Vikraman's site https://vikraman.org/notes/ Vikraman's personal website quarto-1.10.18 Tue, 21 Jul 2026 00:00:00 GMT System L Vikraman Choudhury https://vikraman.org/notes/system-l.html

This is a fine-grained presentation of call-by-value System L.

Syntax

Definition 1 (Types).

Definition 2 (Contexts).

Definition 3 (Judgements).

Definition 4 (Commands).

Definition 5 (Values).

Definition 6 (Terms).

Definition 7 (Coterms).

Weakening & Substitution

Definition 8 (Weakening).

Lemma 1 (Weakening). Given and , the following rules are admissible:

Definition 9 (Substitution).

Lemma 2 (Substitution). Given and , the substitution pair has admissible substitution rules:

Equational Theory

Definition 10 (Let Bindings).

Definition 11 (Equational Theory). Each judgement has an equivalence relation and congruence for every formation rule in Definition 4, Definition 5, Definition 6, Definition 7.

Eta.

Beta (cut elimination).

Strong Normalisation

We show strong normalisation by reducibility candidates.

Definition 12 (Reduction). We orient the beta-eta equations of Definition 11 left-to-right to define a type-preserving one-step reduction relation on commands.

Definition 13 (Strong Normalisation).

Definition 14 (Reducibility Candidates). By induction on , mutually define for , and for :

Lemma 3 (Reducibility is Stable Under Weakening). Given and : if at then at ; dually, if at then at .

Lemma 4 (Orthogonality). If and , then .

Definition 15 (Reducible Substitutions). For and :

Lemma 5 (Reducible Substitutions are Stable and Extendable). and are stable under weakening; and if , , then , and dually for and .

Theorem 1 (Fundamental Lemma). Given , with and :

  • if , then ;
  • if , then for every , ;
  • if , then ;
  • if , then .

Theorem 2 (Strong Normalisation). For every , .

Definition 16 (Normal Forms). , and is the reflexive-transitive closure of .

Theorem 3 (Progress). For every command : either , or for some .

Theorem 4 (Normalisation). For every , there exists with and .

Back to top

Citation

BibTeX citation:
@online{choudhury2026,
  author = {Choudhury, Vikraman},
  title = {System {L}},
  date = {2026-07-21},
  url = {https://vikraman.org/notes/system-l.html},
  langid = {en}
}
For attribution, please cite this work as:
Choudhury, Vikraman. 2026. “System L.” July 21. https://vikraman.org/notes/system-l.html.
]]> https://vikraman.org/notes/system-l.html Tue, 21 Jul 2026 00:00:00 GMT Degeneracy theorems Vikraman Choudhury https://vikraman.org/notes/degeneracy.html

Some interesting degeneracy theorems that I always forget the precise statements of.

Lemma 1 If is a cartesian closed category with an initial object, then any two maps are equal. Dually, if is a cocartesian coclosed category with a terminal object, then any two maps are equal.

Proof. , since preserves colimits. Hence, , and so are equal. If , then .

Lemma 2 If is a cartesian closed category and a cocartesian coclosed category, then is a poset.

Proof. Suppose are two maps in . Then , and so by Lemma 1, .

Back to top

Citation

BibTeX citation:
@online{choudhury2026,
  author = {Choudhury, Vikraman},
  title = {Degeneracy Theorems},
  date = {2026-06-20},
  url = {https://vikraman.org/notes/degeneracy.html},
  langid = {en}
}
For attribution, please cite this work as:
Choudhury, Vikraman. 2026. “Degeneracy Theorems.” June 20. https://vikraman.org/notes/degeneracy.html.
]]> https://vikraman.org/notes/degeneracy.html Sat, 20 Jun 2026 00:00:00 GMT Normalization of sums Vikraman Choudhury https://vikraman.org/notes/sums.html

Definition 1 A posite is a poset with a binary relation such that:

  • if and , then ;

  • if and , then there exists such that , and for all , there exists such that .

