Non-Hausdorff Mapping Cylinder

• Non-Hausdorff mapping cylinder is a construction in algebraic topology and poset theory that encodes relationships between finite T₀-spaces via order relations that violate Hausdorff separation.
• It uses the Alexandrov topology where minimal open neighborhoods act as down-sets, enabling functorial mappings and canonical inclusions that extend order-preserving maps.
• Its homotopical properties allow the structure to collapse onto constituent spaces when inverse or forward fibers are contractible, thereby generalizing Quillen’s Theorem A and nerve theorem applications.

A non-Hausdorff mapping cylinder is a construction—originating in algebraic topology and poset theory—that generalizes the classical mapping cylinder to settings where the underlying spaces, often finite posets or Alexandroff spaces, do not satisfy Hausdorff separation axioms. The central idea is to represent the interplay between two finite T₀-spaces (or posets) linked by a relation, yielding a topological or combinatorial object that encodes both the original spaces and their interrelation while often failing to be Hausdorff. This construction is fundamental for generalizations of key homotopical results, including Quillen’s Theorem A and modern Nerve theorems, and allows a uniform combinatorial framework for topology and applied studies such as Mapper theory.

1. Construction of the Non-Hausdorff Mapping Cylinder

Given finite posets $(X, \le_X)$ and $(Y, \le_Y)$ and any relation $R\subseteq X\times Y$, the non-Hausdorff mapping cylinder (or, equivalently, the "relation cylinder" $B(R)$) is defined as the poset whose underlying set is the disjoint union $B(R)=X\sqcup Y$. The ordering is as follows:

• On elements of $X$ and $Y$, inherit their original partial orders.
• For $x\in X$ and $y\in Y$, declare $x\leq y$ in $B(R)$ whenever there exist $x'\geq_X x$ and $y'\leq_Y y$ such that $(x',y')\in R$.

This construction extends the classical mapping cylinder of an order-preserving map $f:X\to Y$, which is recovered when $R$ is the graph $\Gamma(f)=\{(x, f(x))\}$, yielding $B(f)$ with $x\leq y$ if and only if $f(x)\leq_Y y$ in addition to the native orders on $X$ and $Y$ (Fernández et al., 2018, Das et al., 2024).

The topology on $B(R)$ is the Alexandrov topology: minimal open neighborhoods correspond to down-sets for each point. For $x\in X$, $U_x=\{z: z\leq x\}$; for $y\in Y$, $U_y=\{z: z\leq y\}$. In general, $B(R)$ is not Hausdorff; separation fails when $x\leq y$ is introduced via $R$, forbidding disjoint open neighborhoods for $x$ and $y$.

2. Functoriality and Universal Properties

The assignment $R\mapsto B(R)$ is functorial, defining a functor from the category of posets with relations (Rel(Posets)) to the category of Posets. This functor admits natural transformations—canonical inclusions $i_X: X\to B(R)$ and $j_Y:Y\to B(R)$. For any poset $Q$ and order-preserving maps $\alpha:X\to Q$, $\beta:Y\to Q$ satisfying the compatibility $xRy\implies \alpha(x)\leq_Q \beta(y)$, there exists a unique order-preserving map $\Phi:B(R)\to Q$ extending both $\alpha$ and $\beta$. Thus, $B(R)$ serves as the pushout in the 2-category of posets of the diagram $X\leftarrow R\rightarrow Y$ (Fernández et al., 2018).

3. Homotopical and Collapse Properties

The central homotopical feature of the non-Hausdorff mapping cylinder is its ability to interpolate and relate the homotopy types of $X$ and $Y$. For any $y\in Y$ and $x\in X$, define the fibers:

• $$R^{-1}(U_y)=\{x\in X: xRy'\text{ for some } y'\le_Y y\}$$,
• $R(F_x)=\{y\in Y: x'Ry\text{ for some } x'\ge_X x\}$.

The following key properties hold:

• If each $R^{-1}(U_y)$ is contractible (or collapsible), then $B(R)$ collapse-retracts to $X$ and their order complexes are homotopy equivalent.
• If each $R(F_x)$ is contractible (or collapsible), then $B(R)$ collapses onto $Y$.
• If both fiber families are contractible (or collapsible), $X$, $B(R)$, and $Y$ are all mutually simple-homotopy equivalent (Das et al., 2024).

This collapse mechanism underlies generalizations of Quillen’s Theorem A and is essential in proofs of Nerve theorems for posets and finite spaces (Fernández et al., 2018).

