I’m Sanath Devalapurkar, an undergraduate at MIT starting Fall 2016. I love mathematics. I have conducted some research in algebraic topology under Marcy Robertson at UCLA, where I have taken some classes. My academic webpage is here. I do not post frequently on this blog anymore, but do post semi-frequently on a group (some friends from the math category at ISEF ’15) blog here: erdosninth.wordpress.com.

## Intel STS, Day 4

(21:31) I just came back from a very exciting and fun day. Today we had our poster judging, and mine actually went pretty well. I waited for a judge for about an hour, and my first judge was a mathematician. He began by asking me whether $\mathfrak{q}(\mathscr{C})_n$ (in my paper) was trivial, and I asked if he meant contractible. I told him that it indeed was in the version of the draft that I submitted to STS, but I actually ended up fixing that problem.

He then asked me about what the algebraic K-theory of a general ring is. I went into my usual mode, and said that if $R$ is a ring, then we can consider $\mathrm{Mod}^\mathrm{proj}_R$ is the category of projective $R$-modules. Then this is an abelian category (whose definition I reviewed), and you can choose an exact category structure. Then you do the Quillen Q-construction, and taking the geoemtric realization of the nerve of this category gives a topological space, whose homotopy groups are the K-groups.

I was then asked to compute $K_2(\mathbf{Z}/p\mathbf{Z})$, and I couldn’t do it on the spot, so he asked me to compute $K_1(\mathbf{Z}/p\mathbf{Z})$, and I initially said $\mathbf{Z}/p\mathbf{Z}$, but then later said that it was actually $\mathbf{Z}/(p-1)\mathbf{Z}$, which comes from Quillen’s computation. Then he asked me what $K_1(\mathbf{Z})$, but when I couldn’t do that on the spot either, he pointed to a table I made on my poster and asked me what it meant (it was a table comparing $\mathbf{Z}$-algebras, $\mathbf{S}$-algebras (algebras over the sphere spectrum), and $\mathrm{Sp}$-algebras).

Once I began talking about the sphere spectrum, he asked me what a spectrum was. I told him the usual definition, and he asked me why spectra were “stable”, and I replied the Freudenthal suspension theorem. He also asked me for an example of a spectrum that’s not $\mathbf{S}$, and I said $\mathrm{H}A$ for an abelian group $A$. He asked me to define Eilenberg-Maclane spaces (I told him the definitions via cohomology and the usual homotopy groups definition), and show that they form a spectrum (which was easy). Then he asked me to prove that Eilenberg-Maclane spaces are unique up to homotopy equivalence, which was slightly harder since I didn’t recall the proof. I said that perhaps we should start with a general construction of Eilenberg-Maclane spaces, choose a $(n-1)$-connected space (I realized now that I should have said that this could be a wedge of $n$-spheres), and attach cells to kill higher homotopy groups – denote this space by $X$. He then told me that that was very good, but that I hadn’t proven that they’re unique up to homotopy equivalence. I said that we might have to use Whitehead’s theorem, which says that if $X,Y$ are CW-complexes and the map $X\to Y$ induces isomorphisms on all homotopy groups, then $X$ and $Y$ are homotopy equivalent (he asked me to find $X$ and $Y$ such that their homotopy groups are all isomorphic but $X$ and $Y$ are not homotopy equivalent, and since I wasn’t able to think of such an example fast, he asked me to continue); so in this case, we would have to define a map $X\to K(A, n)$, which he then asked me to do. Very soon after asking me the question, though, he said that he had run out of time, and that I shouldn’t worry because he was asking me incredibly hard questions. (After I had my next judge, I went up to him (this judge) and told him $K_2(\mathbf{Z}/p\mathbf{Z})$ is $0$ since $2$ is even, and that $K_{2n}(\mathbf{Z}/p\mathbf{Z})$ is isomorphic to $0$ in general.)

