nLab regular directed complex

Idea

The notion of regular directed complex, introduced by Hadzihasanovic, is a notion of pasting diagram shape. There, a pasting diagram is encoded as a kind of “directed cell complex”, that is, a cell complex together with an orientation on each cell, which divides its boundary into two halves: an input half and an output half.

Definition

In this framework, a pasting diagram will be encoded as an oriented graded poset (Def. ), which is a poset, whose covering diagram is graded, and each edge possess an orientation. To see this consider the following pasting diagram (referred in this page as the running pasting diagram) and the covering diagram associated to it

Basic order theory

Of course, not all oriented graded posets will lead to a well-formed pasting diagrams: before defining the one that will count, we introduce some terminology.

Definition

(oriented graded poset)
A graded poset is a poset together with a rank function $\dim \colon P \to \mathbb{N}$ such that

• if $x \lt y$, then $\dim x \lt \dim y$,
• if $x$ covers $y$, then $\dim x = \dim y + 1$.
• if $x$ covers no element, then $\dim x = 0$.

An oriented graded poset is a graded poset together with a sign $\alpha \in \{ -, + \}$ for each edge in its covering diagram.

Remark

The grading function of an oriented graded poset defines a notion of dimension of an element. For instance, in the running pasting diagram, the dimensions of $x, f, s$ are repectively $0, 1, 2$. If $P$ is an oriented graded poset, we write $P_n$ for its elements of dimension $n$. We speak also of the dimension of an oriented graded poset to mean the maximal of the dimension of its elements.

Definition

(faces and cofaces)
Let $P$ be an oriented graded poset. For all $x$ in $P$, and $\alpha = -$ (resp. $\alpha = +$), we define the input (resp. output) faces

$$\Delta^\alpha x \coloneqq \{ y \in P \mid x \;\text{covers}\; y \;\text{with orientation}\; \alpha \}.$$

Dually, we define the input (reps. output) cofaces

$$\nabla^\alpha x \coloneqq \{ y \in P \mid y \;\text{covers}\; x \;\text{with orientation}\; \alpha \}.$$

Definition

($\mathbf{ogPos})$
With that, we can define a morphism $f \colon P \to Q$ of oriented graded posets to be a function of the underlying sets such that, for all $x \in P$ and $\alpha \in \{ -, + \}$, $f$ induces a bijection between $\Delta^\alpha x$ and $\Delta^\alpha f(x)$.

Oriented graded posets and morphisms form the category $\mathbf{ogPos}$.

Proposition

Morphisms of oriented graded posets are order-preserving, closed and dimension-preserving.

Definition

(maximal element)
An element $x$ of an oriented graded poset is maximal if for $\alpha \in \{ -, + \}$, $\nabla^\alpha x = \emptyset$, that is, $x$ is maximal if it is covered by no element.

Definition

(boundaries)
Let $U$ be a closed subset of an oriented graded poset, let $\alpha = -$ (resp. $\alpha = +$), and let $n \in \mathbb{N}$, let

$$\Delta^\alpha_n U \coloneqq \{ x \in U_n \mid \nabla^{- \alpha} x \cap U = \emptyset \}.$$

The input (reps. output) $n$-boundary is the closed subset

$$\partial^\alpha_n U \coloneqq \mathrm{cl}(\Delta^\alpha_n U) \cup \bigcup_{k \lt n} \mathrm{cl}(\mathrm{Max} U)_k.$$

Example

In the running pasting diagram, calling it $U$, we have

$$\partial^-_1 U = \{ x, f, y, g, z \},$$

and

$$\partial^+_0 U = \{ z \}.$$

Notation

We write

$$\partial_n U \coloneqq \partial^-_n U \cup \partial^+_n U ,$$

$$\Delta U \coloneqq \Delta^- U \cup \Delta^+ U,$$

etc… The general convention is that if the superscript $\alpha$ is not provided to an operator, it is the union of the operator with $\alpha = -$, and with $\alpha = +$.

Similarly, if the subscript $n$ is not provided, it means “the dimension of the oriented graded poset minus 1”, that is, we write:

$$\partial^- U \coloneqq \partial^-_{\dim U - 1} U.$$

We can also combine both notations, for instance

$$\partial U = \partial^-_{\dim U - 1} U \cup \partial^+_{\dim U - 1} U.$$

Last, if the index $n$ is negative, the result is the empty set, for instance,

$$\partial^-_{- 1} U \coloneqq \emptyset,$$

etc.

