Re terminology:

[Urs Schreiber <urs.schreiber@googlemail.com>]

Dear Ronnie,

> You mention the process from category to infinity-category. Actually that
> was why we introduced the term infinity-category

This is why I am thinking you could embrace the way the term is used
these days: because it follows precisely your use back then, only
removing the restriction of strictness. And algebraicity can  be
restored. See below...


> The problem is that there is
> no unique choice of such retractions, nor is it clear what might be the
> relations between iterates of such fillers.  These considerations led Keith
> Dakin to the notion of T-complex  for his 1976 thesis; somehow `T-complex'
> has more recently become `complicial set', but nobody asked me. (Groan!
> Groan!) So it seems that the notion of quasi category as a weak Kan complex
> still has not captured something about the basic example; but how to repair
> that is quite unclear.

This has recently been clarified by Thomas Nikolaus in his work on
algebraic Kan complexes (which are essentially simplicial
T-complexes!) and algebraic quasi-categories:

  http://ncatlab.org/nlab/show/model+structure+on+algebraic+fibrant+objects

He shows that the model category/quasi-category/(oo,1)-category (check
preferred term) of all Kan complexes is equivalent to that of all Kan
complexes with all horn fillers chosen.

And analogously: that the model
category/quasi-category/(oo,1)-category (check preferred term) of all
quasi-categories is equivalent to that of all quasi-categories with
all inner horn fillers chosen.

This says that while a Kan complex or quasi-category is not directly
an algebraic model for an oo-groupoid or (oo,1)-category,
respectively, you can immediately turn it into an algebraic model by
making choices, and up to equivalence, the resulting algebraic model
does not depend on these choices.

Best,
Urs


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