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--- blog:2023-03-04-l4l5 [2023/03/05 05:10] – created pzhou
+++ blog:2023-03-04-l4l5 [2023/06/25 15:53] (current) – external edit 127.0.0.1
@@ Line 19: / Line 19: @@
 $$ |X| = \colim_{\Delta^n \to X} \bDelta^n $$
-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 $[0], [1], \cdots$
+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.
+===== L5 =====
+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