---
tag: ZD0GTPSJS1
kind: concept
name: Concrete category of algebras
msc: ["08C05", "18A05"]
specializes: [FYBRP16KAN]
sources:
  - url: https://en.wikipedia.org/wiki/Concrete_category
---

## Summary

A concrete category of algebras is a category whose objects are algebras of a fixed signature and whose morphisms are homomorphisms.
It comes with the [[forgetful functor|WV3BPYCHVG]] to sets.
It is the main example of a [[concrete category|FYBRP16KAN]]: objects are sets with structure, and morphisms are functions that respect the structure.
Varieties, quasivarieties and other classes of algebras are studied in this setting.
There, notions such as monomorphism, epimorphism, free object and reflection can be compared with their set-based meanings.

## Explanations

### L1 human

In a concrete category every object has elements and every arrow is an honest function.
Groups with group homomorphisms, rings with ring homomorphisms and lattices with lattice homomorphisms are all of this kind.
Being concrete makes some categorical notions easy to read: an injective homomorphism is a monomorphism, for instance.
Other notions still surprise, since an epimorphism of rings need not be surjective.