4. Multiple Cylinder of Relations

The concept generalizes to sequences of spaces and relations. Let $X_0,\dots,X_n$ be finite T₀-spaces, with relations $R_i\subseteq X_i\times X_{i+1}$. The multiple cylinder $B(R_0,\dots,R_{n-1};X_0,\dots,X_n)$ is the union $\bigsqcup X_i$ with native ordering on each $X_i$, and comparabilities $x\leq y$ (for $x\in X_i$, $y\in X_{i+1}$) whenever there exist $x'\geq x$, $y'\leq y$ with $x' R_i y'$. No additional cross-level comparabilities are introduced.

If the composite of the relations $R_{n-1}\circ\dots\circ R_0$ has all inverse fibers contractible, then the multiple cylinder collapses to $X_0$; similarly, if all forward fibers are trivial, it collapses to $X_n$. This construction allows the comparison and transfer of homotopical data across chains of spaces (essential in advanced Nerve theorem arguments and complexes arising in Mapper-type constructions) (Das et al., 2024).

5. Comparison with Classical (Hausdorff) Mapping Cylinder and Adjunction Spaces

In classical topology, the mapping cylinder $M(f) = (X\times[0,1])\cup_f Y$ (identifying $(x,1)\sim f(x)$) is Hausdorff when $X$ and $Y$ are, and the gluing is along closed subspaces without boundaries. However, when gluing along a region with boundary or a non-closed subspace, Hausdorffness fails precisely at those boundary points. This behavior is formalized in the adjunction-space theory: Hausdorff violations in $M\cup_\phi N$ occur exactly at pairs of boundary points of the gluing regions (O'Connell, 2020). In the finite (combinatorial) setting, the non-Hausdorff mapping cylinder is inherently non-Hausdorff except in trivial situations. Its up-set/Alexandroff topology reflects this, and no separation axiom beyond $T_0$ typically holds.

The table below contrasts the two approaches:

Aspect	Classical Mapping Cylinder	Non-Hausdorff Mapping Cylinder (Relation Cylinder)
Underlying Set	$X\times[0,1]\cup Y$	$X\sqcup Y$
Topology	Hausdorff (if gluing is “tame”)	Alexandroff; rarely Hausdorff
Gluing Mechanism	Points $(x,1)\sim f(x)$	Cross-relations $x\leq y$ via $R$-links
Homotopy Collapses	Retraction onto $Y$ always possible	Collapses to $X$ or $Y$ under fiber triviality

6. Applications to Homotopy Theory and Nerve Theorems

The non-Hausdorff mapping cylinder provides a framework for generalizing Quillen’s Theorem A to relations beyond order-preserving maps. Theorem 2.6 of Fernández–Minian states: if for all $y\in Y$, $R^{-1}(U_y)$ is contractible, and for all $x\in X$, $R(F_x)$ is contractible, then the classifying complexes of $X$ and $Y$ are simple-homotopy equivalent.

This facilitates new versions of the Nerve Theorem. Given a cover $\{U_i\}$ of a poset $X$, construct a relation $R\subset X\times (\mathrm{Nerve}(I))^{op}$ by $x R \sigma$ iff $x\in \bigcap_{i\in \sigma} U_i$. Even when intersections are not globally contractible but decompose into contractible components, the completion of the nerve (labeling each simplex with a contractible component) achieves equivalence of simple-homotopy types. These principles extend naturally to CW complexes and simplicial complexes via the associated order complexes, providing unification between classical topological theorems, Mapper-style invariants, and their combinatorial analogues (Fernández et al., 2018, Das et al., 2024).

7. Concrete Examples and Structural Features

Non-Hausdorffness is transparent in explicit constructions:

• For $X = \{x_1 < x_2\}$, $Y = \{y_1 < y_2\}$, $R = \{(x_1, y_1), (x_1, y_2)\}$, the cylinder $B(R)$ has $x_1 \leq y_1$, $x_1 \leq y_2$, $x_1 \leq x_2$, $y_1 \leq y_2$, and minimal open neighborhoods for $y_2$ contain both $x_1$ and $y_1$, while $y_1$ is in the closure of $\{x_1\}$—demonstrating inseparability (Fernández et al., 2018).
• For the boundary of a triangle (1-skeleton of $\Delta^2$) and a 2-piece cover with intersections that are not contractible, the completion of the nerve (based on the mapping cylinder) restores the correct simple-homotopy type, while the classical nerve fails.

Maximal Hausdorff subspaces in non-Hausdorff mapping cylinders decompose naturally: one component from the "open cylinder" (e.g., $X\times[0,1)$) and another from the target with the problematic glued-in boundaries removed. This decomposition is described rigorously in the adjunction space formalism (O'Connell, 2020).

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References:

• Fernández, X., Minian, E. G. "The cylinder of a relation and generalized versions of the Nerve Theorem" (Fernández et al., 2018)
• O’Connell, J. "Non-Hausdorff Manifolds via Adjunction Spaces" (O'Connell, 2020)
• Recent developments and multiple-relation cylinders: (Das et al., 2024)