><strong>Definition.</strong> A <em>principal $G$-bundle over $X$</em> is a fibre bundle $P\xrightarrow{\pi}X$ with a continuous right action $P\times G\to P$ such that</p>
<ol>
<li>$G$ acts <em>freely</em>;</li>
<li>$G$ acts <em>transitively</em> on the orbits; and</li>
<li>$G$ acts <em>properly</em>.</li>
</ol>
<p>It is maybe helpful to think of the following ‘dictionary’:</p>
<ul>
<li>free = injective (i.e. $\exists x$ s.t. $gx=hx\implies g=h$)</li>
<li>transitively = surjective (i.e. $\forall x,y$ $\exists g$ s.t. $gx=y$)</li>
<li>properly = something that you care about if you care about infinite sequences or Hausdorffness or things like that (i.e. the inverse image of $G\times X\to X\times X$ given by $(g,x)\mapsto(gx,x)$ preserves compactness)</li>
</ul>
<p>Thus the fibres $F$ are homeomorphic to $G$, and also give the orbits, and the orbit space $P/G$ is homeomorphic to $X$.</p>
<p>Another definition is now useful.</p>
<p><strong>Definition.</strong> A <em>$G$-torsor</em> is a space upon which $G$ acts <em>freely</em> and <em>transitively</em>.</p>
<p><strong>Motto.</strong> $G$-torsors <em>are</em> principal $G$-bundles over a point <em>are</em> affine versions of $G$.</p>
<p>What do we mean by this last ‘equivalence’?
Just that $G$-torsors retain all the structure of $G$, but don’t have some specified point that acts as the identity.
Here are some nice examples.</p>
<ul>
<li>$\mathrm{GL}_r$-torsors are vector spaces; $\mathrm{GL}_r$-bundles are vector bundles.</li>
<li>$O(r)$-torsors are vector spaces with an inner product.</li>
<li>$\mathrm{GL}_r^+$-torsors are oriented vector spaces (where $\mathrm{GL}_r^+$ is the connected component of $\mathrm{GL}_r$ consisting of matrices with determinant strictly positive).</li>
<li>$\mathrm{SL}_r$-torsors are vector spaces with a specified isomorphism $\det V\xrightarrow{\sim} k$, where $\det V:=\wedge_{i=1}^r V$, and $k$ is our base field. Note that this is weaker than a choice of basis: it is a choice of an $\mathrm{SL}_r$-conjugacy class of bases.</li>
</ul>
<hr />
<h1 id="footnotes">Footnotes</h1>
<div class="footnotes">
<ol>
<li id="fn:1">
<p>About which I recently had a nice little Twitter conversation with <a href="https://twitter.com/johncarlosbaez">John Baez</a>; his replies starting <a href="https://twitter.com/johncarlosbaez/status/1056999200125157376">here</a> are really quite nice. P.S. if you are not on Twitter then I would highly recommend it: the maths community is really friendly and interesting, and if you have a little question to ask then chances are you’ll get a bunch of nice responses. Also a chance to talk to people across the globe in a completely different time zone. Don’t get me wrong, Twitter has <em>many</em> problems, but you can ignore most of them and just follow the people that you like. <a href="#fnref:1" class="reversefootnote">↩</a></p>
</li>
<li id="fn:3">
<p>(which I won’t talk about here because (a) I think there are many other places to read about this that are much better than something that I could write; and (b) I should be working on my thesis but I’m sort of using this post as a method of procrastination/searching for inspiration). <a href="#fnref:3" class="reversefootnote">↩</a></p>
</li>
<li id="fn:4">
<p>An earlier version of this post incorrectly said ‘$\mathrm{GL}_r$-torsor’; thanks to <a href="https://twitter.com/BarbaraFantechi/status/1057701336291123200">Barbara Fantechi for pointing this out!</a> <a href="#fnref:4" class="reversefootnote">↩</a></p>
</li>
<li id="fn:2">
<p>In particular we really sort of assume that the reader already knows what one of these is and are just writing this for some mild effort towards self-containedness. (Bonus question for anybody actually reading this: what is the word/phrase I can’t think of that means ‘self-containedness’ but is actually a real word/phrase?) <a href="#fnref:2" class="reversefootnote">↩</a></p>
</li>
</ol>
</div>I haven't posted anything in forever, and one of my supervisor's strong pedagogical beliefs is that 'affine vector spaces should be understood as $G$-torsors, where $G$ is the underlying vector space acting via translation', which makes a nice short topic of discussion, whence this post.Localisation and model categories2018-08-25T00:00:00+00:002018-08-25T00:00:00+00:00https://thosgood.github.io/maths/2018/08/25/localisation-and-model-categories-part-1<p>After some exceptionally enlightening discussions with Eduard Balzin recently, I’ve made some notes on the links between model categories, homotopy categories, and localisation, and how they all tie in together.
There’s nothing particularly riveting or original here, but hopefully these notes can help somebody else who was lost in this mire of ideas.</p>
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<p><em>Notational note:</em> we write $\mathcal{C}(x,y)$ instead of $\mathrm{Hom}_\mathcal{C}(x,y)$.</p>
<h3 id="localisation-of-categories">Localisation of categories</h3>
<p>Let $(\mathcal{C},\mathcal{W})$ be a pair, with $\mathcal{C}$ a category and $\mathcal{W}$ a wide subcategory (that is, a subcategory containing all the objects of $\mathcal{C}$, or, equivalently, a set of morphisms in $\mathcal{C}$).
This data is known as a <em>relative category</em>, which is a weaker version of a category with weak equivalences, or a homotopical category, or other such notions.</p>
<p>Often we want to <em>localise</em> $\mathcal{C}$ along $\mathcal{W}$, i.e. ‘formally invert all morphisms in $\mathcal{W}$’.
A nice way of making this rigorous is by defining the localisation $\mathcal{C}[\mathcal{W}^{-1}]$ (also written $\operatorname{Ho}(\mathcal{C})$ or $W^{-1}\mathcal{C}$)<sup id="fnref:3"><a href="#fn:3" class="footnote">1</a></sup> by a universal property:<sup id="fnref:4"><a href="#fn:4" class="footnote">2</a></sup></p>
<script type="math/tex; mode=display">% <![CDATA[
\begin{array}{lcr}
\mathcal{C} & \xrightarrow{\mathcal{W}\,\mapsto\,\text{iso}_\mathcal{D}} & \mathcal{D}\\
\quad\searrow & & \nearrow_{\exists!}\\
& \mathcal{C}[\mathcal{W}^{-1}] &
\end{array} %]]></script>
<p>That is, any category (along with a functor into it) such that all morphisms in $\mathcal{W}$ become isomorphisms <em>must</em> factor <em>uniquely</em> through $\mathcal{C}[\mathcal{W}^{-1}]$.</p>
<p>Since our definition is in terms of a universal property, <strong>if</strong> the localisation of a category exists then it is unique.</p>
<h4 id="gabriel-zisman">Gabriel-Zisman</h4>
<p>There is a reasonably concrete way of constructing the localisation that is called <em>Gabriel-Zisman</em> (or sometimes <em>zigzag</em>) <em>localisation</em>.
It has a few issues, which we discuss below, after giving a definition.
This is the localisation that most people will first study in the case of constructing the derived category of complexes, or some other such example, in a course on homological algebra or algebraic geometry.</p>
<p>We define the objects of $\mathcal{C}[\mathcal{W}^{-1}]$ to be those of $\mathcal{C}$, and the morphisms to be <em>zigzags</em> of morphisms: a morphism $x\to y$ is given by a directed graph whose vertices are objects of $\mathcal{C}$, and whose edges are labelled by arrows in $\operatorname{Arr}(\mathcal{C})\sqcup\operatorname{Arr}(\mathcal{W}^\text{op})$, <strong>modulo certain equivalence relations</strong>.<sup id="fnref:1"><a href="#fn:1" class="footnote">3</a></sup>
That is, a morphism from $x=a_0$ to $y=a_{n+1}$ is given by a string of objects $a_1,\ldots,a_n\in\mathcal{C}$ with maps between them: either a map $a_i\to a_{i+1}$ in $\mathcal{C}$, or a map $a_i\leftarrow a_{i+1}$ in $\mathcal{W}$.</p>
<p>Note that, if $\mathcal{W}$ contains all identity maps (for example), then we can always insert identity maps in our zigzags to ensure that they are always of the form $f_1g_1\ldots f_ng_n$ with $f_i\in\operatorname{Arr}(\mathcal{C})$ and $g_i\in\operatorname{Arr}(\mathcal{W}^\text{op})$.</p>
<p>As you can see, arbitrary morphisms in this category can be unreasonably large (in terms of the data describing them), and so we might hope that, by placing some conditions on $\mathcal{W}$, we can globally bound the length of the zigzags.
