Quivers with Homotopies

• Quivers with homotopies are algebraic structures that extend directed graphs with homotopical data via relations, sections, and subgroupoids.
• They reconstruct classical homological algebra by replacing traditional homotopy groups with Ext-groups through normalization and Morita equivalence.
• In mutation theory, these quivers govern the deletion of oriented 2-cycles and preserve quotient groupoids, enabling an involutive mutation framework.

“Quivers with homotopies” denotes a family of quiver-enriched formalisms in which a directed graph is supplemented by algebraic data carrying homotopical content. In one direction, classical simplicial, cyclic, cubical, and Hochschild theories are recast as a Q-homotopy theory built from quivers with relations and sections, where normalization is implemented by Morita-equivalent “breaking” operations and homotopy groups are replaced by $\operatorname{Ext}$-groups over a basic algebra (Fei, 2012). In another direction, a loop-free quiver is paired with a normal subgroupoid of its fundamental groupoid; this homotopy controls the deletion of oriented $2$-cycles under mutation, yields an involutive mutation theory, and preserves the quotient groupoid $\pi(Q)/H$ (Li et al., 20 Aug 2025). A further, distinct usage occurs for graded quivers in the homotopy theory of pre-Calabi–Yau morphisms, where homotopies are Maurer–Cartan homotopies in suitable $L_\infty$-algebras (Boucrot, 2024).

1. Terminological scope and basic structures

The expression is not attached to a single universally adopted definition. The principal usages in the cited literature are organized by the additional structure imposed on the quiver and by the mechanism through which “homotopy” is realized.

Framework	Basic data	Homotopical role
Q-homotopy theory	$\mathbb Q=(Q,R,S)$ with relations and sections	normalization, stable-category quotient, $\operatorname{Ext}$-based invariants
Mutation-theoretic quiver with homotopy	$(Q,H)$ with $H\triangleleft \pi(Q)$	controls deletion of $2$-cycles during mutation
Graded-quiver pre-Calabi–Yau theory	graded quiver plus Maurer–Cartan data	homotopy of morphisms via $L_\infty$ path objects

In the Q-homotopy framework, the starting point is a quiver with relations and sections, abbreviated QRS. One fixes

$2$0

where $2$1 is a finite quiver, $2$2 is a set of relations in the path algebra $2$3, and $2$4 is a designated collection of arrows $2$5 satisfying

$2$6

Here $2$7 is called a projection and $2$8 its section. This makes the quiver a categorical skeleton of coface and codegeneracy operators rather than only a combinatorial graph (Fei, 2012).

In the mutation-theoretic framework, a quiver with homotopy is a pair $2$9 where $\pi(Q)/H$0 is loop-free and $\pi(Q)/H$1 is a normal subgroupoid of the free groupoid $\pi(Q)/H$2. The condition

$\pi(Q)/H$3

and the conjugacy rule

$\pi(Q)/H$4

for every $\pi(Q)/H$5 make $\pi(Q)/H$6 a family of normal subgroups transported along paths. The pair is called reduced if it contains no $\pi(Q)/H$7-cycles of $\pi(Q)/H$8 (Li et al., 20 Aug 2025).

A related but formally different setup appears for graded quivers $\pi(Q)/H$9, specified by a set of objects $L_\infty$0 and graded vector spaces $L_\infty$1 for each pair $L_\infty$2. There, pre-Calabi–Yau structures and morphisms are expressed as Maurer–Cartan elements in $L_\infty$3-algebras built from cyclic multilinear operations on the underlying quiver (Boucrot, 2024).

2. Q-homotopy theory via relations, sections, and breaking

The central operation in Q-homotopy theory is breaking at a section. If $L_\infty$4 is a section, then

$L_\infty$5

produces a Morita equivalent algebra in which one vertex has been simplified. Iterating breakings along a sequence $L_\infty$6 gives

$L_\infty$7

The resulting algebra may be crisp or weakly crisp, depending on how completely the sequence reduces the original presentation (Fei, 2012).

