Comma Category

Let $F:\mathcal{A}\to\mathcal{C}$, $G:\mathcal{B}\to\mathcal{C}$ be a pair of Functors. The comma category $F/G$ is a category where

• Objects are triples of $A:\mathcal{A},B:\mathcal{B},\phi:{{F(A)}\to{G(B)}}$

• Morphisms between $\phi$ and $\psi$ are pairs of maps $f:\mathcal{A}(A_{1},A_{2})$, $g:{{B_{1}}\to{B_{2}}}$ that make the following diagram commute

Comma categories are more neatly presented as Displayed Categories over $\mathcal{A}\times\mathcal{B}$, which allows us to remove all of the redundant bundling.

Properties

• The Arrow Category is the comma category $mathrm{Id}{ifx..lse\par i}/mathrm{Id}{ifx..lse\par i}$.

• Comma categories are Discrete Two-Sided Fibrations.

• Comma categories $F/G$ classify Natural Transformations

  More explicitly, for $F,G:\mathcal{C}\to\mathcal{D}$, a Natural Transformation $\alpha:F\to G$ can be regarded as a functor $\mathcal{C}\to F/G$ via $X\mapsto(X,X,\alpha_{X})$.