# We have a blog?! (And a basic overview of Cartesian fibrations)

Whoa. We have a blog? We have a blog!

Sorry for the silence! We’ve been quite involved in senior-year stuff (college and scholarship apps, etc.) I’m currently trying to get over the fact that I submitted my Intel STS paper (parts of it are available on my website here)! Someday, when I have time, I’ll try to give an exposition of what it’s all about. I’ve also uploaded a video to YouTube trying to give an intro to $\infty$-categories. It’s meant to be for a general person interested in science. You can see it here:

Anyway, I want to continue on with giving an exposition on Cartesian fibrations. Let $p:X\to S$ be a map of simplicial sets. It is an inner fibration if it has the right lifting property with respect to the inclusions $\Lambda^n_i\hookrightarrow\Delta^n$ for all $0. Recall:

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.

$p$-Cartesian fibrations are critical to the definition of Cartesian fibrations. There are many reasons to care about their definition. Suppose $p:X\to S$ is an inner fibration. If $S$ is contractible, then $X$ is an $\infty$-category. Inner fibrations are stable under pushouts; hence in general, for all $s\in S$, the fiber $X_s$ given by $X\times_S\{s\}$ is an $\infty$-category. Suppose that $s\to s^\prime$ is a map in $S$. However, the induced $X_{s^\prime}\to X_s$ is \textit{not} a functor between $\infty$-categories (instead it is a “correspondence”). In order to obtain a functor it is necessary to make slight adjustments. This yields the notion of a \textit{Cartesian fibration}.

Definition. Let $p:X\to S$ be a map of simplicial sets. It is a Cartesian fibration if:

1. $p$ is an inner fibration.
2. If $f$ is a map in $S$ whose target is the image of a $0$-simplice of $X$, then $f$ can be pulled back via $p$ to a map in $X$.

Observe that these are two of the conditions appearing in Proposition \ref{charmarkedanodynemaps}. In particular, a marked anodyne map is a map with the left lifting property with respect to any Cartesian fibration $p:X\to S$. It is easy to get examples of Cartesian fibrations. Suppose $\mathscr{C}\to\mathscr{D}$ is a Grothendieck fibration. Then the map $\mathrm{N}(\mathscr{C})\to\mathrm{N}(\mathscr{D})$ is a Cartesian fibration. In fact, a map $\mathrm{N}(\mathscr{C})\to\mathrm{N}(\mathscr{D})$ is a Cartesian fibration if and only if the map $\mathscr{C}\to\mathscr{D}$ is a Grothendieck fibration.

Recall that if $p$ is an inner fibration, then a map $s\to s^\prime$ in $S$ gives a correspondence $X_{s^\prime}\to X_s$. $p$ is Cartesian if and only if this correspondence is determined by a functor. This functor is unique up to homotopy. This means that the essence of a Cartesian fibration is contained in the following statement: Cartesian fibrations with base $S$ are contravariant maps from $S$ to an $\infty$-category of $\infty$-categories. Let us now construct this $\infty$-category.

Let $\mathscr{C}\mathrm{at}_\infty^\mathbf{\Delta}$ be the simplicial category defined as follows.

1. Objects are small $\infty$-categories.
2. Let $\mathscr{C}$ and $\mathscr{D}$ be small $\infty$-categories. Then $\mathrm{Map}_{\mathscr{C}\mathrm{at}_\infty^\mathbf{\Delta}}(\mathscr{C},\mathscr{D})$ is the largest Kan complex contained in $\mathrm{Fun}(\mathscr{C},\mathscr{D})$.

Then the $\infty$-category of $\infty$-categories is the nerve of $\mathscr{C}\mathrm{at}_\infty^\mathbf{\Delta}$. The important thing to note is that $\mathscr{C}\mathrm{at}_\infty^\mathbf{\Delta}$ arises as the category of fibrant-cofibrant objects of a particular model structure on $\mathrm{Set}_\mathbf{\Delta}^+$.

Let $p:X\to S$ be a Cartesian fibration. Define $X^\natural$ to be the marked simplicial set $(X,\Gamma)$, where $\Gamma$ is the collection of $p$-Cartesian edges of $X$.

Next time, we’ll talk about the Cartesian model structure, and maybe the straightening and unstraightening constructions as well! You can see all this stuff in PDF format here, which can also be found on my webpage.

Best,
S.K.D