Grothendieck’s theorem: “Every AB5 category with a generator has an injective cogenerator” (Grothendieck 1957).

This initial statement of Grothendieck’s theorem is now restated as:

Every Grothendieck category 𝒢 has an injective cogenerator.

1. Accents This entry tests names and accents: Schrödinger, Noether, Poincaré, Gödel, Möbius, and naïve coordinate choices.

The TeX-safe form is also common: Schrödinger, Poincaré, Gödel, and Möbius. 2. Smart Quotes This sentence uses pasted curly quotes: “energy is conserved” and ‘phase is relative’.

The TeX form should also work: “energy is conserved” and ‘phase is relative’. 3. Dashes And Minus This tests dash-like characters: classical mechanics – quantum mechanics, Hamiltonian dynamics — Lagrangian dynamics, and the Unicode minus sign -1.

The TeX forms are: classical mechanics – quantum mechanics, Hamiltonian dynamics — Lagrangian dynamics, and −1.

4. Greek And Symbols We test Greek letters in prose: α, β, γ, 𝜃, ϕ, ψ, Ω.

The TeX math form should render cleanly:

5. Mixed Reference A reference with correct TeX quotes:

[1] Griffiths, D. “Introduction to Quantum Mechanics” Prentice Hall, New Jersey, 1995.

A pasted Unicode version:

[2] Schrödinger, E. “Quantisierung als Eigenwertproblem,” Annalen der Physik, 1926.

Next round of tests:

1. Unicode Minus In Math-ish Prose The measured offset was -1, while the TeX-safe expression is −1. 2. Mixed Accents In Reference [3] Poincaré, H. “Sur le problème des trois corps,” Acta Mathematica, 1890.

[4] Möbius, A. “Der barycentrische Calcul,” 1827. 3. Nonbreaking Space / Copy-Paste Space Paste this from a web page if possible, where the space after Planck may be nonbreaking: Planck constant and Boltzmann constant should not break storage or rendering. 4. Greek In Existing Math Direct Unicode inside display math:

5. A “Should Not Auto-Fix” Control Hand-written TeX remains best in prose: α, β, and Ω should render cleanly.