If fact, if $\mathcal{W}$ is a <em><a href="https://ncatlab.org/nlab/show/calculus+of+fractions#definition">calculus of fractions</a></em> then we can show that all the zigzags are actually just (co)roofs (depending on the handedness of the calculus of fractions):</p>
<script type="math/tex; mode=display">x\to a\xleftarrow{\small\mathcal{W}} y \quad\text{or}\quad x\xleftarrow{\small\mathcal{W}}a\to y.</script>
<p>Note that we <strong>still</strong> have an equivalence relation: two morphisms $x\xleftarrow{\mathcal{W}}a\to y$ and $x\xleftarrow{\mathcal{W}}b\to y$ are equivalent if there exists some roof $a\xleftarrow{\mathcal{W}}e\to b$ such that ‘everything commutes’.<sup id="fnref:2"><a href="#fn:2" class="footnote">4</a></sup></p>
<p>One potential problem with this construction (depending on how much you care about these things) is that the localisation might live only in some bigger universe, and so you have to start worrying about that.</p>
<h4 id="dwyer-kan">Dwyer-Kan</h4>
<p>Of course, just constructing a category is not usually enough these days, and we instead want to give it some higher structure.
Enter <em>Dwyer-Kan</em> (or <em>simplicial</em>) localisation.</p>
<p>This is a way of constructing an $(\infty,1)$-category $L_\mathcal{W}\mathcal{C}$, realised as a <em>simplicial category</em>.
We talk more about simplicial categories later on, but first we quote Julia E. Bergner from <a href="https://arxiv.org/abs/math/0406507">“A model category structure on the category of simplicial categories”</a>:</p>
<p><em>Note that the term “simplicial category” is potentially confusing. As we have already stated, by a simplicial category we mean a category enriched over simplicial sets.</em>
<em>If $a$ and $b$ are objects in a simplicial category $\mathcal{C}$, then we denote by $\mathrm{Hom}_\mathcal{C}(a,b)$ the function complex, or simplicial set of maps $a\to b$ in $\mathcal{C}$.</em>
<em>This notion is more restrictive than that of a simplicial object in the category of categories.</em>
<em>Using our definition, a simplicial category is essentially a simplicial object in the category of categories which satisfies the additional condition that all the simplicial operators induce the identity map on the objects of the categories involved.</em></p>
<p>First of all, note that we now require that $(\mathcal{C},\mathcal{W})$ be a <em>category with weak equivalences</em>: all isomorphisms are in $\mathcal{W}$, and if any two of ${f,g,g\circ f}$ are in $\mathcal{W}$ then so too is the third.
For example, any model category or homotopical category is automatically a category with weak equivalences.</p>
<p>Now then, the definition by universal property is (modulo some technical $\infty$-details) what you would expect: $L_\mathcal{W}\mathcal{C}$ is an $(\infty,1)$-category such that $\mathcal{C}$ injects into $L_\mathcal{W}\mathcal{C}$ with every morphism in $\mathcal{W}$ becoming an equivalence (in the $(\infty,1)$-sense) in $L_\mathcal{W}\mathcal{C}$, and such that any other such $(\infty,1)$-category factors ‘uniquely’ through this.</p>
<p>One such way of constructing this localisation is by <em>hammock localisation</em>.
For any $x,y\in\mathcal{C}$ we define their $\mathrm{Hom}$ as the simplicial set $L^\mathrm{H}(x,y)$ given by</p>
<script type="math/tex; mode=display">L^\mathrm{H}(x,y) := \coprod_{n\in\mathbb{N}}\mathcal{N}(\operatorname{H}_n(x,y))/\sim</script>
<p>where $\mathcal{N}$ is the nerve (which sends a category to a simplicial set), and both the categories $\operatorname{H}_n(x,y)$ and the equivalence relation $\sim$ remain to be defined.</p>
<p>For each $n\in\mathbb{N}$ the category $\operatorname{H}_n(x,y)$ has objects being length-$n$ zigzags, as in Gabriel-Zisman localisation<sup id="fnref:5"><a href="#fn:5" class="footnote">5</a></sup>, and the morphisms are ‘hammocks’</p>
<script type="math/tex; mode=display">% <![CDATA[
\begin{array}{ccccccccc}
&&a_1&\to&a_2\xleftarrow{\small\mathcal{W}}&\ldots&a_n&\\
&^{\small\mathcal{W}}\swarrow&&&&&&\searrow\\
x&&\downarrow_{\small\mathcal{W}}&&\downarrow_{\small\mathcal{W}}&\ldots&\downarrow_{\small\mathcal{W}}&&y\\
&_{\small\mathcal{W}}\nwarrow&&&&&&\nearrow\\
&&b_1&\to&b_2\xleftarrow{\small\mathcal{W}}&\ldots&b_n&
\end{array} %]]></script>
<p>i.e. commutative diagrams of zigzags, where the ‘linking’ arrows are all in $\mathcal{W}$.
The equivalence relations are the ‘natural’ ones: we can insert or remove identity maps, and compose any composable morphisms.</p>
<h4 id="comparison">Comparison</h4>
<p>Now then, we can ask how this ‘new’ localisation is related to the ‘old’ one, and we can answer this question with the following lemma.</p>
<p><strong>Lemma.</strong> $\pi_0(L_\mathcal{W}\mathcal{C}(x,y))\simeq\mathcal{C}[\mathcal{W}^{-1}]$.</p>
<p>For this post, that’s it, but my next post will talk about how we can extend these ideas to localise <em>quasi-categories</em>, and how the Bergner model structure on simplicial categories comes into the story.
This will, in particular, let us formalise the fact that taking the homotopy category of a category (whenever this makes sense, e.g. for quasi-categories) is somehow equivalent to localising the category along weak equivalences.
The lemma that we’ll look at is the following (where we’ve yet to define the right-hand side).</p>
<p><strong>Lemma.</strong> $\mathcal{C}[\mathcal{W}^{-1}]\simeq\mathrm{h}L\mathcal{C}$.</p>
<h3 id="references">References</h3>
<ul>
<li>Julia E. Bergner, “A model category structure on the category of simplicial categories”, <a href="https://arxiv.org/abs/math/0406507">arXiv:math/0406507</a>.</li>
<li>V. Hinich, “Dwyer-Kan localization revisited”, <a href="https://arxiv.org/abs/1311.4128">arXiv:1311.4128</a>.</li>
<li>W.G. Dwyer and D.M. Kan, “Calculating simplicial localizations”, <a href="https://www3.nd.edu/~wgd/Dvi/CalculatingSimplicialLocalizations.pdf"><em>available online</em></a>.</li>
<li>Pierre Gabriel, Michel Zisman, “Calculus of Fractions and Homotopy Theory”, <a href="http://web.math.rochester.edu/people/faculty/doug/otherpapers/GZ.pdf"><em>available online</em></a>.</li>
</ul>
<hr />
<h1 id="footnotes">Footnotes</h1>
<div class="footnotes">
<ol>
<li id="fn:3">
<p>There are so many things that ‘homotopy category’ or ‘$\operatorname{Ho}(\mathcal{C})$’ or ‘$\operatorname{h}(\mathcal{C})$’ can mean, so the context is always very important <a href="#fnref:3" class="reversefootnote">↩</a></p>
</li>
<li id="fn:4">
<p>This diagram is horribly formatted. I am lost without <code class="highlighter-rouge">tikz-cd</code>. <a href="#fnref:4" class="reversefootnote">↩</a></p>
</li>
<li id="fn:1">
<p>These are just to ensure that composition and the identity morphism behave as expected. See <a href="https://ncatlab.org/nlab/show/localization#general_construction">the nLab</a> for details. <a href="#fnref:1" class="reversefootnote">↩</a></p>
</li>
<li id="fn:2">
<p>I think of the diagram you want to show commutes as a tiny house of cards, two layers high. <a href="#fnref:2" class="reversefootnote">↩</a></p>
</li>
<li id="fn:5">
<p>But, recalling what we said there, since $\mathcal{W}$ contains all isomorphisms then we can assume that our zigzags always alternate between arrows in $\mathcal{C}$ and arrows in $\mathcal{W}^\text{op}$. <a href="#fnref:5" class="reversefootnote">↩</a></p>
</li>
</ol>
</div>After some exceptionally enlightening discussions with Eduard Balzin recently, I’ve made some notes on the links between model categories, homotopy categories, and localisation, and how they all tie in together. There’s nothing particularly riveting or original here, but hopefully these notes can help somebody else who was lost in this mire of ideas.Categorication of the Dold-Kan correspondence2018-08-16T00:00:00+00:002018-08-16T00:00:00+00:00https://thosgood.github.io/maths/2018/08/16/categorication-of-the-dold-kan-correspondence<p>So I’m currently at the Max Planck Institute for Mathematics in Bonn, Germany, for a conference on ‘Higher algebra and mathematical physics’.
Lots of the talks have gone entirely above my head (reminding me how far behind my physics education has fallen), but have still been very interesting.</p>
<!--more-->
<p>There was a talk yesterday on <a href="http://www.mpim-bonn.mpg.de/node/8635">$\mathbb{F}_1$ things</a> by Matilde Marcolli, and on Tuesday a talk by Bertrand Toën on <a href="http://www.mpim-bonn.mpg.de/node/8633">moduli spaces of connections on open varieties</a> as well as one by Damien Calaque (my co-supervisor) on <a href="http://www.mpim-bonn.mpg.de/node/8617">$\mathbb{E}_n$-algebras and vertex models</a>, all of which I managed to follow at least partially (take what you can get, I guess).
