nLab concrete (2,1)-category

Idea

A generalization of the notion of concrete category from category theory to (2,1)-category theory.

 Definition

Definition

A concrete (2,1)-category is a (2,1)-category $C$ equipped with a 3-surjective functor

$$U \colon C \to Grpd$$

to the large (2,1)-category Grpd of groupoids. We say a (2,1)-category $C$ is concretizable if and only if it admits a 3-surjective functor $U \colon C \to Grpd$.

Examples

• The (2,1)-category $Grpd$ of groupoids is a concrete (2,1)-category.

• The (2,1)-category $MonGrpd$ of monoidal groupoids is a concrete (2,1)-category.

• The (2,1)-category $BraidedMonGrpd$ of braided monoidal groupoids is a concrete (2,1)-category.

• The (2,1)-category $SymmetricMonGrpd$ of symmetric monoidal groupoids is a concrete (2,1)-category.

• The (2,1)-category $RingGrpd$ of ring groupoids is a concrete (2,1)-category.

• The (2,1)-category $SymmetricRingGrpd$ of symmetric ring groupoids is a concrete (2,1)-category.

• The (2,1)-category $2Grp$ of 2-groups is a concrete (2,1)-category.

• The (2,1)-category $Smooth2Grp$ of smooth 2-groups is a concrete (2,1)-category.

• The (2,1)-category $Braided2Grp$ of braided 2-groups is a concrete (2,1)-category.

• The (2,1)-category $Symmetric2Grp$ of symmetric 2-groups is a concrete (2,1)-category.

• The (2,1)-category $Cat$ of weak categories (with functors between these and natural isomorphisms between those) is a concrete (2,1)-category.

Related concepts

• (2,1)-category
• concrete object
• concrete category
• concrete (n,1)-category