// given a "biarrow" AxB --> C, check we can transpose it via the getExponentialDiagram
@Test
public void testExponential() {
MultiArrow<DOT, ARROW> biArrow = fixtures.arrowFooBarToBaz();
List<ARROW> projections = biArrow.getProductDiagram().getProjections();
assertTrue(projections.size() == 2);
DOT a = projections.get(0).getTarget();
DOT b = projections.get(1).getTarget();
DOT c = biArrow.getArrow().getTarget();
ExponentialDiagram<DOT, ARROW> exponential = _topos.getExponentialDiagram(c, b);
// Check evaluation maps C^B x B -> C
MultiArrow<DOT, ARROW> ev = exponential.getEvaluation();
List<ARROW> expProjections = ev.getProductDiagram().getProjections();
assertTrue(expProjections.size() == 2);
assertTrue(expProjections.get(1).getTarget() == b);
assertTrue(ev.getArrow().getTarget() == c);
// and the universal property of evaluation
ARROW transpose = exponential.getTranspose(biArrow); // check this maps A -> B^C
assertTrue(transpose.getSource() == a);
assertTrue(
transpose.getTarget() == expProjections.get(0).getTarget()); // the exponent object itself
// Next, construct the arrow: transpose x 1 : A x B -> C^B x B as the productGetDiagram of A x B
// -> A -> C^B and A x B -> B -> B
ARROW x1 = transpose.compose(projections.get(0));
ARROW x2 = projections.get(1);
List<ARROW> multiplicands = new ArrayList<ARROW>();
multiplicands.add(x1);
multiplicands.add(x2);
ARROW t_x_1 =
ev.getProductDiagram()
.multiplyArrows(biArrow.getProductDiagram().getProduct(), multiplicands);
assertTrue(biArrow.getArrow().equals(ev.getArrow().compose(t_x_1)));
}
@Test
public void testCanonicalExponentialIso() {
DOT dot1 = fixtures.dotFoo();
DOT dot2 = fixtures.dotBar();
ExponentialDiagram<DOT, ARROW> expA = _topos.getExponentialDiagram(dot1, dot2);
ExponentialDiagram<DOT, ARROW> expB = _topos.getExponentialDiagram(dot1, dot2);
ARROW isoAB = _topos.canonicalIso(expA, expB);
ARROW isoBA = _topos.canonicalIso(expB, expA);
DOT dotA = exponentialDot(expA);
DOT dotB = exponentialDot(expB);
assertSame(isoAB.getSource(), dotA);
assertSame(isoAB.getTarget(), dotB);
assertSame(isoBA.getSource(), dotB);
assertSame(isoBA.getTarget(), dotA);
assertEquals(dotA.getIdentity(), isoBA.compose(isoAB));
assertEquals(dotB.getIdentity(), isoAB.compose(isoBA));
}