Today, however, was a particularly interesting talk by Tobias Dyckerhoff on a <a href="http://www.mpim-bonn.mpg.de/node/8648">categorified Dold-Kan correspondence</a>.
I don’t really want to talk about the details (because I’m not at all qualified to do so, and you’d be better served by going directly to the source), but something that I really enjoyed was a ‘dictionary’ that he presented.</p>
<p>Historically, the first step towards ‘groupification’ was the idea of enriching Betti numbers to abelian groups, which gave birth to what we now know as homology — the Betti numbers are just the ranks of the groups.
Similarly, we now have the language to be able to ask that our abelian groups can actually be replaced by stable $\infty$-categories: we can turn homological algebra into <em>categorified homological algebra</em>, and, taking inspiration from Serge Lang’s famous ‘exercise’ in homological algebra, we can pick up any textbook on homological algebra and try to categorify (and then prove) all the theorems.
To do so, we need to know what the classical constructions become in this higher-category language, whence the dictionary.</p>
<table class="bordered-table">
<thead>
<tr>
<th>classical</th>
<th>categorified</th>
</tr>
</thead>
<tbody>
<tr>
<td>abelian group $A$</td>
<td>stable $\infty$-category $\mathscr{A}$</td>
</tr>
<tr>
<td>$x\in A$</td>
<td>$X\in\mathscr{A}$</td>
</tr>
<tr>
<td>$y-x\in A$</td>
<td>$\operatorname{cone}(X\xrightarrow{f}Y)\in\mathscr{A}$</td>
</tr>
<tr>
<td>$\sum_{i=0}^n(-1)^i x_i\in A$</td>
<td>$\operatorname{Tot}(X_0\xrightarrow{f_0}\ldots\xrightarrow{f_{n-1}}X_n)\in\mathscr{A}$</td>
</tr>
<tr>
<td>$C\cong A\oplus B$</td>
<td>$\mathscr{C}\simeq\langle\mathscr{A},\mathscr{B}\rangle$</td>
</tr>
</tbody>
</table>
<p>Note that the difference of two elements $X,Y$ in the categorified language depends on the choice of some morphism between them, so there are lots of ‘values’ of “$X-Y$”, one for each $f\colon X\to Y$.
The last line (corresponding to a direct sum) is a semi-orthogonal decomposition.
Finally, we need to define what $\mathrm{Tot}$ is.</p>
<p>For $n=1$, we have already defined that $\mathrm{Tot}$ is given by $\mathrm{cone}$.
For $n=2$, we draw some punctured (i.e. missing one vertex) cube (see below) and take the colimit (which, thanks to stability, makes the cube actually _bi_cartesian).
For $n=3$ we embed the above cube as the face of a 4-dimensional cube, and take a colimit, etc. etc.</p>
<p><img src="/assets/post-images/2018-08-16-categorication-of-the-dold-kan-correspondence-cube.jpg" alt="Defining Tot for length 3 complexes." class="img-responsive" /></p>So I’m currently at the Max Planck Institute for Mathematics in Bonn, Germany, for a conference on ‘Higher algebra and mathematical physics’. Lots of the talks have gone entirely above my head (reminding me how far behind my physics education has fallen), but have still been very interesting.Nothing really that new2018-07-08T00:00:00+00:002018-07-08T00:00:00+00:00https://thosgood.github.io/maths/2018/07/08/nothing-really-that-new<p>Just a small post to point out that I’ve <a href="https://thosgood.github.io/papers/">uploaded some new notes</a>, including some I took at the Derived Algebraic Geometry in Toulouse (DAGIT) conference last year.
I’ve been hard at work on thesis things, so haven’t been able to write up all the blog stuff that I’ve wanted to, but hopefully will get a chance sometime in the near future.</p>
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<p>To give some mathematical content to this post, here are a few of the questions on math.stackexchange/mathoverflow that I’ve come back to read time and time again, because they just float my boat:</p>
<ul>
<li><a href="https://math.stackexchange.com/questions/38517/in-relatively-simple-words-what-is-an-inverse-limit/38522#38522">In (relatively) simple words: What is an inverse limit?</a></li>
<li><a href="https://math.stackexchange.com/questions/846217/prove-if-a-b-in-g-commute-with-probability-5-8-then-g-is-abelian">Prove: if a,b in G commute with probability > 5/8, then G is abelian</a> and <a href="https://mathoverflow.net/questions/91685/5-8-bound-in-group-theory">5/8 bound in group theory</a>.</li>
<li><a href="https://math.stackexchange.com/questions/782448/optimal-strategy-for-jackpot-rock-paper-scissors">Optimal strategy for Jackpot Rock Paper Scissors</a>.</li>
<li><a href="https://math.stackexchange.com/questions/2118796/how-to-choose-the-smallest-number-not-chosen">How to choose the smallest number not chosen?</a></li>
<li><a href="https://math.stackexchange.com/questions/2647300/riddles-that-can-be-solved-by-meta-assumptions">Riddles that can be solved by meta-assumptions</a>.</li>
<li><a href="https://math.stackexchange.com/questions/446130/quickest-way-to-determine-a-polynomial-with-positive-integer-coefficients">Quickest way to determine a polynomial with positive integer coefficients</a>.</li>
</ul>
<p>(In case you can’t tell, I am fascinated by simple game theory and ‘meta-riddles’.)</p>Just a small post to point out that I’ve uploaded some new notes, including some I took at the Derived Algebraic Geometry in Toulouse (DAGIT) conference last year. I’ve been hard at work on thesis things, so haven’t been able to write up all the blog stuff that I’ve wanted to, but hopefully will get a chance sometime in the near future.Derived, DG, triangulated, and infinity-categories2018-04-26T00:00:00+00:002018-04-26T00:00:00+00:00https://thosgood.github.io/maths/2018/04/26/derived-dg-triangulated-and-infinity-categories<p>This post assumes that you have seen the construction of derived categories and maybe the definitions of dg- and $A_\infty$-categories, and wondered how they all linked together.
In particular, as an undergraduate I was always confused as to what the difference was between the two steps of constructing the derived category of chain complexes was: taking equivalence classes of chain homotopic complexes; and then formally inverting all quasi-isomorphisms.
Both of them seemed to be some sort of quotienting/equivalence-class-like action, so why not do them at the same time?
What different roles were played by each step?</p>
<!--more-->
<p><em>This post could have many errors, and I don’t understand the proofs for many of the things I vaguely imply; I just mildly believe.</em>
<em>Please do let me know if there is anything grossly misleading or just plain wrong.</em></p>
<h1 id="warnings">Warnings</h1>
<p>First of all, there are two points that I want to make about choices of language that I think are really very confusing.</p>
<ol>
<li>When we talk about derived categories, we are using this word in the opposite sense to pretty much every other usage of the word in modern mathematics<sup id="fnref:1"><a href="#fn:1" class="footnote">1</a></sup>: a <strong>derived category</strong> is like a homotopy truncation (i.e. the $\pi_0$) of some thing with much higher homotopical data; a <strong>derived scheme/stack/whatever</strong> is something whose $\pi_0$ is the corresponding classical object.
Derived ‘algebraic/geometric objects’ <em>have</em> homotopy truncations; derived categories <em>are</em> homotopy truncations.</li>
<li>When we quotient the category of chain complexes by the equivalence relation given by chain homotopy, we usually call the resulting category $K(\mathcal{A})$ the <strong>homotopy category of chain complexes</strong>.
This is maybe not the best nomenclature, in some sense, because the category that ‘truly’ deserves this name is the <em>actual</em> homotopy category of chain complexes $\mathrm{Ho}\mathsf{Ch}(\mathcal{A})$, which is, by definition, the derived category $D(\mathcal{A})$.
Because of this, I won’t refer to $K(\mathcal{A})$ as anything but $K(\mathcal{A})$.</li>
</ol>
<h1 id="a-nice-diagram">A nice diagram</h1>
<p>Without further ado, here is a diagram that attempts to explain the relationships between various things that you might have learnt about in various places.</p>
<p><img src="/assets/post-images/2018-04-26-derived-dg-triangulated-and-infinity-categories-1.png" alt="How things all sort of fit together" title="How things all sort of fit together" /></p>
<p>Here are a few words about the above diagram.</p>
<ul>
<li>The structure of a <em>triangulated category</em> is essentially the structure that the category $\mathrm{Ho}(\mathcal{C})$ inherits from the fact that $\mathcal{C}$ is a <em>stable</em> $(\infty,1)$-category.