My next judge was also a mathematician, who immediately asked me if every category is an $\infty$-category, and I said yes, canonically. I told her that this is analogous to every set being a category (although a better example might have been every category being a $2$-category); but it was a very easy question. She then asked me what differed between my definition and Barwick’s definition of K-theory, which was also a straightforward question. She asked me to compute $K(\mathrm{Fin}_\ast)$; a simple argument about $\mathrm{Fin}_\ast$ being the free symmetric monoidal category on a point, and since $\mathbf{S}$ is the free symmetric monoidal groupoid with inverses on a point, it makes sense to say that $K(\mathrm{Fin}_\ast)$ is $\mathbf{S}$. She then asked me how abelian categories relate to stable $\infty$-categories, and I described the derived $\infty$-category construction. I was then asked to explain my project as I would to the media, and both of us laughed a little (this was nice, because I had messed up very bad in the questions in her room yesterday). After I began comparing the letters P and O and saying that they’re “homotopy equivalent”, and saying that O can be viewed as being a “topological ring” in a very crude sense, etc. she said that that was a very great explanation, because it conveyed my project, and showed that I had intuition for these structures.

My last judge was from NASA, and he asked me to explain my project at a high level, and dumb it down when I needed to. He then asked me what applications there were, and I said something about quantum field theories (which was all that popped into my head at the moment). After my whole talk, he said that everything was very eloquent, and said that he looked forward to meeting me again in the future.

We then went to a very nice sandwich place on the next block, which was very good. I enjoyed eating the sandwiches, although I couldn’t recall whether I had ordered a sandwich or a salad (we fill in forms for lunch the day we check in), and had to wait for quite a while for them to finish making the sandwiches. It was a lot like Subway’s, but a lot cooler because all 40 finalists were hanging out and high-fiving each other because judging was finally over! (But that wasn’t entirely true; some of the judges came in to ask finalists questions during public exhibition.)

And it was finally time for public exhibition! I was very excited, but expected to get relatively few people since I have a technical math project, which was only true for the first 15 minutes. Many people came up to me just to pick up a card, and others talked with me for a long time. All of them had varying levels of mathematical knowledge, and I met a lot of MIT people (incoming freshmen and those already at MIT), and Po-Shen Loh. The bulk of people were generally asking me questions like where I was going to college (I really don’t know, because I got into Caltech during STS). Some asked me if I was ready to go to college at the age of 15, and I said that I think I was, especially if I was going to a school with people who are as awesome as the other finalists, because I’m very comfortable around everybody else.

One of the most memorable and happy memories from the public exhibition was a small six or seven year old, who came with her mother and her younger brother. She was initially very scared to talk to me, but then asked me how and what I did my project on; although there were at least fifteen such kids who asked me such questions, she was especially memorable because she had a passion for science in her. After she finished talking to me, her brother began pulling on her mom’s leg, asking her if they could go buy food; but the girl (whose name I forgot) asked her mom if she could go and see more projects. It made me very happy to see such a young kid with a passion for learning! (Soon enough, though, my throat started hurting, because by 4:00 pm, we had already talked for about 5 hours.)

When checking the schedule, I saw that I’d be meeting with Caitlin Sullivan at 9:30 to discuss my speech (since I’m the Seaborg winner); however, they hadn’t told me how long the speech was, and whether I needed a full draft due by 9:30. One of the SSP staff told me that it was a 10 minute speech (ten minutes?!) and I needed a full draft by 9:30, which Caitlin then told me was not true (phew). I did end up writing a draft though, with a lot of help from my roommate, Josh.

We then changed into casual clothes, and went bowling, which was truly one of the most memorable events of STS (although I think I say that about all the events, which is true!). I am very bad at bowling, but the whole experience was very fun(ny) because there werre others who had little to no experience, and near the end, we just randomly bowled (because nobody else actually was playing) and missed everything. Our lane had someone who was much better than us; by the sixth round, all of our scores averaged about 20, and hers was 65, so it was really a (very hilarious) competition for the inexperienced.