Given posites and , a morphism of posites is a monotone function such that:

  • given any , for some , we have ;

  • if , , then for some , , we have ;

  • if , then .

Back to top

Citation

BibTeX citation:
@online{choudhury2026,
  author = {Choudhury, Vikraman},
  title = {Normalization of Sums},
  date = {2026-05-15},
  url = {https://vikraman.org/notes/sums.html},
  langid = {en}
}
For attribution, please cite this work as:
Choudhury, Vikraman. 2026. “Normalization of Sums.” May 15. https://vikraman.org/notes/sums.html.
]]> https://vikraman.org/notes/sums.html Fri, 15 May 2026 00:00:00 GMT List objects Vikraman Choudhury https://vikraman.org/notes/lists.html

List Objects

Let be a strict monoidal category, and be an object of .

A list algebra on is an object with a pair of maps .

A list object on is a list algebra , such that:

  • for any -parameterised list algebra , there is a unique map satisfying:

A monoid object in is a pair of maps satisfying:

Lemma 1 is a monoid structure on .

Theorem 1 The list object construction gives .

Symmetric List Objects

Now, we switch to being symmetric strict monoidal, with symmetry .

A symmetric list object on is a list object satisfying:

Lemma 2 is a commutative monoid structure on .

Lemma 3 If is a list object on , the following are equivalent:

  • is a symmetric list object on .
  • is a commutative monoid.
  • is the free commutative monoid on .

Theorem 2 The symmetric list object construction gives .

Lemma 4 If is symmetric monoidal closed and has the relevant colimits, the following are equivalent:

  • is a symmetric list object on .
  • is the sequential colimit of the symmetric powers of .
Back to top

Citation

BibTeX citation:
@online{choudhury2026,
  author = {Choudhury, Vikraman},
  title = {List Objects},
  date = {2026-04-22},
  url = {https://vikraman.org/notes/lists.html},
  langid = {en}
}
For attribution, please cite this work as:
Choudhury, Vikraman. 2026. “List Objects.” April 22. https://vikraman.org/notes/lists.html.
]]> https://vikraman.org/notes/lists.html Wed, 22 Apr 2026 00:00:00 GMT Sandbox Vikraman Choudhury https://vikraman.org/notes/sandbox.html

Introduction

This is a sandbox document to test typesetting and layout. This website and notes are built using Quarto1.

1 Quarto has a website and a Github repository.

Math

This is a math character , and this is some inline math .

This is a prooftree in display math mode2.

2 This is another prooftree but in the margin.

Theorems

This is a theorem with proof, labeled and cross-referenced Theorem 1.

Theorem 1 (Yoneda lemma). For a locally small category , a functor , and an object , there is a natural isomorphism

Proof. Obvious.

Typography

This paragraph checks ligatures, swashes, and old-style figures. Standard ligatures should join fi, fl, ffi, and ffl into single glyphs, and discretionary ligatures may join Th, ct, st, and fj. Old-style figures should show varying heights and depths in a run of digits like 1234567890.

Quixotic zealots waltz through the frozen jungle — in italic, swash capitals, if the font provides them, should show decorative flourishes on the initial letters Q, Z, W, and J.

This paragraph checks the monospace font: inline code like let x = (n, k) => factorial(n) / (factorial(k) * factorial(n - k)); should stay legible next to prose.

# comment in italic-ish monospace
def binomial(n, k):
    return factorial(n) // (factorial(k) * factorial(n - k))

Diagrams

This renders diagrams in markdown Figure 1.

This is some imagified LaTeX math.

This tests a note-specific macro from header-includes.

A TikZ picture Figure 1 (a).

(a)

A commutative diagram Figure 1 (b).

(b)
Figure 1

This is a quiver iframe.

Back to top ]]> https://vikraman.org/notes/sandbox.html Sun, 09 Nov 2025 00:00:00 GMT