Martin Escardo on Nostr: nprofile1q…ufa4k writes "A commutative algebraist studies commutative rings while ...
nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqknzsux7p6lzwzdedp3m8c3c92z0swzc0xyy5glvse58txj5e9ztqaufa4k (nprofile…fa4k) writes
"A commutative algebraist studies commutative rings while an algebraic geometer might work in the opposite category and think of them as affine schemes. "
Similarly, algebraists work with frames, whereas topologists work in the opposite category, called the category of locales, and think of them as point-free spaces.
In reality, though, everybody working in this field is a bit of an algebraist and a bit of a topologist at the same time. The interaction goes both sways, and is very productive.
Also, the algebraic side is often understood as a logical side. (Cf. Steve Vickers "topology via logic" and Abramsky's "logic of observable properties".)
And of course Grothendieck toposes generalize locales, and there, too, we have "geometric morphisms", which in the opposite category are called "logical morphisms" (but perhaps they could also be called "algebraic morphisms", although nobody does that).
"A commutative algebraist studies commutative rings while an algebraic geometer might work in the opposite category and think of them as affine schemes. "
Similarly, algebraists work with frames, whereas topologists work in the opposite category, called the category of locales, and think of them as point-free spaces.
In reality, though, everybody working in this field is a bit of an algebraist and a bit of a topologist at the same time. The interaction goes both sways, and is very productive.
Also, the algebraic side is often understood as a logical side. (Cf. Steve Vickers "topology via logic" and Abramsky's "logic of observable properties".)
And of course Grothendieck toposes generalize locales, and there, too, we have "geometric morphisms", which in the opposite category are called "logical morphisms" (but perhaps they could also be called "algebraic morphisms", although nobody does that).