Background
Basic concepts
equivalences in/of $(\infty,1)$-categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
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
A symmetric monoidal (∞,1)-category $(C,\otimes)$ is closed if for each object $X \in C$ the (∞,1)-functor
given by forming the tensor product with $C$ has a right adjoint (∞,1)-functor
(In cases where the monoidal structure is not assumed symmetric, the property of possessing a right adjoint to tensoring on the left (resp. right) is called left (resp. right) closed, while closed is used for the properties jointly (Def. Definition 4.1.1.15 of Higher Algebra).)
Every (∞,1)-topos with its structure of a cartesian monoidal (∞,1)-category is closed. See there for details.
The (∞,1)-category of (∞,1)-modules over an E-∞ ring is closed.
symmetric monoidal category, symmetric monoidal (∞,1)-category
closed monoidal category, closed monoidal $(\infty,1)$-category
Last revised on May 26, 2022 at 18:07:59. See the history of this page for a list of all contributions to it.