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André Henriques
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Consider the following 2-category:

• It objects are concrete categories, i.e., categories equipped with a faithful functor to $Set$.

• A 1-morphism between $(C_1,U_1)$ and $(C_2,U_2)$ consist of a functor $F:C_1\to C_2$ and a natural transformation $z:U_1\Rightarrow U_2\circ F$.

It'sIts 2-morphisms are the obvious thing.

Question: Is there a name for that notion of functor between concrete categories?

... the pair $(F,z)$ is a  [insert adjective]  functor from $C_1$ to $C_2$ ...

Consider the following 2-category:

• It objects are concrete categories, i.e., categories equipped with a faithful functor to $Set$.

• A 1-morphism between $(C_1,U_1)$ and $(C_2,U_2)$ consist of a functor $F:C_1\to C_2$ and a natural transformation $z:U_1\Rightarrow U_2\circ F$.

It's 2-morphisms are the obvious thing.

Question: Is there a name for that notion of functor between concrete categories?

... the pair $(F,z)$ is a  [insert adjective]  functor from $C_1$ to $C_2$ ...

Consider the following 2-category:

• It objects are concrete categories, i.e., categories equipped with a faithful functor to $Set$.

• A 1-morphism between $(C_1,U_1)$ and $(C_2,U_2)$ consist of a functor $F:C_1\to C_2$ and a natural transformation $z:U_1\Rightarrow U_2\circ F$.

Its 2-morphisms are the obvious thing.

Question: Is there a name for that notion of functor between concrete categories?

... the pair $(F,z)$ is a  [insert adjective]  functor from $C_1$ to $C_2$ ...

edited body
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André Henriques
  • 43.2k
  • 5
  • 130
  • 264

Consider the following 2-category:

• It objects are concrete categories, i.e., categories equipped with a faithful functor to $Set$.

• A 1-morphism between $(C_1,U_1)$ and $(C_2,U_2)$ consist of a functor $F:C_1\to C_2$ and a natural transformation $z:U_1\Rightarrow U_2\circ F$.

• It's 2-morpihsmsmorphisms are the obvious thing.

Question: Is there a name for that notion of functor between concrete categories?

... the pair $(F,z)$ is a  [insert adjective]  functor from $C_1$ to $C_2$ ...

Consider the following 2-category:

• It objects are concrete categories, i.e., categories equipped with a faithful functor to $Set$.

• A 1-morphism between $(C_1,U_1)$ and $(C_2,U_2)$ consist of a functor $F:C_1\to C_2$ and a natural transformation $z:U_1\Rightarrow U_2\circ F$.

• It's 2-morpihsms are the obvious thing.

Question: Is there a name for that notion of functor between concrete categories?

... the pair $(F,z)$ is a  [insert adjective]  functor from $C_1$ to $C_2$ ...

Consider the following 2-category:

• It objects are concrete categories, i.e., categories equipped with a faithful functor to $Set$.

• A 1-morphism between $(C_1,U_1)$ and $(C_2,U_2)$ consist of a functor $F:C_1\to C_2$ and a natural transformation $z:U_1\Rightarrow U_2\circ F$.

• It's 2-morphisms are the obvious thing.

Question: Is there a name for that notion of functor between concrete categories?

... the pair $(F,z)$ is a  [insert adjective]  functor from $C_1$ to $C_2$ ...

Source Link
André Henriques
  • 43.2k
  • 5
  • 130
  • 264

Category of concrete categories

Consider the following 2-category:

• It objects are concrete categories, i.e., categories equipped with a faithful functor to $Set$.

• A 1-morphism between $(C_1,U_1)$ and $(C_2,U_2)$ consist of a functor $F:C_1\to C_2$ and a natural transformation $z:U_1\Rightarrow U_2\circ F$.

• It's 2-morpihsms are the obvious thing.

Question: Is there a name for that notion of functor between concrete categories?

... the pair $(F,z)$ is a  [insert adjective]  functor from $C_1$ to $C_2$ ...