Remark

For any $\alpha, n$, $\partial^\alpha_n U$, $\partial_n U$, etc.. are subsets of $U$, and the inclusion they defines

$$\partial^\alpha_n U \hookrightarrow U \quad\quad \partial_n U \hookrightarrow U \quad\quad \text{etc},$$

are all morphisms of oriented graded posets.

The last bit of terminology we introduce is roundness. Consider a topological $n$-ball $U$, and its boundary the $n$-sphere $\partial U$. The latter is made of the gluing of two $(n - 1)$-balls $\partial^- U$ and $\partial^+ U$ along an $(n - 1)$-sphere $\partial_{n - 2} U$, in particular:

$$\partial^- U \cap \partial^+ U = \partial_{n - 2} U.$$

Similarly, $\partial U$ is an $(n - 1)$-sphere, which is made of a copie of two $(n - 2)$-balls which intersect at an $(n - 2)$-sphere, etc.. We will say that an oriented graded poset is round if it satisfies similar intersection conditions.

Definition

(round)
Let $P$ be an oriented graded poset of dimension $n$. We say that $P$ is round if for all $0 \le k \lt n$, we have

$$\partial^-_k P \cap \partial^+_k P = \partial_{k - 1} P.$$

Inductive construction of molecules

We can now construct inductively a well-formed class of oriented graded posets, called the molecules, which model composable pasting diagrams. There are 3 constructors:

• (Point) the oriented graded poset $\mathbf{1}$ with a single point is a molecule.
• (Paste) if $U, V$ are two molecules of dimensions $\gt n$ such that $\partial^+_n U \cong \partial^-_n V$, we construct the following pushout:

Then $U \#_n V$ is a molecule.

• (Atom) if two molecules are round, of the same dimension $n$, such that there is an isomorphism $\partial U \cong \partial V$ which restricts to isomorphisms $\partial^\alpha U \cong \partial^\alpha V$ for $\alpha \in \{ -, + \}$, we can glue the boundaries together to form an “$n$-sphere”, i.e. we construct the pushout

Then, we add a top element $\top$ to $\partial(U \Rightarrow V)$ with orientation such that $\Delta^- \top \coloneqq \mathrm{Max} U$ and $\Delta^+ \top \coloneqq \mathrm{Max} V$. This defines $U \Rightarrow V$, which is a molecule.

Examples

If we take two copies of (Point) and apply (Atom) to them, we obtain the arrow $\vec I \coloneqq (\mathbf{1} \Rightarrow \mathbf{1})$: Then, we can use (Paste) with $\vec I$ and $\vec I$ on the 0-boundary to obtain $\vec I \#_0 \vec I$:

Now we use (Atom) to construct $(\vec I \#_0 \vec I) \Rightarrow \vec I$: (which is also the second oriental).

Definition

(atom)
An atom is a molecule with a maximal element.

Lemma

If a molecule $U$ is an atom, then $U$ was produced by either (Point) or (Atom).

Definition

(regular directed complex)
A regular directed complex is an oriented graded poset $P$ such that for all $x$ in $P$, $\mathrm{cl}(x)$ is an atom.

Of course, every molecule is a regular directed complex (see Lemma ), but for instance, is a regular directed complex but not a molecule.

Properties

It may seem that the above construction of molecules depends on the choice of an isomorphism during the construction. However we prove already:

Proposition

Isomorphisms of molecules are unique.

Proof

See (Corollary 3.4.12)

We will write $U = V$ to mean that there exists a unique isomorphism between $U$ and $V$.

Proposition

The underlying poset of any regular directed complex $P$ is oriented thin. This means that any interval $[x, y]$ in $P$ with $\dim x + 2 = \dim y$ contains exactly 4 elements $x \lt t, t' \lt y$, with orientation satisfying

Remark

The above Proposition is still still true when considering the augmentation of $P$ (see Def. ). This additional result translates into the fact that any 1-dimensional element in $P$ has exactly one negative face and one positive face.

