Exit paths and entry paths through $\infty$-categories

Let $X$ be a topological space, $(A,\leqslant)$ a poset, and $f: X\to (A,\leqslant)$ a continuous map.

Definition: An exit path in an $A$-stratified space $X$ is a continuous map $\sigma: |\Delta^n|\to X$ for which there exists a chain $a_0\leqslant \cdots \leqslant a_n$ in $A$ such that $f(\sigma(t_0,\dots,t_i,0,\dots,0))=a_i$ for $t_i\neq 0$. An entry path is a continuous map $\tau: |\Delta^n|\to X$ for which there exists a chain $b_0\leqslant \cdots \leqslant b_n$ in $A$ such that $f(\tau(0,\dots,0,t_i,\dots,t_n))=b_i$ for $t_i\neq 0$.

Up to reordering of vertices of $\Delta^n$ and induced reordering of the realization $|\Delta^n|$, an exit path is the same as an entry path. The next example describes this equivalence.

Example: The standard 2-simplex $|\Delta^2|$ is uniquely an exit path and an entry path with a chain of 3 distinct elements, stratfied in the ways described below.

Recall the following algebraic constructions, through Joyal's quasi-category model:

• A simplicial set is a functor $\Delta^{op}\to \Set$.
• A Kan complex is a simplicial set satisfying the inner horn condition for all $0\leqslant k\leqslant n$. That is, the $k$th $n$-horn lifts (can be filled in) to a map on $\Delta^n$.
• An $\infty$-category is a simplicial set satisfying the inner horn condition for all $0<k<n$.

Moreover, if the lift is unique, then the Kan complex is the nerve of some category. Recall also the category $\Sing(X) = \{$continuous $\sigma: |\Delta^n|\to X\}$, which can be combined with the stratification $f: X\to A$ of $X$

Remark: The subcategory $\Sing^A(X)$ of exit paths and the subcategory $\Sing_A(X)$ of entry paths are full subcategories of $\Sing(X)$, with $(\Sing^A(X))^{op} = \Sing_A(X)$. If the stratification is conical, then these two categories are $\infty$-categories.

Recall the nerve construction of a category. Here we are interested in the nerve of the category $SC$ of simplicial complexes, so $N(SC)_n = \{$sequences of $n$ composable simplicial maps$\}$. Recall the $k$th $n$-horns, which are compatible diagrams of elements of $N(SC)_n$. In general, they are colimits of a diagram in the category $\Delta$. That is, $$\Lambda^n_k := \colim \left(\bigsqcup_{0\leqslant i<j\leqslant n} \Delta^{n-2} \rightrightarrows \bigsqcup_{0\leqslant i\leqslant n \atop i\neq k} \Delta^{n-1}\right).$$ Example: The images of the 3 different types of 2-horns and 4 different types of 3-horns in $SC$ are given below. Note that they are not unique, and depend on the choice of simplices $S_i$ (equivalently, on the choice of functor $\Delta^{op}\to SC$).

For example, the 0th 2-horn $\Lambda^2_0$ can be filled in if there exists a simplicial map $h: S_1\to S_2$ in $SC$ (that is, an element of $N(SC)_1$) such that $h\circ f = g$. Similarly, the 1st 3-horn $\Lambda^3_1$ can be filled in if there exists a functor $F: [0<1<2]\to SC$ for which $F(0<1)=f_{02}$, $F(0<2)=f_{03}$, and $F(1<2)=f_{23}$ (equivalently, a compatible collection of elements of $N(SC)_2$).

Definition: Let $A,B$ be $\infty$-categories. A functor $F: A\to B$ is a morphism of the simplicial sets $A,B$. That is, $F:A\to B$ is a natural transformation for $A,B\in \text{Fun}(\Delta^{op},\Set)$.

A functor of simplicial sets of a particular type can be identified with a functor of 1-categories. Recall the nerve of a 1-category, which turns it into an $\infty$-category. This construction has a left adjoint.

Definition: Let $\mathcal C$ be an $\infty$-category. The homotopy category $h\mathcal C$ of $\mathcal C$ has objects $\mathcal C_0$ and morphisms $\Hom_{h\mathcal C}(X,Y) = \pi_0(\text{Map}_{\mathcal C}(X,Y))$.

By Lurie, $h$ is left-adjoint to $N$. That is, $h : \sSet \rightleftarrows \text{Cat} : N$, or $\text{Map}_{\sSet}(\mathcal C,N(\mathcal D)) \cong \text{Map}_{\text{Cat}}(h\mathcal C, \mathcal D)$, for any $\infty$-category $\mathcal C$ and any 1-category $\mathcal D$. Our next goal is to describe a functor $\Sing_A(X)\to N(SC)$, maybe through this adjunction, where $SC$ is the 1-category of simplicial complexes and simplicial maps.

References: Lurie (Higher topos theory, Sections 1.1.3 and 1.2.3), Lurie (Higher algebra, Appendix A.6), Goerss and Jardine (Simplicial homotopy theory, Section I.3), Joyal (Quasi-categories and Kan complexes)