After coming back to the hotel, Caitlin told us about how we’d practice for the gala tomorrow, and certain rules, etc. After she was done, and she had left to the other office with a draft of my speech, 10 of us gathered in a circle to play Cards against Humanity – but we had to close it down because it would have looked really bad if some SSP staff walked in and saw us playing such a game. We then played Taboo, which was really fun – I was partnered with Vikul, and although this was the first time playing the game, we ended up coming second (by one point!). Tomorrow we’ll be visiting Arlington, VA, for a TechShop to help middle-schoolers build stuff. I think it’ll be fun, but with my clumsy hands, it’ll probably be more funny. (I don’t know if I’ll be able to post anything for the remaining days because I’m usually incredibly tired after the day’s events.)

## Intel STS, Day 3

(9:36) I just came back from my first round of judging. I had a mathematician and two biologist. The mathematician asked me two questions. The first was to compute $\pi_3(S^2)$. I said that it’s generated by the Hopf fibration, $S^3\to S^2$. Then I said that the fiber is $S^1$, and we get the long exact sequence in homotopy groups, and by the homotopy groups of $S^1$ and $\pi_n(S^n)\simeq\mathbf{Z}$, we get the desired isomorphism $\pi_3(S^2)\simeq\mathbf{Z}$. He then asked me to construct the Hopf fibration. I didn’t recall the exact construction, so he gave me a hint: namely, consider the dimension of $\mathbf{CP}^n$. I said that in the case $n=1$, the dimension is $2$, and by the CW-structure, it’s basically $S^2$. Then recognize that $\mathbf{CP}^1$ is the collection of lines through the origin in $\mathbf{C}^2$, and $\mathbf{C}^2$ is $\mathbf{R}^4$. By definition, $S^3\subset \mathbf{R}^4$, which leads to the Hopf fibration. Then he asked me the following question: if you have $100$ switches, all turned on, and you turn on all the switches labeled as multiples of $2$, then toggle those which are multiples of $3$, etc. once the whole thing is done, which ones are left? I wasn’t able to think properly, and he asked me to consider $25$; I then realized that each prime factor must appear at least twice in the prime factorization. He asked me to consider $27$, and I then realized that the ones left turned on are the squares.

(11:35) Just came back from my second round of judging. My first judge was from NASA, and he asked me to explain why space suits are important, and key features about them. I was also asked to explain improvements which I’d make to the current design – I just said that the bulk of the weight was probably electronics, and the size would decrease and speed would increase, although I didn’t actually say any key additions I’d make (oops). I was then asked about tidal waves, and he asked my why they occured. I talked about the gravitational pull of the moon, and then explained high and low tides. He then asked me why there are two high tides a day. Think of the earth’s water bodies as being deformed from a $2$-sphere to an ellipsoid; this gives the required answer. He then asked me about snowflakes, and why they were formed. I told him that water, in gaseous state, would condense. When he asked me what object they would condense on, I told him that water condensing on dust particles gives fog, but I didn’t know about clouds – and he said that they did indeed condense on dust, and we laughed a little. He then asked me why snowflakes form a hexagon, and I said that it’s maybe because of the crystalline structure of ice, which also explains why ice is less dense than water.

My second judge began by pulling out a big tube filled with a liquid, which she said I could assume was water, and inside the tube there were small glasses in the shape of tops,filled with colored liquid. These glasses had a small ring attached to them, which had temperatures labeled on them. She asked me why this worked; what I said was that it was maybe because of vapor pressure, and she told me that what was inside the glass didn’t matter. I then suggested that perhaps as temperature increases the volume of the air inside the glass would increase, which was maybe the reason for why they’d rise or fall. I think that was absolutely wrong, though, and she just boxed some words that she wrote on her paper. Whoops. She told me, “OK, let’s go to another question”. She asked me if there were any theoretical limits to a human’s age. I said that I know that brain cells cannot regenerate, and so eventually there’d be too less of brain left for the human body to continue functioning. I then said that maybe other organs could be regrown, via stem cells (she became very happy once I said that). She asked me to explain stem cells, and I told her that they could be used to regrow damaged parts of certain organs, and explained their limitations and how they were different from human embryonic cells.

