symmetric monoidal (∞,1)-category of spectra
natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
categorification
monoid theory in algebra:
A version of the k-tuply monoidal n-groupoid but for $A_n$ spaces.
Invoking the homotopy hypothesis, we define a $k$-tuply associative $n$-groupoid to be an $A_k$-$n$-truncated type: a topological space which is a homotopy n-type with a multiplication that is associative up to higher homotopies involving up to nn variables.
There is a periodic table of $k$-tuply monoidal $n$-groupoids:
$k \,\backslash\, n$ | $-1$ | $0$ | $1$ | $2$ | $\cdots$ | $\infty$ |
---|---|---|---|---|---|---|
$0$ | trivial | pointed set | pointed groupoid | pointed 2-groupoid | $\cdots$ | pointed $\infty$-groupoid |
$1$ | trivial | pointed set | pointed groupoid | pointed 2-groupoid | $\cdots$ | pointed $\infty$-groupoid $A_1$-space |
$2$ | trivial | unital magma | magmoidal groupoid with unit | magmoidal 2-groupoid with unit? | $\cdots$ | H-space/A2-space |
$3$ | “ | monoid | H-monoidal groupoid | H-monoidal 2-groupoid? | $\cdots$ | H-monoid?/A3-space |
$4$ | “ | “ | monoidal groupoid | A4 2-groupoid? | $\cdots$ | A4-space |
$5$ | “ | “ | “ | monoidal 2-groupoid? | $\cdots$ | A5-space |
$\vdots$ | “ | “ | “ | “ | $\ddots$ | $\vdots$ |
$\infty$ | trivial | monoid | monoidal groupoid | monoidal 2-groupoid? | $\cdots$ | $A_\infty$-space |
Last revised on July 14, 2022 at 08:49:15. See the history of this page for a list of all contributions to it.