The equivalent notion in the more general case (the left-hand side of the diagram) is the structure inherent to $\mathrm{h}(\mathcal{C})$ from an $(\infty,1)$-category viewed as a quasi-category (and thus, in particular, simplicial set).</li>
<li>A <em>stable dg-category</em> is also known as an <em>enhanced triangulated category</em> or a <em>pretriangulated category</em>.</li>
<li>If we wanted to draw a column in the middle for <em>stable</em> $(\infty,1)$-categories that <em>aren’t necessarily linear</em>, then we’d arrive at the notion of a <em>spectral category</em> instead of a dg-category.</li>
</ul>
<p>Now here is a picture of where the derived category of chain complexes fits in to all this.</p>
<p><img src="/assets/post-images/2018-04-26-derived-dg-triangulated-and-infinity-categories-2.png" alt="Where does the derived category fit in?" title="Where does the derived category fit in?" /></p>
<p>Here are a few words about the above diagram.</p>
<ul>
<li>We can understand the definition of <em>chain homotopies</em> much better if we understand the idea of the <a href="https://ncatlab.org/nlab/show/interval+object+in+chain+complexes">interval object in chain complexes</a>.</li>
<li>
<p>As said at the beginning of this post, there <em>is</em> a difference between passing from $\mathsf{Ch}(\mathcal{A})$ to $K(\mathcal{A})$ and from $K(\mathcal{A})$ to $D(\mathcal{A})$, i.e. quotienting and localising have no reason to behave similarly<sup id="fnref:2"><a href="#fn:2" class="footnote">2</a></sup>.
<em>But</em>, there is a model structure that we can put on $\mathsf{Ch}(\mathcal{A})$, and so there <em>is</em> some link between the two: to quote from <a href="https://mathoverflow.net/a/188199/73622">a mathoverflow answer by Simon Henry</a>,
<em>“The reason why they give the same things in a lot of example (including chain complexes), giving the idea that they should be related, is because these examples are Quillen model categories and that it is the main result of Quillen’s “Homotopical Algebra” (where he defined model categories) that for Quillen model category the localization by weak equialence can be constructed as a quotient of the full subcategory of fibrant-cofibrant objects.”</em></p>
<p>There is, however, the slightly confusing fact that the construction of $K(\mathcal{A})$ as a quotient <em>does</em> agree with a localisation construction (as explained <a href="https://math.stackexchange.com/a/1128937/71510">here</a>).</p>
</li>
</ul>
<h1 id="unresolved-confusions">Unresolved confusions</h1>
<p>I still don’t fully see the link between the homotopy category $\mathrm{h}(\mathcal{C})$ and the homotopy category $\mathrm{Ho}(\mathcal{C})$ and localisation in general (e.g. I ‘know’ that $\mathrm{Ho}(\mathcal{C})(a,b)\simeq\pi_0(\mathrm{L}\mathcal{C}(a,b))$, where $\mathrm{L}$ is the simplicial localisation, but that’s about it).</p>
<p>You know what?
There are a lot of things that I still don’t understand, but oh my goodness am I having fun trying to figure it all out.</p>
<hr />
<h1 id="footnotes">Footnotes</h1>
<div class="footnotes">
<ol>
<li id="fn:1">
<p>Namely <em>derived algebraic geometry</em>, and its siblings, and its cousins, and etc. <a href="#fnref:1" class="reversefootnote">↩</a></p>
</li>
<li id="fn:2">
<p>Think of rings (for some reason localisations and quotients of rings don’t look as confusingly similar to me as the categorical versions do). <a href="#fnref:2" class="reversefootnote">↩</a></p>
</li>
</ol>
</div>This post assumes that you have seen the construction of derived categories and maybe the definitions of dg- and $A_\infty$-categories, and wondered how they all linked together. In particular, as an undergraduate I was always confused as to what the difference was between the two steps of constructing the derived category of chain complexes was: taking equivalence classes of chain homotopic complexes; and then formally inverting all quasi-isomorphisms. Both of them seemed to be some sort of quotienting/equivalence-class-like action, so why not do them at the same time? What different roles were played by each step?Triangulations of products of triangulations2018-04-11T00:00:00+00:002018-04-11T00:00:00+00:00https://thosgood.github.io/maths/2018/04/11/triangulation-of-products<p>At a conference this week, I ended up having a conversation with Nicolas Vichery and Eduard Balzin about why simplices are the prevalent choice of geometric shape for higher structure, as opposed to e.g. cubes or globes.</p>
<!--more-->
<p>I know there are a bunch of nice properties that simplices have that other shapes don’t have<sup id="fnref:1"><a href="#fn:1" class="footnote">1</a></sup>, and they seem to be the natural choice when you start thinking about higher categories as quasi-categories and filling of horns etc., but I always thought that simplices were badly behaved with respect to taking products.<sup id="fnref:4"><a href="#fn:4" class="footnote">2</a></sup>
Nicolas, however, showed me a great little calculation that solves this problem.
Neither I nor some of the other people I spoke to had seen this before, so I thought it would be worth spreading the word.</p>
<p>If we take the (geometric realisation of the) 1-cube (the interval $[0,1]$), then we can take the product with itself to get $[0,1]\times[0,1]$, which is automatically the 2-cube.
More generally, any product of cubes will itself be a cube.
For simplices, however, this is not at all immediate: think of the problem of trying to find a triangulation for a product of spaces when calculating simplicial homology in an undergraduate course, for example.
Explicitly, if we take the product of the (geometric realisations of) two 1-simplices (the interval $[0,1]$ again), then we, as above, get the space $[0,1]\times[0,1]$, which is not a $\Delta$-complex (in the language of Hatcher).
To triangulate this space, we need to add a 1-simplex along the diagonal of the square.</p>
<p><img src="/assets/post-images/2018-04-11-triangulation-of-products-cubes-work.jpg" alt="Cubes work, simplices don't" title="Cubes work, simplices don't" /></p>
<p>This isn’t too hard to see, but for arbitrary triangulations of spaces, it’s much harder to see what extra simplices we need to add to their product to recover a triangulation.
Even if we <em>can</em> figure out how to do it for hard examples, then we still would need a way to describe an algorithm for doing it to <em>any</em> product space.
But there is a ‘trick’, which works as follows: write your simplices as simplicial sets, <strong>including degeneracies</strong>,<sup id="fnref:2"><a href="#fn:2" class="footnote">3</a></sup> and then take the product in the category of simplicial sets.</p>
<p>Consider our example of $\Delta^1\times\Delta^1$, i.e. $[0,1]\times[0,1]$.
Define the simplicial set $X_\bullet$ by</p>
<script type="math/tex; mode=display">% <![CDATA[
\begin{align*}
X_0 &= \{[0], [1]\}\\
X_1 &= \{[0,0],[0,1],[1,1]\}\\
X_2 &= \{[0,0,0],[0,0,1],[0,1,1],[1,1,1]\}\\
&\vdots
\end{align*} %]]></script>
<p>that is, all simplices in degree 2 and higher are degenerate (and we include all of them); the 0-simplices correspond to the points 0 and 1; and the 1-simplices correspond to the line from 0 to 1, as well as the two degenerate ‘lines’ from 0 to 0 and from 1 to 1_.</p>
<p>When we apply geometric realisation, these degenerate simplices will ‘vanish’, and so we can just look at what the non-degenerate 1-simplices of $X_\bullet\times X_\bullet$ are to see what happens in terms of triangulation.
There are nine 1-simplices in the product:</p>
<script type="math/tex; mode=display">% <![CDATA[
\begin{align*}
&[0,0]\times[0,0], [0,0]\times[0,1], [0,0]\times[1,1],\\
&[0,1]\times[0,0], [0,1]\times[0,1], [0,1]\times[1,1],\\
&[1,1]\times[0,0], [1,1]\times[0,1], [1,1]\times[1,1].
\end{align*} %]]></script>
<p>Five of these are non-degenerate,<sup id="fnref:3"><a href="#fn:3" class="footnote">4</a></sup> and the one that interests us is $[0,1]\times[0,1]$: when we apply geometric realisation, this will be a 1-simplex along the diagonal of the square.</p>
<p><img src="/assets/post-images/2018-04-11-triangulation-of-products-result.jpg" alt="The final result" title="The final result" /></p>
<p>In essence, the idea is simple, but it’s a trick that I’d never seen before, and it really makes me ‘believe’ in simplices.</p>
<hr />
<h1 id="footnotes">Footnotes</h1>
<div class="footnotes">
<ol>
<li id="fn:1">
<p>Or, at least, not a priori. This is sort of exampled by the fact that the cube category is a test category that is <em>not</em> strict, but becomes strict when we consider cubes with connections. See (as always) the <a href="https://ncatlab.org/nlab/show/test+category#examples">nLab page on test categories</a>. <a href="#fnref:1" class="reversefootnote">↩</a></p>
</li>
<li id="fn:4">
<p>I mean, what follows doesn’t really answer what we were actually discussing at all (why simplices are the ‘right’ choice, if they even are), but just assuaged one of my doubts about the niceness of simplices that originally led me to wonder about this problem in the first place. <a href="#fnref:4" class="reversefootnote">↩</a></p>
</li>
<li id="fn:2">
<p>Geometrically, this means we take ‘lines’ that are actually points, ‘triangles’ that are actually lines or points, etc. Algebraically, this means that we take simplices in the image of the degeneracy maps $s_i^p\colon\Delta_p\to\Delta_{p+1}$ given by $s_i^p\colon[0,1,\ldots,p]\mapsto[0,1,\ldots,i,i,\ldots,p]$. <a href="#fnref:2" class="reversefootnote">↩</a></p>
</li>
<li id="fn:3">
<p>The ones not in the corners of our 3-by-3 list, i.e. those that have at least one component non-degenerate. <a href="#fnref:3" class="reversefootnote">↩</a></p>
</li>
</ol>
</div>At a conference this week, I ended up having a conversation with Nicolas Vichery and Eduard Balzin about why simplices are the prevalent choice of geometric shape for higher structure, as opposed to e.g. cubes or globes.Quantum circuits2018-03-31T00:00:00+00:002018-03-31T00:00:00+00:00https://thosgood.github.io/maths/2018/03/31/quantum-circuits-part-1<p>I am not at all a physicist, and my knowledge of quantum physics in particular comes solely from undergraduate courses that I followed years ago, and any reading I can get done when feeling mathematical but not inclined to work on my thesis.