(3:08) Came back from my third interview, which went really bad. My first judge asked me to describe modules over $\mathbf{C}[x]$, and I blanked out. She then suggested that I should consider $\mathbf{C}^n$; I was able to define the multiplication, but made a very foolish mistake which I then fixed. I then had a judge who asked me some ecology questions, namely whether biodiversity is more at the poles or at the equator. I said that it would be at the equator, but I was probably wrong. She then asked me what about comparing the equator and the temperature zones, and I said the temperate zones because of temperature variation. My last judge asked me about optics – and here I failed miserably. I forgot absolutely everything about optics, and mumbled a lot, probably false things, about light and refraction. At least tomorrow we will be on territory more familiar to us.

(10:27) After the whole judging process, we voted for our Seaborg winner. We had about 4 ties, and in the end, I ended up winning! It’s really exciting; I’m very happy to be given the opportunity to represent the STS class, and on the 75th anniversary of STS. We then went to an awesome Italian restaurant, where Demetri had a toast (with water, of course) to celebrate me winning Seaborg. It was really fun, and the food was really filling. (Michael Li and I were joking around, asking people for martinis.) We then visited the World War II memorial, and the Martin Luther King statue. Then we visited the Lincoln memorial, which was quite ethereal (to use a friend’s phrasing of the whole area). We came back at about 10 pm; and we have to get up at 6:30 tomorrow – and daylight saving only reduces the amount of sleeping time, so I guess I’ll go to sleep now.

(technically day 4, but 1:07 am) I’ve been reading up on a whole lot of homotopy theory, but there’s just so much! I chatted with my roommate, Josh, for a while, and he explained his project to me, while I tried to explain mine to him. I always thought that the straightening of (co)Cartesian fibrations was crazy technical but it really isn’t so bad :P.

## Intel STS, Day 2

(5:30) I was actually awake at 4:30 am (probably a bad idea to have gone to sleep at 12:20 am), and just lied down in my bed for about half an hour – I’m very excited and nervous for today, because it’s judging. Yesterday night, I was in Demetri’s room with four other awesome finalists, trying to answer questions and learn stuff from each other. It turns out that I don’t know much biology (which isn’t surprising, actually), so I’m really hoping that if I get any questions on biology (I know I will) then they won’t require too much technical knowledge. I was also at the quad, playing cards against humanity (I’m not the best at it). I think I have two judging sessions today, and three tomorrow. Right now, I’m just trying to review some stuff by browsing Wikipedia, although I’ll probably go and shower sometime soon.

(11:40) I just had my first judging interview, and it actually wasn’t very bad. The first judge immediately began; I forgot what the first question was, but here’s the second question (from the first judge): why don’t satellites fall? I told him that they’re in free fall, but then I messed up and started talking about centripetal force. I should have said that the translational velocity is the same as the velocity during free fall. The third question asked by this same judge was: given 100 tennis players, how many matches do you have to play to get the winner? I told him that the optimal strategy (there could be mistakes, but diregard those) was 99 matches. The reason is, there are 99 losers, and one loser per match – so 99 matches is the optimal strategy.

The second judge asked me about friendship networks, and I told her that they were directed graphs. Then she asked me to compare it to the food chain, which was not too hard. She then asked me to describe how I would represent such a graph in a computer. I told her: given an input (a person), assign him/her a node (two people, say 0 and 1), based on whether or not they consider the other person a friend. This would give you a graph which a computer could understand. She then asked me, how would you write this as a matrix? I told her that we do the same thing as above, except with a square $n\times n$-matrix, where the $(i,j)$th entry is $1$ if person $i$ and person $j$ are friends, and $0$ otherwise. She then asked me, given two people, count the number of mutual friends, including the trivial friendship between the two, if it exists. I gave a rather trivial bound, namely it has to be some integer between $0$ and $n-2$ (because there are two people in consideration).

