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
monoid theory in algebra:
With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
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 $k$-tuply monoidal $n$-categories, as opposed to simply of $n$-categories, but in today’s terminology the periodic table of $k$-tuply monoidal $n$-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:
$k$↓\$n$→ | $-2$ | $-1$ | $0$ | $1$ | $2$ | ... |
---|---|---|---|---|---|---|
$0$ | trivial | truth value | set | category | 2-category | ... |
$1$ | " | trivial | monoid | monoidal category | monoidal 2-category | ... |
$2$ | " | " | abelian monoid | braided monoidal category | braided monoidal 2-category | ... |
$3$ | " | " | " | symmetric monoidal category | sylleptic monoidal 2-category | ... |
$4$ | " | " | " | " | symmetric monoidal 2-category | ... |
⋮ | " | " | " | " | " | ⋱ |
The columns with $n = -1$ and $n = -2$ 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 $(k-1)$-simply connected $(n+k)$-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.