A white object under a blue light will appear blue. However, your brain knows that the light is blue, and takes that into account when interpreting the scene. You'll probably still "know" the object is white.
It's possible for stage technicians to take advantage of this and make you think that blue objects are white, by lighting them with blue light. (This is more common with red and green: lighting a green object with green light makes you think it's white; lighting it with red light will make it look black instead. This is a way to suddenly reveal figures that had previously been hidden on stage.)
Why do you call it a "white object" ? Could it be because it looks white under white light ... light composed of every color ? Well then, the object must reflect every color. So it appears to be the color of whatever light you illuminate it with. Yellow under yellow light, etc.
The object will appear white because it reflects all colors of light equally, resulting in a white appearance.
The object absorbs all colors of light except for red, which is reflected off the object and is what we perceive as the color red.
If an object is white in sunlight it will be green in green light. A white object reflects all wavelengths (colors) of light that shine on it. If only green light shines on it, that color will be reflected and the object will look green.
A white object would appear to be red under a red light because the object reflects red light and absorbs other colors. This is because the color of an object depends on the wavelengths of light it reflects.
A blue object under white light will appear as blue because it reflects blue light wavelengths and absorbs other colors. The white light contains all the colors of the visible spectrum, but the blue wavelengths are the ones reflected by the object, giving it its blue color.
There are two ways that the color white occurs, as light and as the color of an object. When you see white light it is a combination of many colors of light, because white light is a combination of all light. When you see a white object, that object is reflecting all of the colors of light back at you, because that surface doesn't absorb any of the colors of light.
If an object has no color of its own and light hits it, the object will appear white. This is because the object reflects all visible light wavelengths equally, resulting in white light being perceived by our eyes.
A yellow object under a yellow light will appear very bright and vibrant, as the object will reflect the light in a consistent color. This can potentially make it difficult to differentiate between the object and the surrounding light source.
The color of the object that you see will depend on the wavelengths of light that are reflected. Objects appear a certain color because they absorb certain wavelengths of light and reflect others. White light contains all colors of the spectrum, so the color you see is the result of the wavelengths that are reflected by the object.
The reason a black object is described as "a black object" is becasuse even under white light, where every color is available, the thing absorbs all of them, and doesn't reflect any color. Illuminating it with light of only a single color isn't going to change that. It'll still appear black.
When white light hits an object, the object absorbs certain colors of light and reflects others. The colors that are reflected off the object are what we perceive as the color of the object.