I have moved the paragraphs on weak Cartesian morphisms out of the middle of the definition of Cartesian morphisms and gave them their own Remark-environment, now here.

]]>Fixed discrepancy between ’slightly more explicit’ definition and accompanying diagram

Joseph Martin

]]>Fixed discrepancy between ’slightly more explicit’ definition and accompanying diagram

Simon Dima

]]>Removed an old discussion:

David Roberts: There would surely be an anafunctor version of this, that would require no choices whatsoever. It is unlikely that I would be able to find time to write this up, so my plea goes out to those in the know…

I imagine that there would then be an $(\infty,1)$-version using whatever passes as anafunctors in that setting (dratted memory, failing at the first gate)

Mike Shulman: Yes, there would surely be such a version. (-: The simplest way would be to take the specifications $|f^*|$ for the anafunctor $f^*$ to be the cartesian morphisms over $f$, with domain and codomain giving the functions $\sigma$ and $\tau$. Unique factorization would give you the values of morphisms.

David Roberts: just stumbled on this old comment - I’m reading Makkai more closely, and I’m convinced that basically anything defined by a universal property is given by a saturated anafunctor. So this is a heads up for posterity, that a map is a fibration iff the fairly obvious span of functors defines a saturated anafunctor.

]]>Correction: a Cartesian morphism is the special case of a strictly final lift of a structured sink when the sink is a singleton.

]]>Added alternative equivalent slightly more explicit definition.

]]>Changed some of the language in my additions to make the overall presentation flow more smoothly.

]]>Thanks for the additions.

Yes, the `tikzcd`

-functionality is not part of Instiki (which is the software platform the $n$Lab rendering engine is using) but is an external add-on that Richard kindly added a while back, so that we can have decent diagram typesetting at all.

My understanding is that there are plans to improve on this and other aspects, but I gather the priority right now is to get the $n$Lab migrated from its server at CMU (which is running out) to a cloud provider.

As usual, if anyone is interested in lending Richard a hand with coding desired functionality for the $n$Lab, drop him a note and he may tell you what you could do!

]]>It’s weird that you can’t do tikzcd inbetween dollar-sign pairs, and you have to actually use the centre environment thing.

]]>Okay, I think I got it

]]>I felt like the (∞,1) section should give an abstract description rather than a model specific one, so I’ve done so, and proved in the abstract the equivalence between the hom-space and slice-category characterizations.

It feels like cheating to invoke the Grothendieck construction for it; can the argument be made just as cleanly without it?

… I’m having trouble with the formatting, so I’m going to do some bisection to track down the issue….

]]>