Common properties of molecules include:

Proposition

Let $U$ be a molecule. Then

• for all $\alpha \in \{ -, + \}$ and $n \geq \dim U$, $\partial^\a_n U = U$. In fact, $\dim U = \min \{ n \geq 0 \mid \partial^\a_n U = U \}$.
• for all $\alpha \in \{ -, + \}$ and $n \geq 0$, $\partial^\a_n U$ is a molecule, and if $n \le \dim U$, then $\dim \partial^\a_n U = n$;
• $U$ is connected, that is, $U$ is non-empty, and for all closed subsets $V, W \subseteq U$ such that $U = V \cup W$ and $V \cap W = \emptyset$, then $V = \emptyset$ or $W = \emptyset$;
• if $U$ is round, then $\mathrm{Max}(U) = U_{\dim U}$ (we say that $U$ is pure).
• if $U$ is an atom, then $U$ is round.

Strict $\omega$-categorical structure

The category of molecule, with pastings $\#_n$ and boundaries $\partial^\alpha_n$ satisfies all the axioms of strict $\omega$-categories. More precisely, we have the following results.

Proposition

(globularity)
Any molecule $U$ is globular, that is, for all $\alpha, \beta \in \{ -, + \}$ and all $k \lt n$, we have

$$\partial^\alpha_k(\partial^\beta_n U) = \partial^\alpha_k U.$$

Proposition

(boundaries)
Let $U, V$ be molecules such that $U \#_k V$ is defined, let $\alpha \in \{ -, + \}$ and $n \geq 0$, then

$$\partial^\alpha_n (U \#_k V) = \begin{cases} \partial^\alpha_n U = \partial^\alpha_n V & \text{if}\; n \lt k, \\ \partial^-_k U &\text{if}\; n = k, \alpha = -, \\ \partial^+_k V &\text{if}\; n = k, \alpha = +, \\ \partial^\alpha_n U \#_k \partial^\alpha_n V &\text{if}\; n \gt k. \end{cases}$$

Proposition

(unitality)
Let $U$ be a molecule of dimension $n$, then for all $k \lt n$, we have

$$\partial^-_k U \#_k U = U = U \#_k \partial^+_k U.$$

Remark

Notice that in the constructor (Paste), we require that $\dim U, \dim V \gt k$ for $U \#_k V$ to be defined. However, as long as $\partial^+_k U = \partial^-_k V$, the pushout in (Paste) makes sense, so we slightly overload the notation.

Proposition

(associativity)
Let $U, V, W$ be molecules such that $(U \#_k V) \#_k W$ or $U \#_k (V \#_k W)$ is defined, then

$$(U \#_k V) \#_k W = U \#_k (V \#_k W).$$

Proposition

(exchange)
Let $U, V, U', V'$ be molecules, let $k \lt n$ such that $(U \#_n U') \#_k (V \#_n V')$ is defined, then

$$(U \#_n U') \#_k (V \#_n V') = (U \#_k V) \#_n (U' \#_k V').$$

Corollary

The set of all molecules is a strict $\omega$-category, with

• source and target given by the boundaries;
• composition given by the pasting operations $\#_n$;

In fact, we can do better, for $P$ an oriented graded poset, we let $\mathrm{Mol} / P$ to be the category whose objects are (the isomorphism classes of) morphisms $f \colon U \to P$, for $U$ a molecule. Precomposing $f$ by $\partial^\alpha_n U \hookrightarrow U$ defines $\partial^\alpha_n f$, while if $g \colon V \to P$ is in $\mathrm{Mol} / P$, such that $\partial^+_n f = \partial^-_n g$, then the universal property of the pushout defines a unique element $f \#_k g \colon U \#_n V \to P$ in $\mathrm{Mol} / P$. Hence:

Proposition

For all oriented graded poset $P$, $\mathrm{Mol} / P$ is a strict $\omega$-category. Furthermore, any morphism $f \colon P \to Q$ induces a strict $\omega$-functor

$$\mathrm{Mol} / f \colon \mathrm{Mol} / P \to \mathrm{Mol} / Q$$

sending $U \to P$ to $U \to P \stackrel{f}{\to} Q$. This defines a functor $\mathrm{Mol} / - \colon \mathbf{ogPos} \to \mathbf{\omega Cat}$.

Submolecule and layering

Submolecule

Definition

(submolecule inclusion) The class of submolecule inclusions is the smallest subclass of inclusions of molecules containing all isomorphisms, closed under composition, and for all molecules $U, V$ such that $U \#_n V$ is defined, both $U \hookrightarrow U \#_n V$ and $V \hookrightarrow U \#_n V$ are submolecule inclusions.