This normalization process leads to a basic algebra $L_\infty$8 and a normalized quiver complex functor

$L_\infty$9

Passing to the stable category produces the Q-homotopy functor

$\mathbb Q=(Q,R,S)$0

The associated homotopy and cohomotopy groups are defined by

$\mathbb Q=(Q,R,S)$1

These formulas replace the classical derivation of homotopy and cohomology from simplicial objects by a representation-theoretic construction over the normalized algebra (Fei, 2012).

The breaking functor is constructive at the level of modules. For a module $\mathbb Q=(Q,R,S)$2,

$\mathbb Q=(Q,R,S)$3

The inverse Morita functor is expressed as $\mathbb Q=(Q,R,S)$4, where $\mathbb Q=(Q,R,S)$5 is built from a product of elementary matrices $\mathbb Q=(Q,R,S)$6. This explicit control is part of the reason the framework can reproduce classical normalization procedures while remaining internal to quiver representation theory (Fei, 2012).

Orthogonal projection through an exceptional object $\mathbb Q=(Q,R,S)$7 yields a reflective subcategory $\mathbb Q=(Q,R,S)$8, together with universal exact sequence diagrams built from a universal homomorphism and universal extension. The fundamental relations

$\mathbb Q=(Q,R,S)$9

play the role of quiver-categorical simplicial identities. This suggests that, in this formalism, homotopy is implemented by orthogonal projections, sections, Morita equivalences, and derived functors rather than by a prior topological realization.

3. Reconstruction of classical homological algebra from quivers

The flagship example is the reconstruction of the simplicial category $\operatorname{Ext}$0 from type $\operatorname{Ext}$1 quivers. The objects $\operatorname{Ext}$2 of $\operatorname{Ext}$3 are identified with module categories of the linearly oriented quivers

$\operatorname{Ext}$4

The coface maps $\operatorname{Ext}$5 are interpreted as orthogonal projections associated with the simple module $\operatorname{Ext}$6, and the codegeneracies $\operatorname{Ext}$7 as the exact embedding right adjoint to $\operatorname{Ext}$8. The simplicial relations become quiver relations: $\operatorname{Ext}$9

$(Q,H)$0

$(Q,H)$1

This is summarized by the statement that classical simplicial theory is a Q-homotopy theory of type $(Q,H)$2 (Fei, 2012).

The Dold–Kan correspondence is treated as a Morita-theoretic equivalence between the algebra of simplicial objects and the algebra of chain complexes. A chain complex is represented by the linear quiver

$(Q,H)$3

with relation $(Q,H)$4, and the truncated correspondence is induced by

$(Q,H)$5

where $(Q,H)$6 is an explicit projective module built from an upper-triangular matrix $(Q,H)$7 called Jia Xian’s triangle (Fei, 2012).

Hochschild chains are likewise recast as a quiver complex. For an algebra $(Q,H)$8, the quiver $(Q,H)$9 with dimension vector $H\triangleleft \pi(Q)$0 encodes a tensor $H\triangleleft \pi(Q)$1, and the face and degeneracy maps are

$H\triangleleft \pi(Q)$2

$H\triangleleft \pi(Q)$3

$H\triangleleft \pi(Q)$4

The Hochschild complex is therefore presented as the quiver complex of the simplicial QEC, with Hochschild homology and cohomology obtained by tensor or $H\triangleleft \pi(Q)$5 with coefficients (Fei, 2012).

The same program extends to cyclic and cubical theories. In the cyclic case one adds a loop $H\triangleleft \pi(Q)$6 and imposes

$H\triangleleft \pi(Q)$7

together with

$H\triangleleft \pi(Q)$8

while the related algebra is governed by mixed-complex relations

$H\triangleleft \pi(Q)$9

In the cubical case, arrows are indexed by $2$0, the relations distinguish the two directions of faces, and the version with permutations includes transpositions $2$1 satisfying

$2$2

The paper also constructs non-type-$2$3 examples, notably type $2$4 QECwd’s with explicit quiver relations and delooped versions such as

$2$5

used to recover spectral-sequence-like long exact sequences (Fei, 2012).