However, after scanning through some papers by <a href="https://arxiv.org/abs/math/0512103">Bartlett</a>, <a href="https://arxiv.org/abs/0908.2469">Baez, Lauda</a>, and <a href="https://arxiv.org/abs/0905.0465">Lurie</a>, my interest in quantum physics, and quantum computing especially, has come back with a vengeance.</p>
<!--more-->
<p><em>The following is aimed at people who have maybe read some things about quantum physics before and know some linear algebra.</em></p>
<p>Quantum computing has fascinated me ever since I first studied it at university, despite being a self-proclaimed ‘non-physicist’, swearing to do only ‘mathematical maths’ and none of this applied stuff (a shameful phase that I am really glad I outgrew).
This is without a doubt almost entirely due to the incredible teaching of <a href="http://www.arturekert.org">Artur Ekert</a>, whose lectures were probably the best that I ever experienced as an undergrad/masters student.
Really, most of what I’m going to say comes from notes I took in these lectures, so hopefully I don’t distort them too much.</p>
<p>One of the reasons that I’m posting this, apart from to help me to not forget it, is that quantum physics has such a strong link to $n$-category theory, but I don’t know at all really how quantum computing fits into this <a href="https://ncatlab.org/nlab/show/nPOV">nPOV</a> picture.
As a naive guess, I would think that there is some `string diagram’ approach to all this, but I’d need to sit down and have a think about what exactly that would be…
So if anybody has any comments or references then I’d be more that interested to hear them!</p>
<h1 id="foundational-ideas">Foundational ideas</h1>
<p>Before jumping in to the jacuzzi that is quantum circuitry, let’s dip our toes in the vast ocean of <strong>quantum physics axiomatisation</strong>.
First of all though, a note on quantum computers in practice.
There are a bunch of companies that make ‘quantum computers’ (for example the <a href="https://www.dwavesys.com">D-Wave</a>) but these are of a completely different species to those that we’re going to have a look at.
These computers work using <a href="https://en.wikipedia.org/wiki/Quantum_annealing">quantum annealing</a> – a sort of quantum version of <a href="http://www.wikiwand.com/en/Simulated_annealing">thermal annealing</a>; what I’m going to talk about are <a href="http://www.wikiwand.com/en/Quantum_Turing_machine">universal quantum computers</a>.
Quantum annealing lets you implement so-called hill-climbing algorithms, but many of the famous quantum algorithms, like Shor’s, cannot be implemented in this way.Quantum annealing lets you implement so-called hill-climbing algorithms, but many of the more well-known quantum algorithms, such as <a href="https://en.wikipedia.org/wiki/Shor%27s_algorithm?oldformat=true">Shor’s</a>, cannot be implemented in this way.
I hate to be a cynic about something in which I am far from an expert, but it seems to me that calling systems like the D-Wave ‘quantum computers’ is a way to boost sales by using not-entirely-false-but-definitely-misleading buzzwords.<sup id="fnref:2"><a href="#fn:2" class="footnote">1</a></sup></p>
<p>Another very important point is that <strong>quantum algorithm complexity</strong> is by no means trivial.
Said in another way, there are some things that quantum computers can do so much better than classical ones (e.g. prime factorisation), but they are neither ‘better’ nor ‘faster’ overall: there are some things that they are much <em>worse</em> at than classical computers.
Expecting a quantum computer to be able to run arbitrary programs quicker than a classical computer is somewhat akin to expecting an otter to be able to drink a pint of water quicker than a pigeon could, simply because an otter swims faster in water.<sup id="fnref:3"><a href="#fn:3" class="footnote">2</a></sup></p>
<p>Anyway, with that aside, let’s go get our feet wet.</p>
<h2 id="quantum-interference">Quantum interference</h2>
<p>Here is an experimental fact that you can read about in who-knows-how-many places and find numerous videos of: if we send a particle through a barrier with two tiny (really really tiny) holes (or <em>slits</em>, to agree with the name <strong>double-slit experiment</strong>) and place a detector on the other side, it is <em>not necessarily</em> true that the particle reaches the detector with probability $p_1+p_2$, where $p_i$ is the probability of the particle going through the $i$-th hole.
In fact, the probability is given by</p>
<script type="math/tex; mode=display">p_1 + p_2 + 2\sqrt{p_1p_2}\cos(\varphi_1-\varphi_2)</script>
<p>where $\varphi_i$ are numbers, called the <strong>phase</strong>, associated to the $p_i$ (we describe what exactly they are in a bit).
This means that the total probability depends on the <strong>relative phase</strong> of the system: the value $\vert\varphi_1-\varphi_2\vert$.
Sometimes this <em>will</em> agree with $p_1+p_2$, and sometimes <em>not</em>.</p>
<p>The point is, the <strong>Kolmogorov additivity axiom</strong> (the probability of two <em>mutually exclusive</em> events is the sum of the probabilities of the constituent events) seems to fail – but why?
As a result of decades of thinking and experiments and failed ideas, the theory that seems the ‘most likely’ is that of <strong>quantum theory</strong>, which says in particular that <em>the particle passes through both slits simultaneously</em>, and so the events of passing through each slit individually are <em>not</em> mutually exclusive at all.
This does seem to explain the formula for the probability of detection that we described above: the only way that it can depend on $\varphi_1-\varphi_2$ is if the particle ‘interacts’ with both paths each time it passes through the barrier.</p>
<p>I’m not going to talk much more about actual quantum physics, since there are so many great references written about it already, and I am also not a physicist, and even less so an experimental quantum physicist.
However, it’s good to have this basic understanding of quantum physics to really understand how quantum computers work: in some sense that we will make precise, they follow <em>all</em> computational paths at the same time, and give an answer that depends on <em>all</em> of the results.</p>
<p>It is very natural, reading this, to wonder why it is that we don’t experience quantum effects in our daily lives.<sup id="fnref:1"><a href="#fn:1" class="footnote">3</a></sup>
You can try throwing a ball at a wall with two open windows as much as you want, but you’ll never have it go through both windows simultaneously.
The answer is the very thing that’s stopping us from being able to actually build quantum computers with our current technology: <strong>decoherence</strong>.
Simply put, no physical system is really truly separate from the external environment – there is always gravity of far away objects pulling on things, and even being able to <em>look</em> at a physical system means that there are photons passing through and touching things and generally being a nuisance.
Decoherence, then, is when the external environment ‘gets in the way’ of our very delicate teeny tiny quantum systems, and influences the <strong>interference term</strong> (the term with the relative phase) in such a way that we can’t recover it.
This also ties in with the fact that, given any ‘quantum theorem’ (i.e. formula or equation for a quantum system), if we make all the ‘quantum terms’ tend to zero then we should recover the corresponding ‘classical theorem’.
Later on we will talk about decoherence again, and look at what it means explicitly in terms of quantum circuits, but to begin with we just sort of ignore it and work in an ideal world where we can get rid of it entirely.</p>
<h2 id="axiomatics">Axiomatics</h2>
<p>In our discussion of the double-slit experiment we mentioned how there were ‘extra’ numbers $\varphi_i$ associated to the probabilities $p_i$, so let’s try to formalise that now.
Quantum theory suggests that we work with <strong>probability amplitudes</strong> instead of <strong>probabilities</strong>.
That is, rather than just saying an event happens with probability $p$ for some real number $0\leqslant p\leqslant1$, we say that it has <strong>associated probability amplitude $z$</strong>, where $z$ is some complex number such that $\vert z\vert\leqslant1$, and then the probability of the event occurring is exactly $|z|$.
The way that amplitudes behave are very similar to they ways that probabilities behave: the amplitude associated to two events occurring one after the other is the <em>product</em> $z_1z_2$ of their probability amplitudes; the amplitude associated to an event that can occur in two different ways it the sum $z_1+z_2$ of the amplitudes associated to the individual ways.
In fact, writing our complex numbers as $z=|z|e^{i\varphi}$, we can recover the probability in the double-slit experiment:</p>
<script type="math/tex; mode=display">|z_1+z_2|^2 = |z_1|^2 + |z_2|^2 + z_1^*z_2 + z_1z_2^* = p_1 + p_2 + 2\sqrt{p_1p_2}\cos(\varphi_1-\varphi_2).</script>
<p>The other generalisation of classical things that we work with is the idea of a <strong>qubit</strong>.