Not all inclusions of molecule are submolecule inclusion, but notably, we have the following:

Proposition

• Any boundary inclusion $\partial^\alpha_n U \hookrightarrow U$ is a submolecule inclusion;
• If $V$ is an atom, then any inclusion $V \hookrightarrow U$ is a submolecule inclusion.

Notation

Let $U, V$ be two molecules such that $U \subseteq V$. We write $U \sqsubseteq V$ to signify that the canonical inclusion $U \hookrightarrow V$ is a submolecule inclusion. For instance, by the above Proposition, for all $x \in V$, $\mathrm{cl}(x) \sqsubseteq V$.

Definition

(rewritable submolecule)
Let $U$ be a molecule. A submolecule $V \sqsubseteq U$ is rewritable if $V$ is round (Def. ) and $\dim V = \dim U$.

Rewritable submolecule can be substituted: if $V \sqsubseteq U$ is rewritable and if $W$ is another molecule such that $V \Rightarrow W$ is defined, then we can “replace” $V$ by $W$ inside $U$, whose result we write $U[W / V]$. We can construct this formally in two steps: first we take the following pushout where $\iota$ is the submolecule inclusion $V \sqsubseteq U$. Second, $U \cup (V \Rightarrow W)$ can be interpreted as a higher-dimensional rewrite rule that matches $V$ in $U$ and transforms it into $W$, so we define the substition to be the “result” of this rewrite rule, i.e.:

$$U[W / V] \coloneqq \partial^+_n U \cup (V \Rightarrow W).$$

The choice of terminology “substitution” and “rewriting”, is more than a mere analogy: rewritable submolecules and their interpretations for rewriting system are discussed in there.

Layering

Definition

(layering)
Let $U$ be a molecule, let $-1 \leq k \lt \dim U$, and

$$m \coloneqq \card(\bigcup_{i \gt k} (\mathrm{Max} U)_i).$$

A $k$-layering of $U$ is a sequence $(U^{(i)})_{1 \le i \le m}$ such that

$$U = U^{(1)} \#_k \ldots \#_k U^{(m)}$$

and $\dim U^{(i)} \gt k$.

Example

For instance let $O^1$ be $\mathbf{1} \Rightarrow \mathbf{1}$ (i.e. $O^1 = \vec I$) and $O^2$ be $O^1 \Rightarrow O^1$, then the molecule has no -1-layering, one 0-layering

$$O^2 \#_0 O^1 \#_0 O^2,$$

and two 1-layerings: calling $U \coloneqq (O^2 \#_0 O^1 \#_0 O^1)$ and $U' \coloneqq (O^1 \#_0 O^1 \#_0 O^2)$, then the molecule is both equal to $U \#_1 U'$ and $U' \#_1 U$.

The running pasting diagram has only on 0-layering.

Definition

(layering dimension)
Let $U$ be a molecule, the layering dimension of $U$ is the integer

$$\mathrm{lydim} U \coloneqq \min \{ k \geq -1 \mid \card(\bigcup_{i \gt k + 1} (\mathrm{Max} U)_i) \le 1 \}$$

The main theorem about layerings is as follows:

Theorem

Let $U$ be a molecule, $\mathrm{lydim} \le k \lt \dim U$. Then $U$ admits a $k$-layering.

Other results include:

Proposition

Let $U$ be a molecule, and let $(U^{(i)})_{1 \le i \le m}$ be a $k$-layering of $U$. Then

• $k = -1$ if an only if $U$ is an atom;
• if $k \geq 0$, then $m \gt 1$ and $\mathrm{lydim} U^{(i)} \lt k$;
• $U$ admits an $l$ layering for $k \le l \lt \dim U$.

Furthermore, if $V$ is another molecule such that $U \#_n V$ is defined, then

$$\mathrm{lydim} (U \#_n V) \geq \max \{ \mathrm{lydim} U, \mathrm{lydim} V, n \}.$$

Operations

Regular directed complexes are closed under many relevant operations used for higher category theory.

