Matpic
mathematics, in pictures
Concept

Relation

An equation a presentation imposes on words in its generators, saying that some product is the identity. Adding one is what distinguishes a quotient from a free object, and in topology it is what gluing a disc along a loop does to a fundamental group.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

j is one more than i, counting round — as a grid, with both quantifier readings. A grid of marks for a relation, with the row and column facts the two quantifier orders ask about.

Every row, or one column

For every person there is someone who loves them, and there is someone who loves everyone, are the same six words in a different order. Draw the relation as a grid and they become two obviously different questions — one about rows, one about columns.

logic · Quantifiers Three spaces, cut into pieces. 3 panels, drawn from: a sphere split into two caps meeting along a circle, two circles joined at a point, and a square whose opposite edges are glued into a torus. Each carries the fundamental group that van Kampen's theorem computes for it.

Cutting a space to find its group

A space assembled from two pieces has a fundamental group assembled from theirs, and the recipe is exact — take everything both groups offer and impose the relations the overlap forces. Almost every fundamental group anybody knows is computed this way, including all of the surfaces.

topology · Homotopy

Named alongside it

The objects these essays reach for when they reach for this one.

CounterexampleFree groupFundamental groupGeneratorGluingGroup presentationHomotopyNegationPredicate logicQuantifierQuantifier orderSurface

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