The classical idea of a <strong>bit</strong> is that we can store information by encoding things in binary and then, for example, inducing a charge between the plates of a capacitor to represent a 1, and leaving it uncharged to represent a 0.
To make this quantum, we can look at storing our data not on capacitors but on electrons, or using polarisation of light.
For example, we could say that an electron in its <strong>ground state</strong> (its lowest energy state) represents a 0, and if it’s in a higher energy state then it represents a 1.
So far so good, but what’s new?
Well, by shining light of varying frequencies and brightness for varying durations, we can change the electron’s energy state, but if we shine light at, for example, half the brightness, then we will find that our electron is now in a state that can’t be described classically: it’s in a <strong>coherent superposition</strong> of states.
These are states where the electron is in <em>both energy levels at once</em>, similar to how the particle went through both slits at once.</p>
<p>Again, I’m not really the best person to explain this, but the idea is that, experimentally, we can prepare particles in such a way that, when we measure their energy, sometimes we find that they’re in their ground state, and other times they’re in their higher energy state (and anything that we can prepare consistently like this we call a <strong>qubit</strong>; we call one of the states $\vert0\rangle$ and the other $\vert1\rangle$).<sup id="fnref:4"><a href="#fn:4" class="footnote">4</a></sup>
But, you may be saying, how do we know that it isn’t simply the case that some of the particles we prepare are already in one state, and the others in another state?
That is, imagine the two following scenarios:</p>
<ol>
<li>We have an electron in an ‘equal’ superposition (it’s in both states ‘in equal amounts’) and we measure its energy</li>
<li>We have a bag containing two electrons, one in state $\vert0\rangle$ and the other in state $\vert1\rangle$. We pick one randomly and measure its energy.</li>
</ol>
<p>In both cases, when we measure the energy of the particle we will find it to be in state $\vert0\rangle$ or state $\vert1\rangle$ with equal probability (50%), so how does this supposed ‘superposition particle’ differ from a particle with a randomly picked state?
Well, experimentally, we can show a difference, but maybe more convincingly we can give a good mathematical reason why these two situations are completely different.
To do so <em>properly</em>, we need at least the basics of quantum circuitry, but we can describe the proof right now.</p>
<p>If it were the case that sending a pulse of light at the electron that puts it in a 50/50 superposition were the same as simply setting the electron to state $\vert0\rangle$ or $\vert1\rangle$ randomly, then firing this same beam of light <em>twice</em> in a row at the same electron should be the same as firing it once.
In other words, if we randomly set it to 0 or 1, and then randomly set it <em>again</em> to 0 or 1, it’s the same as randomly setting it to 0 or 1 just once.
<em>But</em> (and here’s the kicker), we can show that <em>every single time</em> you start with an electron in state $\vert0\rangle$ and hit it with this pulse of light two times in a row, what you actually end up with is an electron in state $\vert1\rangle$.
<em>Every single time.</em>
It’s a hard thing to understand when you first meet it (and still even after the second and third encounters), but coherent superpositions are not at all the same as a particle being in a randomly chosen state – it really is a particle that is in both states at once.</p>
<p>There is a bunch of really exciting ‘quantum philosophy’ that asks questions like, “if quantum theory is wrong in some way, then can any of it still be right, or does it all break down if we find one problem?”, and “how can we tell that there isn’t a <em>hidden variable</em>?”, i.e. “how do we know that the result of our measurement of a particle isn’t predetermined?”.
What is really astonishing is that quantum theory is in some sense more general and powerful that quantum theory.
What do I mean by this?
Well, as an example that I still think about in awe, even if quantum theory is completely entirely ‘wrong’, in that the universe works by some entirely different theory, then we can <em>still</em> use quantum physics to produce truly random data.
This doesn’t sound very exciting or big, but if you say it in a different way you can see why people would get so worked up about it: <em>if all of our lab equipment is made by some private corporation that we don’t trust, and we want to produce a random number, then using quantum physics we can do so with <strong>complete certainty</strong>,<sup id="fnref:10"><a href="#fn:10" class="footnote">5</a></sup> even if the equipment has been tampered with to produce certain states of particles more often than others, or to produce seemingly random states that are actually just following a predetermined order.</em></p>
<p>I don’t want to be turning quantum physics into sensationalism now, so this comes with the obvious caveat that doing things in practice is usually really hard.
A quote I remember hearing from Artur is that <em>“in theory there is no difference between theory and practice, but in practice there is”</em>.
But, even so, wow this stuff is cool.</p>
<p>Awed ravings aside,<sup id="fnref:5"><a href="#fn:5" class="footnote">6</a></sup> let’s briefly end this section on axiomatisation by actually stating what our axiomatisation is.
This isn’t going to be entirely self-contained for those new to maths I’m afraid – I’m going to freely talk about vector spaces and complex numbers and matrices and the like (basically, <em>linear algebra over $\mathbb{C}$</em>).</p>
<p>The following is all experimentally inspired and guided, but it still a purely <em>mathematical</em> formalism.
So far it seems to agree with every physical experiment we can throw at it, but we won’t worry too much about this, and instead just see what happens if we play around with the maths, and leave physical implementations to the physicists (though we won’t forget that, whenever something seems to pop up out of thin air, it’s almost always because it’s been suggested to exist by some experimental experiences).</p>
<hr />
<ol>
<li>
<p>A <strong>qubit</strong> is any object that can be reliably prepared to be in one of two states (or any superposition therein), manipulated, and measured.
Any qubit can be described by a <strong>state vector</strong>:<sup id="fnref:6"><a href="#fn:6" class="footnote">7</a></sup> a qubit in state $\vert0\rangle$ with probability amplitude $\alpha_0$ and state $\vert1\rangle$ with probability amplitude $\alpha_1$ is represented by the unit vector with entries $\alpha_0$ and $\alpha_1$</p>
<script type="math/tex; mode=display">\alpha_0\vert1\rangle + \alpha\vert1\rangle \longleftrightarrow \begin{pmatrix}\alpha_0\\\alpha_1\end{pmatrix}.</script>
<p>This means that the qubit is in an $\alpha_0\colon\alpha_1$ coherent superposition, and so when measured it will be in state $\vert0\rangle$ with probability $\vert\alpha_0\vert^2$ and in state $\vert1\rangle$ with probability $\vert\alpha_1\vert^2$.</p>
<p>Note that we ignore <strong>global phase</strong>: if we have a qubit in the state $e^{i\vartheta}(\alpha_0\vert1\rangle + \alpha\vert1\rangle)$ then this is the same as a qubit in the state $\alpha_0\vert1\rangle + \alpha\vert1\rangle$.
We do <strong>not</strong>, however, ignore <strong>local phase</strong>: $\alpha_0\vert1\rangle + e^{i\vartheta}\alpha\vert1\rangle$ is <strong>not</strong> the same as $\alpha_0\vert1\rangle + \alpha\vert1\rangle$.</p>
</li>
<li>
<p>Qubits are modified by <strong>quantum evolutions</strong>: redistributions of the amplitudes between the states (for example, the changes of energy induced by pulses of light on an electron qubit).
Admissible quantum evolutions are described by complex isometries, in particular by <strong>unitary matrices</strong>.
For example, the matrix</p>
<script type="math/tex; mode=display">% <![CDATA[
\begin{pmatrix}0&-1\\1&0\end{pmatrix} %]]></script>
<p>sends the basis state $\vert0\rangle$ to $\vert1\rangle$, and the basis state $\vert1\rangle$ to $-\vert0\rangle$, and extends linearly to act on any superposition $\vert\Psi\rangle = \alpha_0\vert0\rangle + \alpha_1\vert1\rangle$.</p>
</li>
<li>
<p>We take <strong>measurements</strong> corresponding to our choice of basis states $\vert0\rangle$ and $\vert1\rangle$.
As mentioned above, the probability of the outcomes of any measurement are given by the moduli of the probability amplitudes, and the choice of basis consists of distinguishable states, so the outcomes of any measurement are mutually exclusive.<sup id="fnref:7"><a href="#fn:7" class="footnote">8</a></sup></p>
</li>
<li>
<p>For multiple qubits (and composite systems in general), the corresponding state vectors are the unit vectors in the tensor product of the Hilbert spaces associated to each individual system, e.g. a system consisting of two qubits is described by the complex Hilbert space $\mathbb{C}^2\otimes\mathbb{C}^2$, and so state vectors are of the form</p>
<script type="math/tex; mode=display">\begin{pmatrix}\alpha_0\\\alpha_1\end{pmatrix}\otimes\begin{pmatrix}\beta_0\\\beta_1\end{pmatrix}.</script>
<p>The evolutions are then described by tensor products of the relevant unitary matrices.</p>
<p><strong>Edit.</strong> More about this at the end of the post.</p>
</li>
</ol>
<hr />
<p> </p>
<p>Finally, for this section, an important note on measurements and the troublesome notion of <strong>collapsing</strong>.