The third judge began by asking me why, if a car is put outside, in a temperature that’s not freezing, there could be frost. Uhhh … I didn’t know. I asked him if the windows were open, if it’s been to a cold place before, etc. but he told me that it has nothing to do with the state of the car. He gave me a hint, saying that there were no clouds, but I still didn’t get it. He then asked me another question: look at this carbonated drink. It’s bound tight. When I open it, why does the carbon bubble up? I told him that there’s high pressure inside the bottle before it’s open; when it’s open, the air rushes out because there’s low air pressure outside. To fill up this gap in the air’s volume, the carbon forms bubbles and comes out of the drink in the form of $\mathrm{CO}_2$. All in all, it didn’t go as bad as I thought it would,  although I do think I messed up in the very first judge’s questions. Well, I’ve got another judging session at 3:00, so I’ll continue later.

(13:00) I just remembered the first question from the first judge. Given a point A, a river below A (say in $\mathbf{R}^2$), and another point B above the river, what is the shortest path from A to the river to B? I said that you choose the point C on the river such that the angle ACB is $90^\circ$; the path ACB should be this path. The judge nodded and continued with his next question; during lunch, Meena told me that that wasn’t always the optimal solution. Rather, what one should do is reflect B across the river to get a point B’, and draw the shortest path from A to B’; this gives the shortest path as required by the question.

(18:21) So I just finished my second round of judging; it was pretty fun! The first judge asked me how many ways there were to schedule the 40 finalists in the 5 judging rooms. I stumbled, and he told me that nobody’s been able to solve it today. I did a key step which he said nobody’s done, and then he turned it over to the other judge. The second judge began by telling me that when a food company removed glucose from food, the food was still healthy. But when the put fructose, people began dying. I said that they have a different ring structure, but didn’t conclude that the human body couldn’t metabolize fructose, I wasted a lot of time in this question. He then asked me about why baking powder made breads grow; I told him that it was an acid-base reaction, and he told me that that was exactly right, and was very happy. The last question was how neurons communicate with one another, and I answered that they do so via pulses of energy. I made an analogy to computers. He then asked me why neurons are faster than computers; this was quite easy, because neurons have more dendrites than computers. He seemed satisfied, and told me that I did pretty well. This was my last judging session for today, so I can relax a bit until tomorrow, when the first room has a judge in pure math (but I’ve heard that he asks combinatorial questions, which is very hard for me).

(21:10) I just reached my room after a very enjoyable dinner, where we ate with the judges. I was seated with the math judges, which was really cool. It was great to see that the judges weren’t as intimidating as you’d expect them to be in real life, after the whole judging interview stuff. There were two nice speeches about what lies ahead for us (for life in general), and it was inspiring to hear stories from a previous STS finalist who was a health policy advisor for Obama. We then had a professor from MIT’s Lincoln Lab, who told us that all of us would be getting planets – with the exception of those who already got some (so me, Demetri, Amol just hung out and ate the Capitol Building’s dome and the White House’s roof (made out of milk chocolate!) when others were getting their planet certificates).

## Intel STS, Day 1

I’m at the Intel Science Talent Search in Washington, D.C. For those of you unfamiliar with it, it is a contest where 40 finalists are selected after a rigorous process (including reading your papers, mentor recommendations, standardized test scores, etc) to come to Washington. I’ve decided to try liveblogging this event.

(11:15) I reached the airport at about 6:45 AM LA time, and I finished all the security checks by 7:30. I met up with one of my friends, George Hou, who’s also an STS finalist; both of us were in the math category last year at the LA County Science Fair. I can’t wait to meet all the other finalists!