4. Quivers with homotopies as normal subgroupoids and mutation data

For mutation theory, one begins with a quiver $2$6, its double quiver $2$7, and the free groupoid $2$8 whose morphisms are reduced walks in $2$9. This groupoid is naturally isomorphic to the fundamental groupoid $L_\infty$0 of the underlying $L_\infty$1-complex. A homotopy is then a normal subgroupoid of $L_\infty$2, and the quotient groupoid $L_\infty$3 identifies morphisms $L_\infty$4 whenever

$L_\infty$5

If $L_\infty$6 is generated by a finite set $L_\infty$7 of reduced cyclic walks, attaching $L_\infty$8-cells along the loops in $L_\infty$9 produces a $2$00-complex $2$01 with

$2$02

so the homotopy has an explicit geometric realization (Li et al., 20 Aug 2025).

Given a quiver with reduced homotopy $2$03 and a vertex $2$04, pre-mutation first forms the ordinary pre-mutation quiver $2$05: for each $2$06 and $2$07, one adds a new arrow

$2$08

and one reverses every arrow incident to $2$09. The homotopy is transported by a functor

$2$10

defined by

$2$11

and by $2$12 on all other arrows. One then sets

$2$13

At this stage $2$14 may contain oriented $2$15-cycles (Li et al., 20 Aug 2025).

Mutation is obtained by deleting a maximal collection of $2$16-cycles subject to the homotopy criterion that a pair $2$17 is removed precisely when it lies in $2$18. The resulting quiver is denoted $2$19, and its homotopy is

$2$20

The mutation is then

$2$21

Although the deletion step involves choices, the outcome is well-defined up to quiver isomorphism (Li et al., 20 Aug 2025).

Two structural facts organize the theory. First, mutation is involutive: $2$22 Second, mutation preserves the quotient groupoid, since the maps

$2$23

are isomorphisms. Thus the invariant object is not the quiver alone but the quotient groupoid determined by the homotopy (Li et al., 20 Aug 2025).

The framework recovers Fomin–Zelevinsky mutation when $2$24 is $2$25-acyclic and one chooses the maximal homotopy

$2$26

Then $2$27 is trivial, the reduction step deletes exactly the $2$28-cycles that would be deleted in classical mutation, and any mutation sequence with homotopies produces the same quivers as ordinary FZ mutation (Li et al., 20 Aug 2025).

5. Coverings, infinite mutation sequences, and surface realizations

The mutation formalism is compatible with orbit mutations coming from coverings. If

$2$29

is a weakly admissible covering, where $2$30 is $2$31-acyclic, $2$32 is loop-free, and no arrow in $2$33 connects vertices in the same orbit of the deck group $2$34, then an orbit mutation at an orbit $2$35 is obtained by mutating all vertices in $2$36 once and then deleting $2$37-cycles in a $2$38-compatible way. When $2$39 is $2$40-mutable, the orbit mutation induces a quotient quiver isomorphic to the homotopy mutation $2$41 for

$2$42

more precisely through a quiver isomorphism $2$43 such that

$2$44

This identifies homotopy mutation as the quotient-level shadow of orbit mutation (Li et al., 20 Aug 2025).

A notable feature is that homotopy mutations can always be performed indefinitely, even when orbit mutations cease to be weakly admissible after finitely many steps because loops appear in the quotient. A useful sufficient condition for global weak admissibility is

$2$45

where $2$46 is the normal subgroupoid generated by squares of cyclic walks. The same paper proves that for an acyclic quiver $2$47, any homotopy is equivalent to the trivial homotopy, and hence any quiver $2$48 obtained by FZ mutation from an acyclic quiver satisfies

$2$49

These statements show that the homotopy datum is strongest precisely when $2$50-cycles and nontrivial fundamental-group information are present (Li et al., 20 Aug 2025).

