**homotopy theory, (∞,1)-category theory, homotopy type theory**

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also **algebraic topology**

**Introductions**

**Definitions**

**Paths and cylinders**

**Homotopy groups**

**Basic facts**

**Theorems**

The operation $V \mapsto V^*$ of forming dual vector spaces extends to a contravariant functor.

The **dual linear map** or **transpose map** of a linear map $A\colon V\to W$, is the linear map $A^* = A^T\colon W^*\to V^*$, given by

$\langle{A^*(w), v}\rangle = \langle{w, A(v)}\rangle$

for all $w$ in $W^*$ and $v$ in $V$.

This functor is, of course, the representable functor represented by $K$ as a vector space over itself (a line).

This construction is the notion of dual morphism applied in the monoidal category Vect with its tensor product monoidal structure.

Last revised on April 30, 2019 at 10:00:16. See the history of this page for a list of all contributions to it.