Twisted Arrow Category

Let $\mathcal{C}$ be a category. The twisted arrow category $\mathrm{Tw}(\mathcal{C})$ is defined as:

• Objects are triples $X:\mathcal{C},Y:\mathcal{C},f:{{X}\to{Y}}$

• Morphisms are squares of the following form:

Note that this is only one possible convention: we could use squares that "go the other way".

Properties

• The twisted arrow category is a Discrete Cocartesian Fibration when viewed as a Displayed Category over $\mathcal{C}^{mathrm{op}}\times\mathcal{C}$. Note that the alternative convention yields a Discrete Cartesian Fibration over $\mathcal{C}\times\mathcal{C}^{op}$.
• The twisted arrow category classifies Wedges, insofar that for every $F:\mathcal{C}^{mathrm{op}}\times\mathcal{C}\to\mathcal{D}$ Extranatural Transformations $W\to F$ are the same as Natural Transformations $W\to F\circ\pi_{\mathcal{C}}$, where $\pi_{\mathcal{C}}:\mathrm{Tw}(\mathcal{C})\to\mathcal{C}^{mathrm{op}}\times \mathcal{C}$.
• The twisted arrow category can be constructed as the Category of Elements of the Hom Functor of $\mathcal{C}$. This is a useful trick in infinity category theory, as describing the hom functor explicitly can be quite difficult!

References

• https://ncatlab.org/nlab/show/twisted+arrow+category