Gray product

Definition

(Gray product)
Let $P, Q$ be oriented graded posets. The Gray product of $P$ and $Q$ is the oriented graded poset $P \otimes Q$ whose

• underlying graded poset is the cartesian product $P \times Q$,
• orientation is defined, for all $(x, y) \in P \times Q$ and $\alpha \in \{ -, + \}$, by the equation $\Delta^\alpha(x, y) = \Delta^\alpha x \times \{ y \} + \{ x \} \times \Delta^{(-)^{\dim x} \alpha} y$.

Example

The Gray product $\vec{I} \otimes \vec{I}$ is

Theorem

Let $P, Q$ be regular directed complexes. Then $P \otimes Q$ is a regular directed complex.

Proof

The proof is extremely combinatorial, and can be found in Section 7.2.

The Gray product is associtive and has unit the point $\mathbf{1}$. It is not symmetrical.

Join

Similarly, we can take the join of regular directed complexes. If the underlying poset of a join is easy to describe, describing the precise orientation can be quite cumbersome; since we have the Gray product, we can take the following shortcut.

Definition

(augmentation and diminution)
Let $P$ be an oriented graded poset.

• The augmentation of and $P$ is the oriented graded poset $P_\bot \coloneqq P \cup \{ \bot \}$ such that $\nabla^+ \bot = P_0$ and $\nabla^- \bot = \emptyset$.
• Suppose $P$ has a minimal element $\bot$, then the diminution of $P$ is the oriented graded poset $P_{\not{\bot}} \coloneqq P \backslash \{ \bot \}$.

Definition

(join)
Let $P, Q$ be an oriented graded poset. The join of $P$ and $Q$ is the oriented graded poset

$$P \star Q \coloneqq (P_{\bot} \otimes Q_{\bot})_{\not\bot}.$$

Proposition

The underlying poset of $P \star Q$ has elements

$$\{ x\star \mid x \in P \} \cup \{ x \star y \mid x \in P, y \in Q \} \cup \{ \star y \mid y \in Q \}.$$

As in topology, it really consist of a copy of $P$ (whose elements are of the form $x\star$), a copy of $Q$ (whose elements are of the form $\star y$), and every element of $x$ of $P$ is linked to every element $y$ of $Q$ with $x \star y$.

Theorem

Let $P, Q$ be regular directed complexes. Then $P \star Q$ is a regular directed complex.

The join is associtive and has unit $\emptyset$, the empty regular directed complex. It is not symmetrical.

Example

The $n$-th iterated join $\mathbf{1} \star \ldots \star \mathbf{1}$ of the point is the $(n - 1)$-th oriental.

Suspension

Definition

(suspension)
Let $P$ be an oriented graded poset. The suspension of $P$ is the oriented graded poset $\mathsf{S}P$ whose

• underlying set is
  $$\{ \mathsf{S}x \mid x \in P \} \cup \{ \bot^-, \bot^+ \},$$
• partial order and orientation are defined by
  $$\nabla^\alpha x \coloneqq \begin{cases} \{ \mathsf{S}y \mid y \in \nabla^\alpha x' \} & \text{if}\; x = \mathsf{S}x', x' \in P, \\ \{ \mathsf{S}y \mid y \in P_0 \} & \text{if}\; x = \bot^\alpha, \\ \emptyset & \text{if}\; x = \bot^{-\alpha}. \end{cases}$$

  for all $x \in \mathsf{S}P$ and $\alpha \in \{ -, + \}$.

Theorem

Let $P$ be a regular directed complex. Then $\mathsf{S}P$ is a regular directed complex.

Proposition

Let $U, V$ be molecules.

• suppose $U \#_k V$ is defined, then $\mathsf{S}(U \#_k V) = \mathsf{S}U \#_{k + 1} \mathsf{S}V$.
• suppose $U \Rightarrow V$ is defined, then $\mathsf{S}(U \Rightarrow V) = (\mathsf{S}U \Rightarrow \mathsf{S}V)$.

Example

The $n$-th iterated suspension $\mathsf{S}\ldots\mathsf{S}\mathbf{1}$ is the $n$-th globe.

Geometric realization

Since every regular directed complex is in particular a poset, we can take its nerve, which is a simplicial set $P^\Delta$, whose $n$-simplices are chains $x_0 \le \ldots \le x_n$ in $P$. Via the classical geometric realization of simplicial sets, we define a functor $|-|$ from regular directed complexes to (a convenient category of) topological spaces.

