--- title: Quivers with Homotopies url: https://www.emergentmind.com/topics/quivers-with-homotopies type: topic --- # Quivers with Homotopies “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 [1211.5789]. 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 $\pi(Q)/H$ [2508.15022]. 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 [2405.20854]. ## 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 \[ \mathbb Q=(Q,R,S), \] where $Q$ is a finite quiver, $R$ is a set of relations in the path algebra $kQ$, and $S$ is a designated collection of arrows $a^r$ satisfying \[ aa^r=e_{ha}. \] Here $a$ is called a projection and $a^r$ its section. This makes the quiver a categorical skeleton of coface and codegeneracy operators rather than only a combinatorial graph [1211.5789]. In the mutation-theoretic framework, a quiver with homotopy is a pair $(Q,H)$ where $Q$ is loop-free and $H$ is a normal subgroupoid of the free groupoid $\pi(Q)$. The condition \[ H(x,y)=\varnothing \text{ unless } x=y \] and the conjugacy rule \[ g^{-1}H(x,x)g=H(y,y) \] for every $g\in G(x,y)$ make $H$ a family of normal subgroups transported along paths. The pair is called reduced if it contains no $2$-cycles of $Q$ [2508.15022]. A related but formally different setup appears for graded quivers $A$, specified by a set of objects $O_A$ and graded vector spaces ${}_xA_y$ for each pair $(x,y)$. There, pre-Calabi–Yau structures and morphisms are expressed as Maurer–Cartan elements in $L_\infty$-algebras built from cyclic multilinear operations on the underlying quiver [2405.20854]. ## 2. Q-homotopy theory via relations, sections, and breaking The central operation in Q-homotopy theory is breaking at a section. If $a^r$ is a section, then \[ Br_a(A)=e'Ae', \qquad e'=e-a^ra \] produces a Morita equivalent algebra in which one vertex has been simplified. Iterating breakings along a sequence $(a_k,\dots,a_1)$ gives \[ Br_{(a_k,\dots,a_1)}=Br_{a_k}\circ\cdots\circ Br_{a_1}. \] The resulting algebra may be crisp or weakly crisp, depending on how completely the sequence reduces the original presentation [1211.5789]. This normalization process leads to a basic algebra $A^b$ and a normalized quiver complex functor \[ \nu: Mod(A)\to Mod(A^b). \] Passing to the stable category produces the Q-homotopy functor \[ \underline{\pi}=q\nu. \] The associated homotopy and cohomotopy groups are defined by \[ \pi_i(T,-)=Ext^i_{A^b}(T,\nu(-)), \qquad \pi^i(-,T)=Ext^i_{A^b}(\nu(-),T). \] These formulas replace the classical derivation of homotopy and cohomology from simplicial objects by a representation-theoretic construction over the normalized algebra [1211.5789]. The breaking functor is constructive at the level of modules. For a module $M$, \[ Br_{(a_k,\dots,a_1)}(M)_v = Coker\!\left(\sum_{ta_i=v} a_i^r\right) = \bigcap_{ta_i=v} Ker(a_i). \] The inverse Morita functor is expressed as $Hom_B(P,-)$, where $P$ is built from a product of elementary matrices $E_i$. This explicit control is part of the reason the framework can reproduce classical normalization procedures while remaining internal to quiver representation theory [1211.5789]. Orthogonal projection through an exceptional object $E$ yields a reflective subcategory $E^\perp$, together with universal exact sequence diagrams built from a universal homomorphism and universal extension. The fundamental relations \[ \tilde{\pi}_F\tilde{\pi}_E=\tilde{\pi}_{E'}\tilde{\pi}_F,\qquad \iota_E\iota_F=\iota_F\iota_{E'},\qquad \tilde{\pi}_F\iota_E=\iota_{E'}\tilde{\pi}_F \] 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 $\Delta$ from type $A$ quivers. The objects $[n]$ of $\Delta^{op}$ are identified with module categories of the linearly oriented quivers \[ A_{n+1}: \quad 1\to 2\to \cdots \to n+1. \] The coface maps $\pi_i$ are interpreted as orthogonal projections associated with the simple module $S_i$, and the codegeneracies $\iota_i$ as the exact embedding right adjoint to $\pi_i$. The simplicial relations become quiver relations: \[ \pi_j\pi_i=\pi_{i-1}\pi_j \quad (ji+1, \end{cases} \] \[ \iota_j\iota_i=\iota_i\iota_{j-1}\quad (j>i). \] This is summarized by the statement that classical simplicial theory is a Q-homotopy theory of type $A$ [1211.5789]. 