When we have some particle in a coherent superposition and then measure it, we find it in state $\vert0\rangle$ with probability $\vert\alpha_0\vert$.
But then, after the measurement, the particle is in state $\vert0\rangle$, <em>full stop</em>; there is now no uncertainty, and we know that we can measure this particle over and over and always find it in state $\vert0\rangle$.
Even more worryingly, the original state $\alpha_0\vert0\rangle+\alpha_1\vert1\rangle$ is completely irretrievable.
This seems to be a contradiction to our assumptions on quantum evolutions: this ‘collapse’ that follows a measurement is not described by a hermitian operator (in particular, it’s not invertible, since we lose the original state), but instead something far more blunt and sudden.</p>
<p>There are many ways of dealing with this problem, but the important thing to note is that measurements can still be described in exactly the same way as all other quantum processes, and aren’t subject to a different axiomatisation or something.<sup id="fnref:8"><a href="#fn:8" class="footnote">9</a></sup>
However, for the moment, it’s a convenient way of thinking that simplifies the maths (and conceptualisation) to just think of measurements as collapses.</p>
<h1 id="classical-circuits">Classical circuits</h1>
<p>Finally for this post, we’ll talk a bit about classical circuits.
Let’s take a single bit, which can be in one of two states: $\vert0\rangle$ (off) and $\vert1\rangle$ (on).
Using the same notation as before, we can use a 2-vector to describe the state of this bit:</p>
<script type="math/tex; mode=display">% <![CDATA[
\begin{align*}
\begin{pmatrix}1\\0\end{pmatrix} \,\,\leftrightarrow\, &\vert0\rangle \leftrightarrow \text{off}\\
\begin{pmatrix}0\\1\end{pmatrix} \,\,\leftrightarrow\, &\vert1\rangle \leftrightarrow \text{on}.
\end{align*} %]]></script>
<p>Then we can describe the four classical computations on a single bit by matrices:</p>
<script type="math/tex; mode=display">% <![CDATA[
\begin{array}{cccc}
\begin{pmatrix}1&0\\0&1\end{pmatrix} & \begin{pmatrix}0&1\\1&0\end{pmatrix} & \begin{pmatrix}1&1\\0&0\end{pmatrix} & \begin{pmatrix}0&0\\1&1\end{pmatrix}\\[1em]
\text{IDENTITY} & \text{NOT} & \text{CONSTANT-0} & \text{CONSTANT-1}
\end{array} %]]></script>
<p>If we have two bits, we can use that $\mathbb{R}^2\otimes\mathbb{R}^2\cong\mathbb{R}^4$ to write state vectors in the form</p>
<script type="math/tex; mode=display">\begin{pmatrix}\delta_a\\1-\delta_a\end{pmatrix}\otimes \begin{pmatrix}\delta_b\\1-\delta_b\end{pmatrix} \longleftrightarrow \begin{pmatrix}\delta_a\otimes\delta_b\\\delta_a\otimes(1-\delta_b)\\(1-\delta_a)\otimes\delta_b\\(1-\delta_a)\otimes(1-\delta_b)\end{pmatrix}</script>
<p>for $\delta_a,\delta_b\in{0,1}$.</p>
<p>Then, for example, a gate that acts as NOT on the first bit and as the CONSTANT1 gate on the second bit would be given by<sup id="fnref:9"><a href="#fn:9" class="footnote">10</a></sup> the matrix</p>
<script type="math/tex; mode=display">% <![CDATA[
\begin{pmatrix}0&1\\1&0\end{pmatrix} \otimes \begin{pmatrix}0&0\\1&1\end{pmatrix} = \begin{pmatrix}0&0&0&0\\0&0&1&1\\0&0&0&0\\1&1&0&0\end{pmatrix}. %]]></script>
<hr />
<p><strong>Edit.</strong> Somehow I forgot to talk about <strong>entanglement</strong>, so let me fix that now with a small update.</p>
<p>When we have a multi-particle system (e.g. two qubits), then the states of this joint system are described by tensor products, as we said in our four-point axiomatisation.
This means that states of the form</p>
<script type="math/tex; mode=display">\vert0\rangle\otimes\vert0\rangle = \vert0\rangle\vert0\rangle = \vert00\rangle</script>
<p>(where the latter two equalities are just us introducing some shorthand notation) describe our system.
However, not every element of a tensor product is a tensor of elements in each component: they are <em>linear combinations</em> of such things.
For example,</p>
<script type="math/tex; mode=display">\alpha\vert00\rangle + \beta\vert01\rangle = \vert0\rangle\otimes(\alpha\vert0\rangle + \beta\vert1\rangle)</script>
<p>but</p>
<script type="math/tex; mode=display">\alpha\vert00\rangle + \beta\vert11\rangle</script>
<p><em>cannot</em> be written in the form $\vert \Psi_a\rangle\otimes\vert \Psi_b\rangle$.</p>
<p>We call states in the former form (tensors of states of the composite systems) <strong>pure states</strong>, and the latter form (states that don’t ‘split’ as a state in each composite system) <strong>entangled states</strong>, because somehow the two systems are interacting with each other in such a way that we can’t ‘pull them apart’.</p>
<hr />
<h1 id="footnotes">Footnotes</h1>
<div class="footnotes">
<ol>
<li id="fn:2">
<p>In general, it seems that quantum physics shares the fate of Gödel’s incompleteness theorems – to be misquoted and misapplied for the purpose of sensationalism. <a href="#fnref:2" class="reversefootnote">↩</a></p>
</li>
<li id="fn:3">
<p>To be fair, I have neither witnessed an otter-pigeon swimming race nor an otter-pigeon drinking race, so this could be a flawed analogy. <a href="#fnref:3" class="reversefootnote">↩</a></p>
</li>
<li id="fn:1">
<p>Well, in some way we <em>do</em>, because quantum theory governs fundamental actions of particles themselves, which sort of make up everything, so of course we’re affected. But what I mean here is that we don’t tend to see things in two places at once, or witness ‘mutually exclusive’ events simultaneously. <a href="#fnref:1" class="reversefootnote">↩</a></p>
</li>
<li id="fn:4">
<p>So, in our example, an electron in its ground state is said to be in state $\vert0\rangle$, and in its higher energy state its said to be in state $\vert1\rangle$, <a href="#fnref:4" class="reversefootnote">↩</a></p>
</li>
<li id="fn:10">
<p>Whatever this might actually mean… <a href="#fnref:10" class="reversefootnote">↩</a></p>
</li>
<li id="fn:5">
<p>Enthusiasm isn’t a substitute for knowledge, so I really hope that I haven’t said anything misleading here. If it were the case that being really excited about everything counted the same as understanding it all, then I’d be a much better mathematician. <a href="#fnref:5" class="reversefootnote">↩</a></p>
</li>
<li id="fn:6">
<p>We always need a <em>choice of basis</em>, because this is equivalent to being able to interpret a qubit ‘as a 0 or a 1’. <a href="#fnref:6" class="reversefootnote">↩</a></p>
</li>
<li id="fn:7">
<p>This is a possible sticky wicket. A particle can be in a superposition of states, but any measurement will result in just <em>one</em> state; we can’t actually <em>observe</em> a superposition. See the comments in a bit about measurement and collapse. <a href="#fnref:7" class="reversefootnote">↩</a></p>
</li>
<li id="fn:8">
<p>What even <em>is</em> measurement? The more you think about it and try to define it, the more you realise what a tricky concept it is. It’s when an observer learns information about a quantum system, but then, what is an observer? And how do we formalise the idea of ‘gaining information about something’? There is a lot of ‘quantum philosophy’ here, and many ways of answering these questions. Personally, I like the approach that captures the ‘measurement collapse’ as ‘lots of decoherence’, and describes decoherence as ‘the universe trying to learn information about the quantum system’. Hopefully I’ll write about this some other time, and try to find some good references by real physicists. <a href="#fnref:8" class="reversefootnote">↩</a></p>
</li>
<li id="fn:9">
<p>The tensor product of matrices is given by the <a href="https://en.wikipedia.org/wiki/Kronecker_product">Kronecker product</a>. <a href="#fnref:9" class="reversefootnote">↩</a></p>
</li>
</ol>
</div>I am not at all a physicist, and my knowledge of quantum physics in particular comes solely from undergraduate courses that I followed years ago, and any reading I can get done when feeling mathematical but not inclined to work on my thesis. However, after scanning through some papers by Bartlett, Baez, Lauda, and Lurie, my interest in quantum physics, and quantum computing especially, has come back with a vengeance.Loop spaces, spectra, and operads2018-03-12T00:00:00+00:002018-03-12T00:00:00+00:00https://thosgood.github.io/maths/2018/03/12/spectra-part-3<p><em>[See part 1 <a href="/maths/2017/12/08/spectra-part-1.html">here</a> and part 2 <a href="/maths/2017/12/11/spectra-part-2.html">here</a>]</em></p>
<p>This post is a weird one: it’s not really aimed at any one audience, but is more of a dump of a bunch of information that I’m trying to process.</p>
<!--more-->
<p>Now would be a great time to talk about operads, but I really could not do a better job than <a href="https://twitter.com/math3ma">Tai-Danae Bradley</a> has already done over at her <a href="http://www.math3ma.com">blog</a> in the post <em>What is an Operad</em>, <a href="http://www.math3ma.com/mathema/2017/10/23/what-is-an-operad-part-1">part 1</a> and <a href="http://www.math3ma.com/mathema/2017/10/30/what-is-an-operad-part-2">part 2</a>.
