Noncommutative algebra – very simple. You draw diagrams which do not commute.
In this economy, we use a bootlegged copy of ``Higher Topos Theory.''
One time, outside the UMD math building, I turned to a sparrow beside me and asked, "What is the deal with weak equivalences?" He looked me directly in the eyes, defecated on a bench, and flew off. In this moment, I understood homotopy categories.
Applied Category Theory '23 was excellent. Met great people and left with interesting problems to work on. Many thanks to the organizers.
I was killed repeatedly in board games afterwards, but this is part of the strategy
Wrote a paper on fuzzy logic two years ago. My father gave me this for my birthday last week.
"M ⊭ φ" read as "M modeln't φ"
You trace a Jordan almond, it forms a Jordan curve. Very simple.
Autocorrect changing "Morleyization" to "Moralization."
This speaks volumes about society.
Nobody knows where \(NSOP_1\) theories came from. The original definition was written down by Shelah at some point. When asked later why he came up with it, he had no idea.
Categorical theories, model categories, and categorical model theory — all distinct concepts.
Moreover, when we consider an $R$-algebra $A$ s.t $A \models F$, then there's a unique map of rings into the Grothendieck ring of definable sets $K_0(F) \to K_0(Def_A)$* ; generally, this is not an isomorphism.
* $K_0(Def_M)$ is the Grothendieck ring of definable sets, see (https://arxiv.org/abs/math/0510133) for a precise definition.
defined by $\eta$ comes from a bijection between $\varphi$ and $\psi$-definable sets. We then can define the Grothendieck group of $F$, denoted $K_0(F)$, as a quotient of the free abelian group on $\sim$-equivalence classes of $\mathcal{L}$-formulae by its subgroup of formulae constructed as $[\varphi \wedge \psi] + [\varphi \vee \psi] − [\varphi] − [\psi]$ (where $\varphi, \psi$ have the same free vars). This becomes a ring by taking $[\varphi] \cdot [\psi]$ to be $[\varphi \wedge \psi]$.
Consider a ring $R$, and let $\mathcal{L}$ denote the language of rings over $R$ (e.g formulae composed of quantifiers and equalities of $R$-valued polynomials). It's possible to take a "Grothendieck group" of an $\mathcal{L}$-theory $F$.
First, define an equivalence relation $\sim$ on $\mathcal{L}$-formulae as $\varphi(x_1,..., x_n) \sim \psi(y_1,..., y_m) \iff$ there exists $\eta(x_1,...,x_n, y_1,..., y_m)$ s.t for every $R$-algebra $A$ where $A \models F$, the subset of $A^(m+n)$ ...
Imagine mentioning "symplectic meanders associated to seaweed" to someone, and imagine them believing it is not part of an elaborate joke. This cannot be done.
Watching a finch outside my window repeatedly pick up/chew/spit out pebbles. "He clearly has not learned" I think to myself. Later, the finch sits on my windowsill as I attempt to prove something about $NSOP_1$ theories, and thinks the same.
Moral of this? Eat pebbles in the garden.
Rodoljub the sparrow, who has made many of my diagrams commute.
...giving an equivalence between the $\infty$-category of commutative algebra objects of $Shv_{Sp}(X)$ and the $\infty$-category of $E^{\infty}$-ring-valued sheaves on X.
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Joined Nov 2018