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 \[ n \xrightarrow{d} n-1 \xrightarrow{d} \cdots \xrightarrow{d} 1 \] with relation $d^2=0$, and the truncated correspondence is induced by \[ R = Hom_{A^b}(P,-): Mod(A^b)\to Mod(A), \] where $P$ is an explicit projective module built from an upper-triangular matrix $T$ called Jia Xian’s triangle [1211.5789]. Hochschild chains are likewise recast as a quiver complex. For an algebra $A$, the quiver $A_{n+1}$ with dimension vector $(1,\dots,1)$ encodes a tensor $a_1\otimes\cdots\otimes a_n$, and the face and degeneracy maps are \[ \partial_1(a_1\otimes \cdots \otimes a_n)=a_2\otimes\cdots\otimes a_n, \] \[ \partial_i(a_1\otimes \cdots \otimes a_n) = a_1\otimes \cdots \otimes a_i a_{i+1}\otimes\cdots\otimes a_n \quad (i>1), \] \[ \sigma_i(a_1\otimes \cdots \otimes a_n) = a_1\otimes\cdots\otimes a_i\otimes 1\otimes a_{i+1}\otimes\cdots\otimes a_n. \] The Hochschild complex is therefore presented as the quiver complex of the simplicial QEC, with Hochschild homology and cohomology obtained by tensor or $\operatorname{Hom}$ with coefficients [1211.5789]. The same program extends to cyclic and cubical theories. In the cyclic case one adds a loop $t$ and imposes \[ \pi_i t=t\pi_{i-1}, \qquad \iota_i t=t\iota_{i-1}, \] together with \[ \pi_1 t=\pi_n,\qquad \iota_1 t=t^2\iota_n, \] while the related algebra is governed by mixed-complex relations \[ d^2=0,\qquad B^2=0,\qquad dB+Bd=0. \] In the cubical case, arrows are indexed by $\pm i$, the relations distinguish the two directions of faces, and the version with permutations includes transpositions $t_i$ satisfying \[ \pi_{i\epsilon} t_i = \pi_{(i+1)\epsilon},\qquad t_i \iota_{i\epsilon} = \iota_{(i+1)\epsilon}. \] The paper also constructs non-type-$A$ examples, notably type $D$ QECwd’s with explicit quiver relations and delooped versions such as \[ cd=ac,\qquad dd=0,\qquad aa=0, \] used to recover spectral-sequence-like long exact sequences [1211.5789]. ## 4. Quivers with homotopies as normal subgroupoids and mutation data For mutation theory, one begins with a quiver $Q=(Q_0,Q_1,s,t)$, its double quiver $\overline Q$, and the free groupoid $\pi(Q)$ whose morphisms are reduced walks in $Q$. This groupoid is naturally isomorphic to the fundamental groupoid $\pi_1(Q,Q_0)$ of the underlying $1$-complex. A homotopy is then a normal subgroupoid of $\pi(Q)$, and the quotient groupoid $G/H$ identifies morphisms $f,g\in G(x,y)$ whenever \[ f^{-1}g\in H. \] If $H$ is generated by a finite set $W$ of reduced cyclic walks, attaching $2$-cells along the loops in $W$ produces a $2$-complex $\Delta(Q,H,W)$ with \[ \pi(Q)/H \cong \pi_1(\Delta(Q,H,W),Q_0), \] so the homotopy has an explicit geometric realization [2508.15022]. Given a quiver with reduced homotopy $(Q,H)$ and a vertex $k$, pre-mutation first forms the ordinary pre-mutation quiver $Q'={}_k(Q)$: for each $\alpha:i\to k$ and $\beta:k\to j$, one adds a new arrow \[ [\beta\alpha]: i\to j, \] and one reverses every arrow incident to $k$. The homotopy is transported by a functor \[ \varphi:\pi(Q')\to \pi(Q)/H \] defined by \[ \alpha^\star\mapsto \alpha^{-1},\qquad \beta^\star\mapsto \beta^{-1},\qquad [\beta\alpha]\mapsto \beta\alpha, \] and by $\gamma\mapsto\gamma$ on all other arrows. One then sets \[ H'=\ker\varphi. \] At this stage $Q'$ may contain oriented $2$-cycles [2508.15022]. Mutation is obtained by deleting a maximal collection of $2$-cycles subject to the homotopy criterion that a pair $\gamma\delta$ is removed precisely when it lies in $H'$. The resulting quiver is denoted $Q^\dagger$, and its homotopy is \[ H^\dagger=\ker\psi,\qquad \psi:\pi(Q^\dagger)\to \pi(Q')/H'. \] The mutation is then \[ \mu_k(Q,H)=(Q^\dagger,H^\dagger). \] Although the deletion step involves choices, the outcome is well-defined up to quiver isomorphism [2508.15022]. Two structural facts organize the theory. First, mutation is involutive: \[ \mu_k^2(Q,H)\cong (Q,H). \] Second, mutation preserves the quotient groupoid, since the maps \[ \pi(Q^\dagger)/H^\dagger \xrightarrow{\ \psi\ } \pi(Q')/H' \xrightarrow{\ \varphi\ } \pi(Q)/H \] are isomorphisms. Thus the invariant object is not the quiver alone but the quotient groupoid determined by the homotopy [2508.15022]. The framework recovers Fomin–Zelevinsky mutation when $Q$ is $2$-acyclic and one chooses the maximal homotopy \[ H=\pi(Q). \] Then $\pi(Q)/H$ is trivial, the reduction step deletes exactly the $2$-cycles that would be deleted in classical mutation, and any mutation sequence with homotopies produces the same quivers as ordinary FZ mutation [2508.15022]. ## 5. Coverings, infinite mutation sequences, and surface realizations The mutation