#homologytheory — Public Fediverse posts

#Mathober #Mathober2025

The prompt for day 4 was 'Strongly'. In linear algebra, the dual of a vector space V is the space V* of linear maps from V to the field. There's a natural inclusion 𝑉⊗𝑉* → Hom(𝑉,𝑉) and we call V *strongly* dualisable if and only if this map has an inverse.

The strongly dualisable vector spaces are precisely the finite dimensional ones. This is useful because it allows us to prove statements about finite dimensional vector spaces in a basis-free way. For example the trace is easy to define for strongly dualisable objects without mentioning any basis.

The same approach works for modules over a commutative ring. The strongly dualisable objects are precisely the finitely-generated projective modules. For example for ℤ this includes free modules like ℤ¹⁰ but not quotients like ℤ/12ℤ.

But if we work in higher algebra something interesting happens! In the category of chain complexes a chain is strongly dualisable if and only if it has finitely many nonzero modules, all of which are themselves strongly dualisable.

The ∞-category of chain complexes include the category of modules by sending V to … → 0 → 0 → 𝑉. But the chain complex … → 0 → 0 → ℤ/12ℤ is equivalent to … → 0 → ℤ → ℤ, which is dualisable!

So using ∞-categories we can extend the methods of finite-dimensional linear algebra to modules that were previously out of reach!

#Math #Maths #Mathematics #CategoryTheory #Homology #HomologyTheory

#Mathober #Mathober2025

The prompt for day 4 was 'Strongly'. In linear algebra, the dual of a vector space V is the space V* of linear maps from V to the field. There's a natural inclusion 𝑉⊗𝑉* → Hom(𝑉,𝑉) and we call V *strongly* dualisable if and only if this map has an inverse.

The strongly dualisable vector spaces are precisely the finite dimensional ones. This is useful because it allows us to prove statements about finite dimensional vector spaces in a basis-free way. For example the trace is easy to define for strongly dualisable objects without mentioning any basis.

The same approach works for modules over a commutative ring. The strongly dualisable objects are precisely the finitely-generated projective modules. For example for ℤ this includes free modules like ℤ¹⁰ but not quotients like ℤ/12ℤ.

But if we work in higher algebra something interesting happens! In the category of chain complexes a chain is strongly dualisable if and only if it has finitely many nonzero modules, all of which are themselves strongly dualisable.

The ∞-category of chain complexes include the category of modules by sending V to … → 0 → 0 → 𝑉. But the chain complex … → 0 → 0 → ℤ/12ℤ is equivalent to … → 0 → ℤ → ℤ, which is dualisable!

So using ∞-categories we can extend the methods of finite-dimensional linear algebra to modules that were previously out of reach!

#Math #Maths #Mathematics #CategoryTheory #Homology #HomologyTheory

#Mathober #Mathober2025

The prompt for day 4 was 'Strongly'. In linear algebra, the dual of a vector space V is the space V* of linear maps from V to the field. There's a natural inclusion 𝑉⊗𝑉* → Hom(𝑉,𝑉) and we call V *strongly* dualisable if and only if this map has an inverse.

The strongly dualisable vector spaces are precisely the finite dimensional ones. This is useful because it allows us to prove statements about finite dimensional vector spaces in a basis-free way. For example the trace is easy to define for strongly dualisable objects without mentioning any basis.

The same approach works for modules over a commutative ring. The strongly dualisable objects are precisely the finitely-generated projective modules. For example for ℤ this includes free modules like ℤ¹⁰ but not quotients like ℤ/12ℤ.

But if we work in higher algebra something interesting happens! In the category of chain complexes a chain is strongly dualisable if and only if it has finitely many nonzero modules, all of which are themselves strongly dualisable.

The ∞-category of chain complexes include the category of modules by sending V to … → 0 → 0 → 𝑉. But the chain complex … → 0 → 0 → ℤ/12ℤ is equivalent to … → 0 → ℤ → ℤ, which is dualisable!

So using ∞-categories we can extend the methods of finite-dimensional linear algebra to modules that were previously out of reach!

#Math #Maths #Mathematics #CategoryTheory #Homology #HomologyTheory

#Mathober #Mathober2025

The prompt for day 4 was 'Strongly'. In linear algebra, the dual of a vector space V is the space V* of linear maps from V to the field. There's a natural inclusion 𝑉⊗𝑉* → Hom(𝑉,𝑉) and we call V *strongly* dualisable if and only if this map has an inverse.

The strongly dualisable vector spaces are precisely the finite dimensional ones. This is useful because it allows us to prove statements about finite dimensional vector spaces in a basis-free way. For example the trace is easy to define for strongly dualisable objects without mentioning any basis.

The same approach works for modules over a commutative ring. The strongly dualisable objects are precisely the finitely-generated projective modules. For example for ℤ this includes free modules like ℤ¹⁰ but not quotients like ℤ/12ℤ.

But if we work in higher algebra something interesting happens! In the category of chain complexes a chain is strongly dualisable if and only if it has finitely many nonzero modules, all of which are themselves strongly dualisable.

The ∞-category of chain complexes include the category of modules by sending V to … → 0 → 0 → 𝑉. But the chain complex … → 0 → 0 → ℤ/12ℤ is equivalent to … → 0 → ℤ → ℤ, which is dualisable!

So using ∞-categories we can extend the methods of finite-dimensional linear algebra to modules that were previously out of reach!

#Math #Maths #Mathematics #CategoryTheory #Homology #HomologyTheory

#Mathober #Mathober2025

The prompt for day 4 was 'Strongly'. In linear algebra, the dual of a vector space V is the space V* of linear maps from V to the field. There's a natural inclusion 𝑉⊗𝑉* → Hom(𝑉,𝑉) and we call V *strongly* dualisable if and only if this map has an inverse.

The strongly dualisable vector spaces are precisely the finite dimensional ones. This is useful because it allows us to prove statements about finite dimensional vector spaces in a basis-free way. For example the trace is easy to define for strongly dualisable objects without mentioning any basis.

The same approach works for modules over a commutative ring. The strongly dualisable objects are precisely the finitely-generated projective modules. For example for ℤ this includes free modules like ℤ¹⁰ but not quotients like ℤ/12ℤ.

But if we work in higher algebra something interesting happens! In the category of chain complexes a chain is strongly dualisable if and only if it has finitely many nonzero modules, all of which are themselves strongly dualisable.

The ∞-category of chain complexes include the category of modules by sending V to … → 0 → 0 → 𝑉. But the chain complex … → 0 → 0 → ℤ/12ℤ is equivalent to … → 0 → ℤ → ℤ, which is dualisable!

So using ∞-categories we can extend the methods of finite-dimensional linear algebra to modules that were previously out of reach!

#Math #Maths #Mathematics #CategoryTheory #Homology #HomologyTheory

two interesting vacation finds at a new bookshop, Hello Books, in #SouthHaven #Michigan #logic #HomologyTheory