(12:52) Once I got onto the plane, I took a few pics, and promptly fell asleep. But I got a good sleep yesterday, and so I was awake within 2 hours. With nothing else to do, I’ll probably continue reading Lurie’s thesis or my own paper. (Lurie’s thesis it is)

(22:05) I met most of the STS finalists; all are incredible people! I’m rooming with an awesome guy, Joshua Choe, whose project is in biology. I’m currently sitting in Demitri’s room, chatting about science. We’re talking about why the Big Bang was not a bang, why DNA is what determines an organism’s structure, etc. Before this, we ate dinner – it was awesome. We then practiced telling our story, i.e., about ourselves and our projects. I tried setting up my board, and messed up a bit (but it wasn’t too bad). Apparently there’s a TV in the bathroom, but we haven’t found it yet. My judging interviews tomorrow are at 11:15 am and 3:00 pm. I guess I’ll continue tomorrow, when there’s a little more to write about.

## Marked Simplicial Sets – I

There are many ways to present the theory of $\infty$-categories. One is via the Joyal model structure on $\mathscr{S}\mathrm{et}_\mathbf{\Delta}$:

Theorem: There exists a model structure on $\mathscr{S}\mathrm{et}_\mathbf{\Delta}$ where:

• The cofibrations are the monomorphisms.
• The weak equivalences are those maps $f:X\to Y$ such that the induced map $\mathfrak{C}[f]:\mathfrak{C}[X]\to\mathfrak{C}[Y]$ is an equivalence.

There is also a presentation via complete Segal spaces, which are simplicial objects of $\mathscr{S}\mathrm{et}_\mathbf{\Delta}$. It is natural to ask for a simpler definition, something which requires us to specify a minimal amount of information. One such model is the theory of \textit{marked simplicial sets}, introduced by Lurie. A marked simplicial set is a simplicial set with a collection of distinguished edges. More precisely:

Definition: Let $X$ be a simplicial set. A marked simplicial set is a pair $(X,\Gamma)$, where $\Gamma$ is a collection of edges of $X$ such that the collection $\mathrm{deg}_1(X)$ of degenerate edges of $X$ is a subcollection of $\Gamma$. The edges in $\Gamma$ are called marked edges.

It should be obvious that there are two canonical ways of presenting any simplicial set as a marked simplicial set. Let $X$ be a simplicial set. Then we can form a marked simplicial set $X^\flat$ where only the degenerate edges are marked. We may form another marked simplicial set $X^\sharp$ where all of the edges of $X$ are marked.

Suppose $(X,\Gamma)$ and $(Y,\Omega)$ are marked simplicial sets. A map from $(X,\Gamma)$ to $(Y,\Gamma)$ is a map $f:X\to Y$ such that $f(\Gamma)\subseteq \Omega$. If it is obvious, we will not specify the collection of marked edges. In other words, we will usually write $X$ instead of $(X,\Gamma)$ if there is no risk of confusion. The collection of marked simplicial sets and maps between them forms a category, $\mathscr{S}\mathrm{et}_\mathbf{\Delta}^+$. Suppose $X$ is a simplicial set. We will write $(\mathscr{S}\mathrm{et}_\mathbf{\Delta}^+)_{/X}$ to denote $(\mathscr{S}\mathrm{et}_\mathbf{\Delta}^+)_{/X^\sharp}$.

In ordinary homotopy theory, anodyne maps play a very important role. Let $\mathscr{S}\mathrm{et}_\mathbf{\Delta}$ denote the model category of simplicial sets with the Kan model structure. A map is called \textit{anodyne} if it has the right lifting property with respect to every fibration. In other words, the collection of anodyne maps is simply the collection of trivial cofibrations in $\mathscr{S}\mathrm{et}_\mathbf{\Delta}$. Recall that the horn inclusions $\Lambda^n_i\hookrightarrow\Delta^n$ for $0\leq i\leq n$ are generating cofibrations for the Kan model structure. This means that a map $X\to {\Delta^0}$ is a fibration if it has the right lifting property with respect to the horn inclusions $\Lambda^n_i\hookrightarrow\Delta^n$ for $0\leq i \leq n$.

