We u
se the term
s ∞-
categories and ∞-
functors to me
an the object
s and morphi
sm
s in
an ∞-
cosmos :
a simplici
ally enriched c
ategory
sati
sfying
a few
axiom
s, remini
scent of
an enriched c
ategory of fibr
ant object
s. Qu
asi-c
ategorie
s, Seg
al c
ategorie
s, complete Seg
al
sp
ace
s, m
arked
simplici
al
set
s, iter
ated complete Seg
al
sp
ace
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ace
s,
and fibered ver
sion
s of e
ach of the
se
are
all ∞-c
ategorie
s in thi
s sen
se. Previou
s work in thi
s serie
s show
s th
at the b
asic c
ategory theory of ∞-c
ategorie
s and ∞-functor
s c
an be developed only in reference to the
axiom
s of
an ∞-co
smo
s; indeed, mo
st of the work i
s intern
al to the
homotopy 2-category,
a strict 2-c
ategory of ∞-c
ategorie
s, ∞-functor
s,
and n
atur
al tr
an
sform
ation
s. In the ∞-co
smo
s of qu
asi-c
ategorie
s, we rec
apture preci
sely the
same c
ategory theory developed by Joy
al
and
Lurie,
although our definition
s are 2-c
ategoric
al in n
atur
al, m
aking no u
se of the combin
atori
al det
ail
s th
at differenti
ate e
ach model.
sp0020">In this paper, we introduce cartesian fibrations, a certain class of ∞-functors, and their groupoidal variants. Cartesian fibrations form a cornerstone in the abstract treatment of “category-like” structures a la Street and play an important role in Lurie's work on quasi-categories. After setting up their basic theory, we state and prove the Yoneda lemma, which has the form of an equivalence between the quasi-category of maps out of a representable fibration and the quasi-category underlying the fiber over its representing element. A companion paper will apply these results to establish a calculus of modules between ∞-categories, which will be used to define and study pointwise Kan extensions along ∞-functors.