As expected, round molecules are round:

Proposition

Let $U$ be a round molecule of dimension $n$, then

• $|U|$ is an $n$-ball,
• $|\partial U|$ is an $n$-sphere.

Furthermore, Gray products, joins, and suspensions are sent to their topological counterpart:

Proposition

Let $P, Q$ be regular directed complexes, then

• $|P \otimes Q| \cong |P| \times |Q|$, where $\times$ is the product of topological spaces;
• $|P \star Q| \cong |P| \star |Q|$, where $\star$ is the join of topological spaces;
• $|\mathsf{S}P| \cong \mathsf{S}|P|$, where $\mathsf{S}$ is the suspension of topological spaces.

Proof techniques

Structural induction

Since molecules are defined inductively, we can use structural induction to prove statement about them. To prove that a statement $\mathcal{P}$ is true for all molecules, we can do as follows:

• We prove that $\mathcal{P}$ is true for the (Point) $\mathbf{1}$.
• We let $U, V$ be molecules such that $U \#_n V$ is defined, and we suppose that $\mathcal{P}$ holds for both $U$ and $V$. Then we prove that $\mathcal{P}$ holds for $U \#_n V$.
• We let $U, V$ be molecules such that $U \Rightarrow V$ is defined, and we suppose that $\mathcal{P}$ holds for both $U$ and $V$. Then we prove that $\mathcal{P}$ holds for $U \Rightarrow V$.

For instance:

Lemma

Any molecule is a regular directed complex.

Proof

Let $U$ be a molecule, we need to prove that for all $x \in U$, $\mathrm{cl}(x)$ is an atom. We proceed by induction on the construction of $U$. If $U$ was produced by (Point), then $x$ must be the unique element of $U$ whose closure is $U$ itself. If $U$ was produced by (Paste), then $U = V \#_k W$ , and $x \in V$ or $x \in W$ ; the inductive hypothesis applies. If $U$ was produced by (Atom), it is equal to $(V \cup W ) + \{ \top \}$, and either $x \in V$ or $x \in W$, in which case the inductive hypothesis applies, or $x = \top$, and $\mathrm{cl}(x) = U$ is an atom by definition.

Other methods

The proof by induction on submolecules allows us to prove a statement $\mathcal{P} \implies \mathcal{Q}$ for all molecules as follows: assume that a molecule $U$ satisfies $\mathcal{P}$, then

• we prove that $\{ x \}$ satisfies $\mathcal{Q}$ for all $x \in U_0$ (the base cases);
• we prove that $U$ satisfies $\mathcal{Q}$ under the assumption that every proper submolecule $V \sqsubseteq U$ satisfies $\mathcal{Q}$

This induction principle is justified since every proper submolecule of $U$ has stricly fewer elements than $U$, and the set of minimal elements for $\sqsubseteq$ is presicely $U_0$.

Example

The proof that molecules are stable under Gray product uses induction on submolecules.

Another proof technique is the induction on the layering dimension: to prove that a property holds for all molecules U, it suffices to

• prove that it holds when $\mathrm{lydim} U = -1$, i.e. when $U$ is an atom (base case);
• prove that it holds when $k \coloneqq \mathrm{lydim} U \geq 0$, assuming that it holds of all the $(U^{(i)})_{1 \le i \le m}$ in any $k$-layering of $U$.

Indeed, by Theorem , any molecule has a layering, and Proposition gives the well-foundedness.

For instance, let us prove the following:

Proposition

Every molecule is made of pasting of atoms.

Proof

Let $U$ be a molecule, we proceed by induction on $k \coloneqq \mathrm{lydim} U$. If $k = -1$, then this follows by definition since $U$ is an atom. Let $k \gt 0$, and take a $k$-layering of $U$

$$U = U^{(1)} \#_k \ldots \#_k U^{(m)},$$

with $\mathrm{lydim} U^{(i)} \lt k$. Then by induction, each $U^{(i)}$ is a composition of atoms, thus so is $U$.

Remark

The proof works equally well by structural induction.

References

The main reference on regular directed complexes:

• Amar Hadzihasanovic, Combinatorics of higher-categorical diagrams, 2024 (link)

For a computational perspective on regular directed complexes:

• Diana Kessler, Amar Hadzihasanovic, Higher-dimensional subdiagram matching, LICS 2023 (link)