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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 $z:U_1\Rightarrow U_2\circ F$.

• 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 $C_2$ ...

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    $\begingroup$ What's the obvious thing, if I may ask? Modifications? $\endgroup$ – Andrej Bauer Aug 22 '17 at 9:31
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    $\begingroup$ A $2$-morphism from $(F_1,z_1)$ to $(F_2,z_2)$ is a natural transformation $\Phi:F_1\Rightarrow F_2$ wish the property that $z_2=(U_2\circ \Phi)\bullet z_1$, where $\circ$ denotes horizontal composition and $\bullet$ denotes vertical composition. $\endgroup$ – André Henriques Aug 22 '17 at 9:40
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    $\begingroup$ This article uses "concrete functor": Porst, What is concrete equivalence? doi.org/10.1007/BF00878502, I think (I can't check right now) $\endgroup$ – David Roberts Aug 22 '17 at 9:53
  • $\begingroup$ @David Roberts: Thanks for the pointer. The article you cite assumes that the natural transformation $U_1\Rightarrow U_2\circ F$ is an equivalence (which is something I don't want to assume). $\endgroup$ – André Henriques Aug 22 '17 at 10:08
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    $\begingroup$ "(co)lax concrete functor (see [Porst])"? $\endgroup$ – Fosco Loregian Aug 22 '17 at 10:12
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Concrete functor is established in the literature for the related notion where the natural transformation is an isomorhpism (see e.g. Porst 1996 Concrete Categories Are Concretely Equivalent if…) — i.e. the sub-2-category of the slice 2-category of CAT over Set on faithful functors.

Your 2-category is similarly the sub-2-category on faithful functors of the colax slice 2-category of CAT over Set. So it seems very natural to call your notion colax concrete functors, though as far as I can find this term hasn’t been used before. A lax concrete functor would be the same thing but with the transformation in the other direction.

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