Anindya wrote:

> possible typo: in previous lectures we've defined profunctors as \\(\mathcal{V}\\)-enriched functors \\(\mathcal{X}^\text{op}\times\mathcal{Y}\rightarrow\mathcal{V}\\) but here we're using \\(\mathcal{X}\times\mathcal{Y}^\text{op}\rightarrow\mathcal{V}\\) – is this a mistake or are we swapping round for a reason?

This is just a mistake - I'll fix it, thanks!

I repeatedly amaze myself by my inability to stick to arbitrary conventions. In my defense I'll mention that there's a heated and pointless debate among category theorists about which convention is better. It doesn't really matter which you use as long as you stick with it, but people who get used to one find the other counterintuitive. I'm _trying_ to use Fong and Spivak's convention \\(\mathcal{X}^\text{op}\times\mathcal{Y}\rightarrow\mathcal{V}\\), and this is actually my favorite convention, mainly because it makes the hom-functor

\[ \text{hom} : \mathcal{C}^{\text{op}} \times \mathcal{C} \to \mathbf{Set} \]

into a profunctor.

(We are currently just studying _enriched_ profunctors between _enriched_ categories, and for a \\(\mathcal{V}\\)-enriched category \\(\mathcal{C}\\) will have an enriched profunctor

\[ \text{hom} : \mathcal{C}^{\text{op}} \times \mathcal{C} \to \mathcal{V}, \]

but the idea is just the same for ordinary categories and ordinary profunctors.)

> possible typo: in previous lectures we've defined profunctors as \\(\mathcal{V}\\)-enriched functors \\(\mathcal{X}^\text{op}\times\mathcal{Y}\rightarrow\mathcal{V}\\) but here we're using \\(\mathcal{X}\times\mathcal{Y}^\text{op}\rightarrow\mathcal{V}\\) – is this a mistake or are we swapping round for a reason?

This is just a mistake - I'll fix it, thanks!

I repeatedly amaze myself by my inability to stick to arbitrary conventions. In my defense I'll mention that there's a heated and pointless debate among category theorists about which convention is better. It doesn't really matter which you use as long as you stick with it, but people who get used to one find the other counterintuitive. I'm _trying_ to use Fong and Spivak's convention \\(\mathcal{X}^\text{op}\times\mathcal{Y}\rightarrow\mathcal{V}\\), and this is actually my favorite convention, mainly because it makes the hom-functor

\[ \text{hom} : \mathcal{C}^{\text{op}} \times \mathcal{C} \to \mathbf{Set} \]

into a profunctor.

(We are currently just studying _enriched_ profunctors between _enriched_ categories, and for a \\(\mathcal{V}\\)-enriched category \\(\mathcal{C}\\) will have an enriched profunctor

\[ \text{hom} : \mathcal{C}^{\text{op}} \times \mathcal{C} \to \mathcal{V}, \]

but the idea is just the same for ordinary categories and ordinary profunctors.)