A very similar construction can be done in the setting of marked simplicial sets:

Definition: The class of marked anodyne maps is the smallest weakly saturated class of maps generated by:

• The inclusions $(\Lambda^n_i)^\flat\hookrightarrow(\Delta^n)^\flat$ for $0.
• The inclusion $(\Lambda^n_n,(\mathrm{deg}_1(\Delta^n)\cup\Delta^{\{n-1,n\}})\cap(\Lambda^n_n)_1)\hookrightarrow(\Delta^n,\mathrm{deg}_1(\Delta^n)\cup\Delta^{\{n-1,n\}})$.
• The inclusion $(\Lambda^2_1)^\sharp\coprod_{(\Lambda^2_1)^\flat}(\Delta^2)^\flat\to(\Delta^2)^\sharp$.
• The map $K^\flat\to K^\sharp$ for every Kan complex $K$.

Let us recall the concept of a $p$-Cartesian morphism for an inner fibration $p:X\to S$. We will elaborate on this more in a future post.

Definition: Suppose $p:X\to S$ is an inner fibration and $f:x\to y$ an edge in $X$. $f$ is said to be $p$-Cartesian if the map $X_{/f}\to X_{/y}\times_{S_{/p(y)}}S_{/p(f)}$ is a trivial Kan fibration.

Proposition: A map $p:X\to S$ is a marked anodyne map if and only if it has the left lifting property with respect to every map $i: Y\to Z$ in $\mathscr{S}\mathrm{et}_\mathbf{\Delta}^+$ such that:

• $i$ is an inner fibration on the underlying simplicial sets.
• An edge of $Y$ is marked if and only if its image under $i$ is and it is $i$-Cartesian.
• If $f$ is a map in $Z$ whose target is the image of a $0$-simplice of $Y$, then $f$ can be pulled back via $i$ to a map in $Y$.

Therefore if $\mathscr{C}$ is an $\infty$-category, $\mathscr{C}^\flat\to{\Delta^0}^\flat$ and $\mathscr{C}^\sharp\to{\Delta^0}^\sharp$ both have the right lifting property with respect to every marked anodyne map. We can attempt to prove this statement in the case when $\mathscr{C}$ is a Kan complex directly as well.

Lemma: There is only one unique map $(\Delta^n)^\flat\to(\Delta^n)^\sharp$.

Proof. There is a canonical map $(\Delta^n)^\flat\to(\Delta^n)^\sharp$ given by the inclusion. It suffices to show that the only map $\Delta^n\to\Delta^n$ is the identity. In the category $\mathbf{\mathbf{\Delta}}$ there is only one map $[n]\to[n]$ given by the identity (the morphisms in $\mathbf{\Delta}$ are nonincreasing maps); since $\Delta^n$ is simply $\mathrm{N}([n])$ we observe that there is therefore only one map from $\Delta^n$ to itself given by the identity.

Theorem: Let $p:X\to S$ be a map of simplicial sets with the right lifting property with respect to $\Lambda^n_i\hookrightarrow\Delta^n$ for $0. Then the maps $p^\sharp$ and $p^\flat$ have the right lifting property with respect to the class of all maps generating the class of marked anodyne maps. In addition, if $p$ has a unique lift with respect to $\Lambda^n_i\hookrightarrow\Delta^n$ for $0, then so do $p^\flat$ and $p^\sharp$.