So throughout this post I’m going to assume that you’ve read this, and I’ll freely reference bits of it without necessarily saying so.</p>
<p>One of the really important points that Tai-Danae makes is talking about the theorem by May that says that</p>
<ol>
<li>$1$-fold loop spaces are (weakly homotopy) equivalent to algebras over the associahedra operad; and</li>
<li>$k$-fold loop spaces (for $k\geqslant2$) are (weakly homotopy) equivalent to algebras over the little $k$-cubes operad.</li>
</ol>
<p>But as well as algebras over an operad, we also have the notion of <strong>spaces</strong> over an operad, where we obtain similar looking equivalences.
This idea is more in the language of <strong>brave new algebra</strong> – a phrase introduced by Waldhausen for topological spaces that have some algebraic structures up to coherent homotopies.
There is a paper, <em>Introduction to Algebra over “Brave New Rings”</em>, by R.M. Vogt (Rendiconti del Circolo Matematico di Palermo, Serie II, Suppl. 59 (1999) pp. 49-82) that is a really great read, if you keep in mind the proviso of slightly different approach: he often works with <strong>non-$\Sigma$-operads</strong>, which are operads without necessarily having the datum of an action by the symmetric group.
Just to save you from finding his exact definition, I paraphrase it here.</p>
<hr />
<p><strong>Definition.</strong> A <strong>non-$\Sigma$-operad</strong> is a $\mathsf{Top}$-enriched category $\mathcal{A}$ with objects in bijection with $\mathbb{N}={0,1,\ldots}$, along with an associative continuous bifunctor $\oplus\colon\mathcal{A}\times\mathcal{A}\to\mathcal{A}$ given on objects by $m\oplus n=m+n$, and such that, for all $k$ and $n$, the morphism</p>
<script type="math/tex; mode=display">% <![CDATA[
\begin{align*}
\coprod_{n=\Sigma r_i} \mathcal{A}(r_1,1)\times\ldots\times\mathcal{A}(r_k,1) &\to \mathcal{A}(n,k)\\
(f_1,\ldots,f_k) &\mapsto f_1\oplus\ldots\oplus f_k
\end{align*} %]]></script>
<p>is a homeomorphism.</p>
<p>There is also the technical assumption that $\mathrm{id}_1\in\mathcal{A}(1,1)$ is a closed fibration, which we will pass over.</p>
<hr />
<p> </p>
<p>An important thing to note is how this corresponds to May’s definition of an operad.
The property satisfied by $\oplus$ on coproducts means that $\mathcal{A}$ is actually uniquely determined by composition and the morphism spaces $\mathcal{A}(n,1)$ for $n\geqslant0$, which is the data that May uses to define (non-$\Sigma$-)operads.</p>
<p>The very first page of Vogt’s paper has a beautiful ‘dictionary’ that I’ll reproduce (in full, even though we won’t talk about some of the more complicated rows) here, as well as adding another column with references to the places where Vogt proves these equivalences.</p>
<table class="bordered-table">
<thead>
<tr>
<th>classical algebra</th>
<th>ref.</th>
<th>brave new algebra</th>
<th>spectra</th>
</tr>
</thead>
<tbody>
<tr>
<td>monoid</td>
<td>4.3</td>
<td>$A_\infty$-space</td>
<td> </td>
</tr>
<tr>
<td>group</td>
<td>4.4</td>
<td>loop space</td>
<td> </td>
</tr>
<tr>
<td>abelian monoid</td>
<td>5.5</td>
<td>$E_\infty$-space</td>
<td> </td>
</tr>
<tr>
<td>abelian group</td>
<td>6.5, 7.5</td>
<td>infinite loop space</td>
<td>$\mathbb{S}$-module spectrum</td>
</tr>
<tr>
<td>ring</td>
<td> </td>
<td>$A_\infty$-ring space</td>
<td>monoid in $\mathsf{Mod}_\mathbb{S}$</td>
</tr>
<tr>
<td>commutative ring</td>
<td> </td>
<td>$E_\infty$-ring space</td>
<td>commutative monoid in $\mathsf{Mod}_\mathbb{S}$</td>
</tr>
<tr>
<td>ground ring $\mathbb{Z}$</td>
<td> </td>
<td>$E_\infty$-ring space $Q(S^0)$</td>
<td>sphere spectrum $\mathbb{S}$</td>
</tr>
<tr>
<td>tensor product $\otimes_\mathbb{Z}$</td>
<td> </td>
<td>?</td>
<td>smash product $\wedge_\mathbb{S}$</td>
</tr>
</tbody>
</table>
<p> </p>
<p>Now, first of all, we need to define <em>what</em> exactly e.g. an $A_\infty$-space is, which means we need to define two things:</p>
<ol>
<li>the operad $\mathcal{A}_\infty$, and</li>
<li>the notion of a space over an operad.</li>
</ol>
<p>For the second point, given some non-$\Sigma$-operad $\mathcal{A}$, we define an <strong>$\mathcal{A}$-space</strong> to be a continuous functor $X\colon\mathcal{B}\to\mathsf{Top}$ that sends $\oplus$ to $\times$.
In particular then, $X(n)=X(1)^n$, and so we often abuse notation and identify $X$ with the space $X(1)$.</p>
<p>For the first point, we let $\mathrm{Assoc}$ be the terminal non-$\Sigma$-operad, so that each $\mathrm{Assoc}(n,1)$ consists of just a single point which we will call {$\mu_n$}.
This forces the composition to satisfy</p>
<script type="math/tex; mode=display">\mu_k\circ(\mu_{r_1}\oplus\ldots\oplus\mu_{r_k}) = \mu_{r_1+\ldots+r_k}.</script>
<p>It can be seen that an $\mathrm{Assoc}$-space is exactly a monoid (this is example 2.4 in Vogt).</p>
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<p><strong>Definition.</strong> Let $\mathcal{B}$ be a non-$\Sigma$-operad.
We say that it is an <strong>$A_\infty$-operad</strong> if the unique functor $\mathcal{B}\to\mathcal{A}_\infty$ is a homotopy equivalence of the underlying morphism spaces.
If this is the case, then we call a $\mathcal{B}$-space an <strong>$A_\infty$-space</strong>.</p>
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<p><strong>Warning.</strong> <em>What follows is nothing more than a sketch of an elephant by a blind man.</em></p>
<p>Again, there is an important note about names and notation.
What is commonly known as the <strong>associative operad</strong> $\mathrm{Assoc}$ is referred to as $\mathcal{A}$ in Vogt, and what is now called $\mathcal{A}_\infty$ is called $\mathcal{W}\mathcal{A}$ in Vogt, because he constructs it as a ‘homotopy universal $A_\infty$ operad’ from taking some free operad and quotienting by certain relations (sections 2.6, 2.7).
In fact, there is the idea of <a href="https://ncatlab.org/nlab/show/Boardman-Vogt+resolution">Boardman-Vogt resolution</a>, which is a specific choice of cofibrant resolution of operads, and the associahedra operad gives such a resolution of $\mathrm{Assoc}$, i.e. a cofibrant replacement:</p>
<script type="math/tex; mode=display">\mathcal{A}_\infty\to\mathrm{Assoc}.</script>
<p>As for the commutative case, section 3 of Vogt explains in detail how the little cubes operad (in one dimension) is an $A_\infty$-operad that acts on loop spaces.
This generalises in section 6 to showing that $n$-fold loop spaces can be acted on by the little $n$-cubes operad, and taking a suitable colimit we find that infinite loop spaces are acted on by some ‘little $\infty$-cubes operad’.
That is, we define $E_n$ to be the little $n$-discs operad[^1], and $E_\infty$ to be the colimit of the $E_n$, which is weakly equivalent to the commutative operad $\mathrm{Comm}$.
As an interesting fact, we can see that $E_n$ is generally homotopically non-trivial (tiny picture below), but $E_\infty$ is contractible.</p>
<p><img src="/assets/post-images/2018-3-12-spectra-part-3-E22.jpg" alt="Proof that $E_2(2)$ is homotopic to the circle" class="img-responsive" /></p>
<p> </p>
<p>I haven’t really talked much about spectra here, but the definition of an <strong>infinite loop space</strong> in Vogt is the same as the definition of an $\Omega$-spectrum in the previous post.
Vogt mentions (5.2) that if $X$ is an infinite loop space then $Y\mapsto[Y,X_n]^\bullet$ defines a cohomology theory, and that any cohomology theory arises in this way.
This all seems really nice, but I’m not entirely certain how complicated a proof of this is, or how trivial this statement is.</p>[See part 1 here and part 2 here] This post is a weird one: it’s not really aimed at any one audience, but is more of a dump of a bunch of information that I’m trying to process.