We finished some elementary discussion of monad, in the usual category.

$\gdef\bDelta{\mathbf{\Delta}}$ $\gdef\colim{\text{colim}}$

We start with simplex category $\Delta$, with objects $[0], [1], \cdots$, and order preserving morphisms.

A simplicial set is a functor $X: \Delta^{op} \to Set$. The set $X([0])$, or denoted as $X_0$, is the set of vertices, $X_1$ the set of edges. Given $[0] \to [1]$ mapping to the first vertices, we have the corresponding $X_1 \to X_0$, extracting the vertex in the edge labelled by $0$. These are called 'pull-back' maps, or face maps.

• $\Delta^n$ is the sSet given by $Map_{\Delta}(-, [n])$.
• Horn: $\Lambda^n_i$ is the sub sSet of $\Delta^n$, that avoid the facet opposite to vertex $i$.
• mapping sSet: given two sSets $X,Y$, we denote $Map(X,Y)_n = Map_{sSet}(\Delta^n \times X, Y)$. The usual mapping set is $Map(X,Y)_0$. This sSet remembers more information.
• $\bDelta^n$ is the geometrical $n$-dim simplex.

We have geometric realization of a simplicial set, it is a functor $|-|: sSet \to Top$, that $$ |X| = \colim_{\Delta^n \to X} \bDelta^n $$

Actually, I am not sure, what's going on here. What is the indexing category? It is like, we have $Map(pt, X)_n$ the mapping sSet from a point to $X$. It is a category with objects $[0], [1], \cdots$, each time we have $[n]$, we consider topological space $\sqcup \bDelta^n \times Map(pt, X)_n$.

Question: is it true $Map(pt, X)_n = X_n$? I think so. $X_n$ also contains all the degenerate simplices.

Ex; Geometric realization is the left-adjoint to singular chain.

• A sSet is called a Kan complex, if it has some horn filling property (?why do we care? and what space has such property?)
• Topological space are too special, geometric realization lose too too much information.

Nerve of a category C, it is the utmost flattening of the category. $N(C)_n: = Fun_{cat}([n], C)$, where $[n]$ is the poset viewed as a category. By definition, we just need a chain of composable morphism. The face map is given by the composition, or subchain, or whatever. So, no information is lost, lots of redundancy.

$Nerve(C)$ satisfies the property that, for internal nodes, there is a unique way to fill in.

DEF: a quasi-category is a weak Kan complex, that, a sSet such that any horn with an internal open facet missing, exists (maybe non-unique) filling.

So, there is no unique composition, but many possible such composition. (who would use such a weird kind of category, why we cannot compose as the usual way?) But any two such filling can be related by an interpolation, which is again not unique, but non-uniqueness is killed by something higher.

So, composition is unique up to isotopy.

What the hell is that $C[\Delta^n]$? A simplicial category (category with hom given by sSet), objects are $0, 1, \cdots, n$, for $i \leq j$, we define $Map_C(i,j)$ be the nerve of the poset of detailed jumpy path from $i$ to $j$, smallest is the 1-step jump, and longest is the smallest jump. Why do we do that? For some reason, I don't care.

OK, colimit now. If we have a functor from one indexing category $I$ to a target category $C$, we need to find universal object $X$ in $C$, such that any other objects that receives the map from $I$ in $C$, factors through $X$.

His note has a typo, that $\Delta^n \star \Delta^m = \Delta^{n+m+1}$.

So, the colimit of $F: I \to C$, is just the initial object in all possible filling. The space of filling, or cone $C_{F/}$. well-defined upto contractible choices, so what. no big deal.

filtered $\infty$-cat. instead of just filling 2 points to a wedge, and a coffee filter shape, one fill $S^n$ to $D^{n+1}$.

filtered colimit and finite limit commute, in the $\infty$ category of spaces. (aha, we indeed replaced set by spaces)

CoCartesian lift. Given a map of sSet $p: U \to D$, and a morphism $f: x \to y$ in $D$. We say a morphism $\wt f: \wt x \to \wt y$ over $f$ is nice, if for any triangle with an edge $f$, there is a unique lift with edge $\wt f$ , if we lift the edge $x \to z$ to $\wt x \to \wt z$.

so, a local system of category, roughly means a co-cartesian fibration (leave the detail to experts...)

relative nerve. Suppose you have a category $J$, and you have a functor $J \to sSet$. A relative nerve is like: a nerve in $J$ but with a section of objects up there, so that all possible fillings choices exists and are made.

sort of ok, just remember someone had defined a 'local system of category over an ordinary category'. Or, just a functor from the base category to $Cat_\infty$. in this language, it is not so fancy.

but why we care about these terminology?

monoidal $\infty$-category. just can tensor.

What is a monoidal $\infty$-category? Why do I need that?

In the old story, if $C$ is a category, then $End(C)$ is a monoidal category, where two objects can compose (monoidal structure), and two endofunctor can define natural transformation.

To define the monoidal structure, you just define the monoidal structure.

I just escaped the confusing $\infty$-cat hell. Did Emily Riehl do a better job of black boxing these? Not sure, still scary table of content.

giving up