The Path Algebra as a Left Adjoint Functor

Abstract

We consider an intermediate category between the category of finite quivers and a certain category of pseudocompact associative algebras whose objects include all pointed finite dimensional algebras. We define the completed path algebra and the Gabriel quiver as functors. We give an explicit quotient of the category of algebras on which these functors form an adjoint pair. We show that these functors respect ideals, obtaining in this way an equivalence between related categories.

A Functorial Approach to Gabriel k-quiver Constructions for Coalgebras and Pseudocompact Algebras

“…We define a natural equivalence relation ∼ on the morphisms of PCog, observe that the functors above can be interpreted as functors between k-Quiv and the quotient category PCog ∼ and show that, interpreted this way, the Gabriel k-quiver functor is left adjoint to the path coalgebra functor (Theorem 4.3). This adjunction improves on the main result of Iusenko and MacQuarrie (2020) in every way: the category of k-quivers presented here is far more natural; there are no finiteness assumptions; there are no hypotheses applied to coalgebra homomorphisms. Using certain dualities, we also give two pairs of adjoint functors where on the algebraic side we have pseudocompact algebras: Firstly, a pair of contravariant functors adjoint on the left between k-Quiv and a quotient PAlg ∼ of the category PAlg of pointed pseudocompact algebras.…”

“…In Iusenko and MacQuarrie (2020), the authors define a pair of covariant adjoint functors between a certain category of finite k-quivers and a category whose objects are pseudocompact pointed algebras A such that A/J 2 (A) is finite dimensional and whose morphisms are (congruence classes of) those algebra homomorphisms α : A → B such that the induced map A/J (A) → B/J (B) is surjective. The adjunctions 4.3 and 5.3 are far more general, because there are no finiteness assumptions and there are no conditions on the algebra homomorphisms.…”

“…The main adjunction from Iusenko and MacQuarrie (2020) can be interpreted as a special case of Theorem 5.5: The subcategory F of ParAlg whose objects are those pairs (A, U ) with both A, U finite dimensional and whose morphisms are those (ϕ 0 , ϕ 1 ) with ϕ 0 surjective, is equivalent to the category of finite pointed quivers given in Iusenko and MacQuarrie (2020). On the algebra side we restrict PAlg ∼ to the category A whose objects are those algebras A in PAlg ∼ with A/J 2 (A) finite dimensional, and whose morphisms are (congruence classes of) those algebra homomorphisms A → B such that the induced map A/J (A) → B/J (B) is surjective.…”

“…In the other direction, beginning with a finite quiver Q, one may construct an associative algebra, the (complete) path algebra of Q. In Iusenko and MacQuarrie (2020), the first two authors of this article describe these constructions as a pair of adjoint functors, utilizing a certain intermediate category of "Vquivers", in which the arrows of a quiver are replaced with vector spaces (in this article, we refer to such objects as "k-quivers" rather than "Vquivers", the former being a more common name in the literature, e.g. (Gabriel (1973), Section 7; Keller (2006), Section 5.1).…”

“…The adjunction presented there thus gives a very precise explanation of what information one can obtain about a finite dimensional algebra in terms of the underlying combinatorial structure. On the other hand, the adjunction has several limitations: firstly, the category of Vquivers presented in Iusenko and MacQuarrie (2020) is rather unnatural, in the sense that the morphisms are not intuitive. Secondly, only finite quivers and (essentially) only finite dimensional algebras are considered.…”