---
tag: 6EWEMZ6FX1
kind: statement
role: proposition
name: Variety as a reflective subcategory
msc: ["08C05", "08B99", "18A40"]
uses: [AJNZJD8442, FC7VB4WSG0, V3BFGKBJF7, DCSXXT4SER, N5KE7FH6CX, KBXSZC6BV1]
about: [AJNZJD8442, FC7VB4WSG0]
sources:
  - url: https://en.wikipedia.org/wiki/Reflective_subcategory
---

## Statement

Let $\sigma$ be an algebraic signature, and let $\mathbf{Alg}(\sigma)$ be the category of all $\sigma$-algebras and their [[homomorphisms|KBXSZC6BV1]].
Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of $\sigma$-algebras, viewed as a [[full subcategory|V3BFGKBJF7]].
For a $\sigma$-algebra $A$, let $\theta_A$ be the intersection of all [[congruences|DCSXXT4SER]] $\theta$ of $A$ with $A / \theta \in \mathcal{V}$, and let $q_A \colon A \to A / \theta_A$ be the [[quotient map|N5KE7FH6CX]].
Then $A / \theta_A \in \mathcal{V}$, and for every $B \in \mathcal{V}$ and every homomorphism $f \colon A \to B$ there is a unique homomorphism $\bar f \colon A / \theta_A \to B$ with $\bar f \circ q_A = f$.
Hence $\mathcal{V}$ is a [[reflective subcategory|FC7VB4WSG0]] of $\mathbf{Alg}(\sigma)$, with reflection $q_A$, and its reflector $A \mapsto A / \theta_A$ is left adjoint to the inclusion.

## Remarks

The algebra $A / \theta_A$ embeds into the product of the algebras $A / \theta$ over all such $\theta$, so it lies in $\mathcal{V}$ by closure under subalgebras and products.
For $f \colon A \to B$ with $B \in \mathcal{V}$, the image $f(A)$ is a subalgebra of $B$, so $A / \ker f \cong f(A) \in \mathcal{V}$.
Thus $\ker f \supseteq \theta_A$, and $f$ factors through $q_A$; the factorization is unique because $q_A$ is surjective.

Only closure under isomorphic copies, subalgebras and products is used, so the same holds for every quasivariety and every [[prevariety|EEAJ7AXBCT]].
Example: for the variety of abelian groups inside groups, $\theta_G$ is the congruence of the commutator subgroup $[G, G]$, and the reflection is the abelianization $G \to G / [G, G]$.
Applied to the absolutely free term algebra on a set $X$, the reflection gives the [[free algebra|GTS4DRRMZR]] of $\mathcal{V}$ on $X$.

## Explanations

### L1 human

Any algebra can be pushed into a given variety in a best possible way: impose just the equations of the variety, and nothing more.
For groups and the variety of abelian groups, this means forcing all elements to commute, which gives the abelianization.
Every homomorphism into an algebra of the variety factors through this best approximation in exactly one way.

### L2 human

The reflection $A \mapsto A / \theta_A$ is a universal construction, and $\theta_A$ is the congruence generated by the pairs $(s(\bar a), t(\bar a))$ for identities $s \approx t$ of $\mathcal{V}$ and tuples $\bar a$ of $A$.
Since the reflector is a left adjoint, it preserves colimits: the free product in $\mathcal{V}$ of two members is the reflection of their free product in $\mathbf{Alg}(\sigma)$.
Composing with the free functor of $\mathbf{Alg}(\sigma)$ recovers the [[free functor|N353JF8RBK]] of $\mathcal{V}$.
Reflective subcategories are closed under limits, which matches the closure of varieties under products and under equalizers, which are subalgebras.