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2 Added reference

Here's a pretty abstract answer. Bill Lawvere, working on his ideas about "axiomatic cohesion", often talks about a Nullstellensatz that at first sight has nothing whatsoever to do with zeros of polynomials.

Axiomatic cohesion is about pinning down the properties that a category of "spaces" should have. Here the word "space" is up for negotiation; the word "cohesion" is to indicate that a space should somehow cohere to itself. The typical situation when you have a category Sp of spaces (in whatever sense) is that there's a string of adjoint functors $$\pi_0 \dashv D \dashv U \dashv I$$ between Sp and Set, where

• $\pi_0$ gives the set of connected components of a space
• $D$ gives the discrete space on a set
• $U$ gives the set of points of a space
• $I$ gives the indiscrete or codiscrete space on a set.

A couple of axioms are imposed on these adjunctions. Under those axioms, there are canonical natural transformations $U \to \pi_0$ and $D \to I$, and the former is an epimorphism iff the latter is a monomorphism. Lawvere calls this property (that $U \to \pi_0$ is an epimorphism) the Nullstellensatz.

In concrete terms, this says something like: for a space $X$, the quotient map from $X$ to its set of connected-components is surjective. Why call that the Nullstellensatz? I have no idea. Here's the reference (top of p.44).

In similar usage, Colin McLarty says (bottom of p.125) that a topos satisfies the Nullstellensatz if for every nonempty object $X$ there is at least one map $1 \to X$. Again I don't understand the usage. Maybe someone else will wander along and help out.

Update: Peter Johnstone has just published a paper all about this. He prefers the term "punctual local connectedness" to "Nullstellensatz". Here it is: http://tac.mta.ca/tac/volumes/25/3/25-03abs.html

1

Here's a pretty abstract answer. Bill Lawvere, working on his ideas about "axiomatic cohesion", often talks about a Nullstellensatz that at first sight has nothing whatsoever to do with zeros of polynomials.

Axiomatic cohesion is about pinning down the properties that a category of "spaces" should have. Here the word "space" is up for negotiation; the word "cohesion" is to indicate that a space should somehow cohere to itself. The typical situation when you have a category Sp of spaces (in whatever sense) is that there's a string of adjoint functors $$\pi_0 \dashv D \dashv U \dashv I$$ between Sp and Set, where

• $\pi_0$ gives the set of connected components of a space
• $D$ gives the discrete space on a set
• $U$ gives the set of points of a space
• $I$ gives the indiscrete or codiscrete space on a set.

A couple of axioms are imposed on these adjunctions. Under those axioms, there are canonical natural transformations $U \to \pi_0$ and $D \to I$, and the former is an epimorphism iff the latter is a monomorphism. Lawvere calls this property (that $U \to \pi_0$ is an epimorphism) the Nullstellensatz.

In concrete terms, this says something like: for a space $X$, the quotient map from $X$ to its set of connected-components is surjective. Why call that the Nullstellensatz? I have no idea. Here's the reference (top of p.44).

In similar usage, Colin McLarty says (bottom of p.125) that a topos satisfies the Nullstellensatz if for every nonempty object $X$ there is at least one map $1 \to X$. Again I don't understand the usage. Maybe someone else will wander along and help out.
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