The surface model extends the Fomin–Shapiro–Thurston construction to marked bordered surfaces with punctures of two colors. A colored marked surface is written $2$51 with $2$52. From a triangulation $2$53, one builds a preliminary quiver $2$54 using the usual FST puzzle pieces together with a new piece arising from a self-folded triangle enclosing a $2$55-puncture. The final quiver $2$56 is obtained by deleting certain $2$57-cycles only in the case of an $2$58-puncture configuration; $2$59-cycles around $2$60-punctures are retained (Li et al., 20 Aug 2025).

The homotopy $2$61 is generated by oriented $2$62-cycles from puzzle pieces and by oriented cycles $2$63 around each $2$64-puncture of valency at least $2$65. The induced homotopy $2$66 on $2$67 is characterized by

$2$68

The associated $2$69-complex $2$70 satisfies

$2$71

and

$2$72

The tagged arc complex is a pseudomanifold: every maximal tagged triangulation has size

$2$73

every codimension-$2$74 face lies in exactly two maximal faces, and each tagged triangulation may be flipped at any tagged arc. The main theorem states that if $2$75 is the flip of $2$76 at arc $2$77, then

$2$78

This provides a surface-theoretic model for mutation in the presence of oriented $2$79-cycles (Li et al., 20 Aug 2025).

6. Related graded-quiver homotopy formalisms and conceptual distinctions

A different homotopical use of quivers appears in the theory of pre-Calabi–Yau morphisms. There the underlying datum is a graded quiver $2$80 with objects $2$81 and graded vector spaces $2$82, and a $2$83-pre-Calabi–Yau structure is an element

$2$84

of degree $2$85 satisfying the Maurer–Cartan equation

$2$86

Pre-Calabi–Yau morphisms are Maurer–Cartan elements in larger $2$87-algebras built from source, target, and morphism components, and two homotopy notions are defined: weak homotopy for morphisms whose underlying graded-quiver maps agree, and homotopy for morphisms between fixed pre-Calabi–Yau categories (Boucrot, 2024).

The chain-level path object is the standard $2$88 extension $2$89. Two Maurer–Cartan elements are homotopic if there exists

$2$90

with $2$91, $2$92, and

$2$93

Within this formalism, homotopy is stable under left and right composition, homotopy equivalences are quasi-isomorphisms, and the functor

$2$94

sends homotopic pre-Calabi–Yau morphisms to weakly homotopic $2$95-morphisms (Boucrot, 2024).

A common source of confusion is the assumption that “homotopy of a quiver” has a uniform meaning across these literatures. It does not. In Q-homotopy theory, homotopy is extracted from breaking, normalization, stable-category quotients, and $2$96 over a basic algebra (Fei, 2012). In mutation theory, a homotopy is literally a normal subgroupoid of the fundamental groupoid of a loop-free quiver, and the preserved invariant is the quotient groupoid $2$97 (Li et al., 20 Aug 2025). In the pre-Calabi–Yau setting, homotopy refers to Maurer–Cartan homotopy in an $2$98 path object attached to graded-quiver operations (Boucrot, 2024).

Taken together, these constructions show that quivers support several non-equivalent homotopical architectures. One architecture reconstructs classical simplicial and homological algebra from quiver representation theory; another extends quiver mutation beyond the $2$99-acyclic regime by recording which cycles are homotopically trivial; a third organizes higher-categorical morphism theory on graded quivers through Maurer–Cartan deformation theory. A plausible implication is that “quivers with homotopies” functions less as the name of a single theory than as a quiver-theoretic paradigm in which homotopical information is encoded algebraically, either through relations and sections, through subgroupoids of path groupoids, or through higher brackets on graded quiver data.