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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeOct 22nd 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexCopying old query box here from [[pseudofunctor]] (having incorporated its content into the entry): [[Tim]]: in specifying a pseudo functor $F$ you have to decide whether the isomorphism goes from $F(g f)$ to $F(g) F(f)$ or in the other direction. Of course they are equivalent as each will be inverse to the other. You might say that one is lax and pseudo the other op-lax and pseudo. When specifying the Grothendieck construction for such a functor, which is to be preferred? Both are about equally represented in the literature that I have seen which gets confusing. (In other words, I'm confused!) _Toby_: As you suggest, the two versions are equivalent, so in a way it doesn\'t make a difference. But it might be nice to settle a convention in case we need it. [[Tim]]: I have been using (for the Menagerie) the idea that there are pseudofunctors presented in two equivalent flavours lax pseudofunctor and oplax ones. [[Mike Shulman|Mike]]: Well, the natural comparison maps that you get in a [[Grothendieck fibration]] go in the "lax" direction $F(g) F(f) \to F(g f)$, since they are induced by the universal property of cartesian arrows. In particular, if you have a functor with "weakly cartesian" liftings that don't compose, then it is a lax functor. Not a very strong argument, but if we just want _some_ convention it might be a reason to pick lax. I think that making too big a deal out of the difference would be misleading, though.

   Copying old query box here from pseudofunctor (having incorporated its content into the entry):

   Tim: in specifying a pseudo functor $F$ you have to decide whether the isomorphism goes from $F(g f)$ to $F(g) F(f)$ or in the other direction. Of course they are equivalent as each will be inverse to the other. You might say that one is lax and pseudo the other op-lax and pseudo. When specifying the Grothendieck construction for such a functor, which is to be preferred?

   Both are about equally represented in the literature that I have seen which gets confusing. (In other words, I’m confused!)

   Toby: As you suggest, the two versions are equivalent, so in a way it doesn't make a difference. But it might be nice to settle a convention in case we need it.

   Tim: I have been using (for the Menagerie) the idea that there are pseudofunctors presented in two equivalent flavours lax pseudofunctor and oplax ones.

   Mike: Well, the natural comparison maps that you get in a Grothendieck fibration go in the “lax” direction $F(g) F(f) \to F(g f)$, since they are induced by the universal property of cartesian arrows. In particular, if you have a functor with “weakly cartesian” liftings that don’t compose, then it is a lax functor. Not a very strong argument, but if we just want some convention it might be a reason to pick lax. I think that making too big a deal out of the difference would be misleading, though.