Doriath
Stephen Britton DB Test Page

I'll be using Doriath for experimenting on a database of n-categories. Suggestions from Dr. Urs Schreiber and Dr. Andrew Stacey are welcomed. I also would appreciate if the pages that I am currently working on were not deleted until I am officially done experimenting (which I intend on indicating).

• (∞,1)Cat
• (∞,1)-category of (∞,1)-categories
• (∞,1)-category of (∞,1)-functors
• (∞,1)-category of (∞,1)-presheaves
• (∞,1)-category of (∞,1)-sheaves
• (∞,1)-category of cartesian sections
• (∞,n)-category of cobordisms
• stable (∞,1)-category of spectra
• ASSet
• Ab
• Alg
• BoolAlg
• CAT
• CartSp
• Cat
• CoDGCA
• CocommCoalg
• CompBoolAlg
• CompLat
• dgAlg
• DiGraph
• Diff
• DistLat
• FinSet
• Frm
• Grp
• Grpd
• HeytAlg
• Hilb
• Ho(Cat)
• Ho(Top)
• ∞Grpd
• Lat
• Lex
• LieAlg
• Loc
• MSet
• Meas
• Mod
• MultiSet
• Pr(∞,1)Cat
• Prof
• Rel
• Ring
• SDiff
• SSet Cat
• SemiLat
• Set
• SimpAb
• SimpSet
• Spec
• SupLat
• Top
• Vect
• category of chain complexes
• category of representations
• nCob

Bord(∞,n)

Object Morphism Table

Category		Objects		Morphisms
(∞,1)Cat		All small (∞,1)-categories		(∞,1)-functors
(∞,1)-category of (∞,1)-categories , Homotopy theory of homotopy theories		(∞,1)-categories		(∞,1)-functors
(∞,1)-category of (∞,1)-functors		(∞,1)-functors		natural transformations of (∞,1)-functors
(∞,1)-category of (∞,1)-presheaves		(∞,1)-presheaves		natural transformations of (∞,1)-presheaves
(∞,1)-category of (∞,1)-sheaves		(∞,1)-sheaves		natural transformations of (∞,1)-sheaves
(∞,1)-category of cartesian sections		Foo		Foo
(∞,n)-category of cobordisms, ${\mathrm{Bord}}_{(\mathrm{\infty },n)}$		0-dimensional (oriented) manifolds (disjoint unions of (oriented) points)		Foo
stable (∞,1)-category of spectra		Foo		Foo
ASSet		Foo		Foo
Ab		abelian groups		group homomorphisms
Alg		algebras		algebra homomorphisms
BoolAlg		Boolean algebras		lattice homomorphisms $\subset$ Boolean algebra homomorphisms
CAT		All 1-categories		All functors and natural transformations over 1-categories
CartSp		Foo		Foo
Cat		All small 1-categories		All functors and natural transformations over small 1-categories
CoDGCA		Foo		Foo
CocommCoalg		graded-cocommutative codifferential coalgebras		Foo
CompBoolAlg		complete Boolean algebras		Boolean algebra homomorphisms
CompLat		complete lattices		complete lattice homomorphisms
dgAlg		dg-algebras		dg-algebra homomorphisms
DiGraph		directed graphs		graph homomorphisms
Diff		smooth manifolds based on cartesian spaces		smooth maps
DistLat		distributive lattices		lattice homomorphisms
FinSet		finite set		functions
Frm		frames		frames homomorphisms
Grp		groups		group homomorphisms
Grpd		groupoids		groupoid functors and natural transformations
HeytAlg		Heyting algebras		Heyting algebra homomorphisms $\subset$ lattice homomorphisms
Hilb		Hilbert spaces		short linear maps
Ho(Cat)		small categories		natural isomorphism classes of functors
Ho(Top)		topological spaces		Foo
∞Grpd		∞-groupoids		(∞,1)-functors
Lat		lattices		lattice homomorphisms
Lex		small finitely complete categories		finite-limit preserving functors and natural transformations
LieAlg		Lie algebras		Lie algebra homomorphisms
Loc		locales		continuous maps
MSet		monoids		equivariant functions
MeasSp		measure spaces		measure-preserving maps
Mod		modules over all commutative rings		pairs of ring homomorphisms and $R$-module morphisms
MultiSet		multisets		multiset functions
Pr(∞,1)Cat		locally presentable (∞,1)-categories		colimit-preserving (∞,1)-functors
Prof		1-categories		profunctors and natural transformations
Rel		sets		binary relations
Ring		rings		ring homomorphisms
SDiff		smooth supermanifolds		smooth maps
SSet Cat		SSet-categories		1-morphisms, 2-morphisms and identity morphisms
SemiLat		semilattices		semilattice homomorphisms
Set		sets		functions
SimpAb		simplicial abelian groups		natural transformations
SimpSet		simplicial sets		natural transformations
Spec		spectra		(∞,1)-functors
SupLat		suplattices		suplattices homomorphisms
Top		topological spaces		continuous maps
Vect		vector spaces		linear maps
category of chain complexes		chain complexes		chain complex homomorphisms
category of representations		representations		natural transformations???
nCob		$(n-1)$-dimensional compact oriented manifolds		$n$-dimensional oriented cobordisms

Definition Table

Category		Definition
CartSp		Definition
Cat		Definition
FinSet		Definition
Ho(Cat)		Definition
Ho(Top)		Definition
LieAlg		Definition
Mod		Definition
Pr(∞,1)Cat		Definition
Prof		Definition
Rel		Definition
Set		Definition
Vect		Definition
category of chain complexes		Definition

(n,r)-Dimension Table

Category		(n,r)-Dimension
(∞,1)Cat		(∞,2)
(∞,1)-category of (∞,1)-categories, Homotopy theory of homotopy theories		(∞,1)
(∞,1)-category of (∞,1)-functors		(∞,1)
(∞,1)-category of (∞,1)-presheaves		(∞,1)
(∞,1)-category of (∞,1)-sheaves		(∞,1)
(∞,1)-category of cartesian sections		(∞,1)
(∞,n)-category of cobordisms		(∞,n)
stable (∞,1)-category of spectra		(∞,1)