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# concrete category

A *concrete category* over a category $\mathcal{B}$ is a category $\mathcal{A}$ together with a
faithful functor $U:\mathcal{A}\to\mathcal{B}$. (The functor $U$ is sometimes called the *forgetful
functor* or the *underlying functor*.)

A concrete category over $\mathbf{Set}$ is called a *construct*. (Here $\mathbf{Set}$ denotes the category of
sets.)

Remarks:

1. An alternative meaning of a

*concrete category*is that of a category with objects that have elements; such objects can be classes, semigroups, monoids, groups, groupoids, topological spaces, and so on.2. Note also the Yoneda-Grothendieck Lemma that relates a category $\mathcal{C}$ to the functor category $\hat{\mathcal{C}}$ of contravariant functors from $\mathcal{C}$ to ${\bf Sets}$, the category of sets.

# References

- 1 J. Adámek, H. Herrlich, and G. Strecker. Abstract and Concrete Categories. Wiley, New York, 1990.

## Mathematics Subject Classification

18A05*no label found*

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## Comments

## construct vs. concrete category

I removed construct from "Also defines" field, for it caused to redirect here totally irrelevant entries.

As far as I remember, some authors use "concrete category" in the sense as "construct" is used in this entry (an in Adamek-Herrlich-Strecker).

Or am I mistaken?