As far as I know everyone is consistent about those, regardless of the choices they make about lax and oplax transformations, so let’s not create additional confusion. So the term analogous to “space” is “category”, the term analogous to “continuous map” is “functor”, and the term analogous to “space of all continuous maps from one space to another” or “function space” is “category of all functors from one category to another” or “functor category”. See the history of this page for a list of all contributions to it. Anyway, this whole discussion started only to sound a warning to the reader that conventions do differ in the literature.
I think you may be right (Mike) and I wrong about which is more common in Australia; certainly Kelly for example goes along with Bénabou. In the end, if D CD^C denotes the functor category, the universal property provides an equivalence between functors X→D CX \to D^C and functors X×C→DX \times C \to D. Pursuing this, let’s figure out what morphisms in D CD^C should look like. My own convention is that of Street: the arrow orientations in a product of parity complexes is to follow the usual sign conventions for chain complexes. \alpha_A \Rightarrow \alpha_B \circ F(f), for each A,BA,B, the α f\alpha_f are the components of a (strict) natural transformation α A,B:(α A) *∘G A,B→˙(α B) *∘F A,B\alpha_{A,B}: Toby: That arrow certainly looks to me like a 22-arrow from α D∘F\alpha_D\circ F to α C∘G\alpha_C\circ G. Unless it has no particular meaning and is just labelling the entire square as α:F⇒G\alpha\colon F \Rightarrow G. In any case, there is the question of what, if anything, goes between α D∘F\alpha_D\circ F and α C∘G\alpha_C\circ G. If nothing, then I would call such a thing a square of functors. Now in general, a vertical transformation between double categories that are proarrow equipments does not quite induce a pseudo natural transformation on horizonal bicategories, but only an oplax natural transformation. How can I extend my camera's battery life?
What do you think?
How practical is a spear-rapier (fencing foil) hybrid? A natural … site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. It is tempting to say we might want α D∘F⇒α C∘G\alpha_D\circ F\Rightarrow\alpha_C\circ G to be a natural transformation in the traditional sense though.
(Maybe inconsistency isn’t what he means by “being flexible,” but then I don’t know what he means.) That 2-arrow is not supposed to be a natural transformation in the traditional sense because it is not a map between functors with the same domain and codomain. I haven’t seen any of the other references. Eric: I hope to pick up where we left off because I was making some progress understanding some of this stuff. Todd: I think you’ve certainly made a good start. I also think there is a very good reason to use that convention. In Cat at least, you can check that monad morphisms as defined on nlab induce a functor between E-M categories, while oplax transformations would give rise to a functor between Kleisli categories instead, which doesn't sound as useful. How do I remember which is which? For instance you can describe oplax naturals as being oplax morphisms somewhere or other. Given topological spaces W,X,Y,ZW,X,Y,Z, we could also be interested in studying spaces (W→X)⇒(Y→Z)(W\to X)\Rightarrow (Y\to Z).
The standard definition of natural transformation involves a map between functors (sharing the same domain and codomain) together with component morphisms. If this is not a natural transformation, is there another name for it? And there should be a universal property which gives the right such category.
When Ross asked Steve if his opinion on this were strong or lax, he replied: “I don’t think I’d say I have a strong opinion.
One or two references use “left lax” and “right lax” instead.
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