formalism is compatible with orbit mutations coming from coverings. If \[ p:\widetilde Q\to Q \] is a weakly admissible covering, where $\widetilde Q$ is $2$-acyclic, $Q$ is loop-free, and no arrow in $\widetilde Q$ connects vertices in the same orbit of the deck group $\Gamma=\mathrm{Deck}(p)$, then an orbit mutation at an orbit $[k]$ is obtained by mutating all vertices in $p^{-1}(k)$ once and then deleting $2$-cycles in a $\Gamma$-compatible way. When $p$ is $[k]$-mutable, the orbit mutation induces a quotient quiver isomorphic to the homotopy mutation $\mu_k(Q,H)$ for \[ H=p_*(\pi(\widetilde Q))), \] more precisely through a quiver isomorphism $f:Q^\ddag\to Q^\dagger$ such that \[ H^\dagger=f_*(H^\ddag). \] This identifies homotopy mutation as the quotient-level shadow of orbit mutation [2508.15022]. 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 \[ \pi(Q)^2 \subseteq p_*(\pi(\widetilde Q)), \] where $\pi(Q)^2$ is the normal subgroupoid generated by squares of cyclic walks. The same paper proves that for an acyclic quiver $Q$, any homotopy is equivalent to the trivial homotopy, and hence any quiver $Q'$ obtained by FZ mutation from an acyclic quiver satisfies \[ \mathrm{rank}\,\pi_1(Q') \ge \mathrm{rank}\,\pi_1(Q). \] These statements show that the homotopy datum is strongest precisely when $2$-cycles and nontrivial fundamental-group information are present [2508.15022]. The surface model extends the Fomin–Shapiro–Thurston construction to marked bordered surfaces with punctures of two colors. A colored marked surface is written $(S,M,c)$ with $c:P\to\{I,II\}$. From a triangulation $T$, one builds a preliminary quiver $\widetilde Q(T)$ using the usual FST puzzle pieces together with a new piece arising from a self-folded triangle enclosing a $II$-puncture. The final quiver $Q(T)$ is obtained by deleting certain $2$-cycles only in the case of an $I$-puncture configuration; $2$-cycles around $II$-punctures are retained [2508.15022]. The homotopy $\widetilde H(T)$ is generated by oriented $3$-cycles from puzzle pieces and by oriented cycles $C_p$ around each $I$-puncture of valency at least $2$. The induced homotopy $H(T)$ on $Q(T)$ is characterized by \[ \pi(Q(T))/H(T)\cong \pi(\widetilde Q(T))/\widetilde H(T). \] The associated $2$-complex $X(T)$ satisfies \[ \pi(X(T),X(T)_0)\cong \pi(\widetilde Q(T))/\widetilde H(T)\cong \pi(Q(T))/H(T), \] and \[ X(T)\simeq S\setminus P_{II}. \] The tagged arc complex is a pseudomanifold: every maximal tagged triangulation has size \[ n=6g+3b+3p+s-6, \] every codimension-$1$ face lies in exactly two maximal faces, and each tagged triangulation may be flipped at any tagged arc. The main theorem states that if $T'$ is the flip of $T$ at arc $k$, then \[ (Q(T'),H(T'))=\mu_k(Q(T),H(T)). \] This provides a surface-theoretic model for mutation in the presence of oriented $2$-cycles [2508.15022]. ## 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 $A$ with objects $O_A$ and graded vector spaces ${}_xA_y$, and a $d$-pre-Calabi–Yau structure is an element \[ s_{d+1}M_A \in Multi^\bullet_d(A)^{C_{lg(\bullet)}}[d+1] \] of degree $1$ satisfying the Maurer–Cartan equation \[ [s_{d+1}M_A,s_{d+1}M_A]_{nec}=0. \] Pre-Calabi–Yau morphisms are Maurer–Cartan elements in larger $L_\infty$-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 [2405.20854]. The chain-level path object is the standard $L_\infty$ extension $\mathcal L[t,dt]$. Two Maurer–Cartan elements are homotopic if there exists \[ a(t)+b(t)\,dt \] with $a(0)=f$, $a(1)=g$, and \[ \frac{\partial a}{\partial t}(t) = \sum_{n\ge1}\frac{1}{(n-1)!}\,\ell^n(a(t),\dots,a(t),b(t)). \] Within this formalism, homotopy is stable under left and right composition, homotopy equivalences are quasi-isomorphisms, and the functor \[ \mathcal P: pre\text{-}CY \to \text{partial }A_\infty\text{-categories} \] sends homotopic pre-Calabi–Yau morphisms to weakly homotopic $A_\infty$-morphisms [2405.20854]. 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 $\operatorname{Ext}$ over a basic algebra [1211.5789]. 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 $\pi(Q)/H$ [2508.15022]. In the pre-Calabi–Yau setting, homotopy refers to Maurer–Cartan homotopy in an $L_\infty$ path object attached to graded-quiver operations [2405.20854]. 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 $2$-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. Source: https://www.emergentmind.com/topics/quivers-with-homotopies