nLab periodic table




Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels


Monoid theory

Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



In higher category theory, higher algebra, homotopy theory, and homotopy type theory there are several periodic tables, analogous to the periodic table of chemical elements. Just as this table allowed Менделеев to predict the existence of undiscovered elements in the table's gaps, so these periodic tables sometimes inspire us to invent new varieties of n-categories, n-groupoids, and n-truncated types.


The first periodic table of n-categories, due to John Baez and James Dolan in BaezDolan95, was the following. Originally, Baez and Dolan referred to it as a periodic table of kk-tuply monoidal nn-categories, as opposed to simply of nn-categories, but in today’s terminology the periodic table of kk-tuply monoidal nn-categories would be the one at the page k-tuply monoidal n-category. That page contains various notes on the table.

Fully filled out, the table looks like this:

k k ↓\ n n 2 -2 1 -1 0 0 1 1 2 2 ...
0 0 trivialtruth valuesetcategory2-category...
1 1 "trivialmonoidmonoidal categorymonoidal 2-category...
2 2 ""abelian monoidbraided monoidal categorybraided monoidal 2-category...
3 3 """symmetric monoidal categorysylleptic monoidal 2-category...
4 4 """"symmetric monoidal 2-category...

The columns with n=1n = -1 and n=2n = -2 were not originally there, but they follow the pattern of the table.

Eugenia Cheng and Nick Gurski wrote a paper observing that the above table does not quite behave ‘correctly’ with respect to higher morphisms if one restricts to (k1)(k-1)-simply connected (n+k)(n+k)-categories, but things do appear to come out correctly if one looks at the ‘pointed’ version of the table at k-tuply monoidal n-category. More on this can be found in the appendix to n-categories and cohomology.



Last revised on May 16, 2022 at 17:36:30. See the history of this page for a list of all contributions to it.