Δ: the simplex category
sSet: the category of simplicial sets, presheaves on Δ
CATΔ: the category of simplicial categories
KAN: the simplicial category of kan complexes, ie all horns have fills
MAP(a,b): the kan complex of maps a->b
Fun(X,Y): the ∞category of functors X->Y
(C)~: The core of C, the subcategory of C spanned by h(C)~, the full subcategory whos morphisms are isomorphisms. The maximal ∞gpd in C qCAT: the simplicial category of infinity categories
∞CAT: the category of infinity categories, ie inner horns have fills, the homotopy coherent nerve of qCAT
∞GPD: the category of infinity groupoids
C/F: The over category for F:K->C, C/Fn = homF(Δn★K, C), that is maps that restrict to F on K SPC: the homotopy coherent nerve of KAN
PrL: The category of presentable ∞categories, morphisms are left adjoints
PrR: The category of presentable ∞categories, morphisms are right adjoints
PrL=PrR(op)
Sp(C): The stabilisation of C (pointed), formally inverting the loop functor
Sp: Sp(SPC), the category of spectra
TW(C): TW(C)n=hom(Δn★Δn(op), C), the twisted arrow category

N: 1CAT->∞CAT(->sSet)
sends a category to its nerve, simplices are compositions
h: sSet->1CAT
takes morphisms up to homotopy
h ⊣ N
|•|:sSet->Top
realises the simplices as spaces and glues with maps
Sing: Top -> sSet
the singular simplicial set of a space
|•| ⊣ Sing
ℭ: sSet -> CATΔ
objects at Δn are 1...n, the Δhom of i,j is the nerve of the poset of subsets of {i...j}, extend by colimits
NΔ: CATΔ->sSet
NΔ(S)(n)=homΔ(ℭ(Δn),S), the homotopy coherent nerve
ℭ ⊣ NΔ
hom(-,-): sSetxsSet(op)->sSet
hom(A,B)(n)=hom(AxΔn,B), the internal hom
Ax(-) ⊣ hom(A,-)
πi:sSet(*) -> Set(Grp)
The ith fundamental group / at 0 just the connected components
(-)★(-): sSetxsSet->sSet
A★B(n) is the union over all A({0...i})xB({i+1...n})
よ: C -> Fun(C(op), SPC)
The yoneda embedding