Proof. Let us first prove that the maps $p^\sharp$ and $p^\flat$ have the right lifting property with respect to the inclusions $(\Lambda^n_i)^\flat\hookrightarrow(\Delta^n)^\flat$ for $0. Assume $p$ is a map that satisfies the conditions in the proposition. It is then obvious that $p^\flat$ has the right lifting property with respect to the inclusions $(\Lambda^n_i)^\flat\hookrightarrow(\Delta^n)^\flat$ for $0. Consider the map $p:X\to S$. The map $p^\sharp$ then has the right lifting property with respect to the inclusions $(\Lambda^n_i)^\sharp\hookrightarrow(\Delta^n)^\sharp$ for $0. There are then inclusions $(\Lambda^n_i)^\flat\hookrightarrow(\Lambda^n_i)^\sharp$ and $(\Delta^n)^\flat\hookrightarrow(\Delta^n)^\sharp$ for $0. We can then construct the lifting $(\Delta^n)^\flat\to X$ as the composition $(\Delta^n)^\flat\hookrightarrow(\Delta^n)^\sharp\to X^\sharp$. Proving that $p^\flat$ and $p^\sharp$ have the right lifting property with respect to the inclusion $(\Lambda^n_n,(\mathrm{deg}_1(\Delta^n)\cup\Delta^{\{n-1,n\}})\cap(\Lambda^n_n)_1)\hookrightarrow(\Delta^n,\mathrm{deg}_1(\Delta^n)\cup\Delta^{\{n-1,n\}})$ is obvious since $p$ has the right lifting property with respect to the inclusions $\Lambda^n_i\hookrightarrow\Delta^n$ for $0.

Let us now prove that $p^\flat$ and $p^\sharp$ have the right lifting property with respect to the inclusion $(\Lambda^2_1)^\sharp\coprod_{(\Lambda^2_1)^\flat}(\Delta^2)^\flat\to(\Delta^2)^\sharp$. There is an inclusion $(\Lambda^2_1)^\sharp\hookrightarrow(\Lambda^2_1)^\sharp\coprod_{(\Lambda^2_1)^\flat}(\Delta^2)^\flat$. Since $(\Lambda^2_1)^\sharp\coprod_{(\Lambda^2_1)^\flat}(\Delta^2)^\flat\hookrightarrow(\Delta^2)^\sharp$ we observe that since $p^\sharp$ has the right lifting property with respect to $(\Lambda^2_1)^\sharp\coprod_{(\Lambda^2_1)^\flat}(\Delta^2)^\flat$. Indeed, this is obvious from the following diagram:

We will now approach the last class of maps, namely the maps $K^\flat\to K^\sharp$ for $K$ a Kan complex. Consider the diagram:

Asking that the lift exists amounts to constructing a map $K^\sharp\to X^\sharp$. Consider the map $K^\flat\to X^\sharp$, and denote by $f$ the map on the underlying simplicial sets. Then the map $K^\sharp\to X^\sharp$ can be chosen to be the map $f^\sharp$, and the diagram commutes.

For the last part we observe that if the lift $\Delta^n\to X$ is unique, then so are the maps $(\Delta^n)^\sharp\to X^\sharp$ and $(\Delta^n)^\flat\to X^\flat$. The above Lemma completes the proof.

Suppose $S={\Delta^0}$. Then $X$ is a Kan complex, and $X^\flat\to{\Delta^0}$ and $X^\sharp\to{\Delta^0}$ both have the right lifting property with respect to every marked anodyne map.

Let $X\to Y$ be a map of marked simplicial sets. It is a cofibration if the underlying map of simplicial sets is a monomorphism. We will conclude this section with an important fact to remember about marked anodyne maps.

Proposition: Marked anodyne maps are closed under smash products with arbitrary cofibrations.

In a future post, we will talk about Cartesian fibrations.

## A collaborative effort

Hi all! I have been silent for an extremely long time because of many things, but I wanted to announce a blog that I started with a few friends at ISEF this year: ErdosNinth. For this blog: I promise to write more soon!