nLab
concept with an attitude

Contents

Contents

Idea

In mathematics it happens at times that one and the same concept is given two different names to indicate a specific perspective, a certain attitude as to what to do with such objects.

Examples

Presheaves

A presheaf is just a contravariant functor.

(More specifically, an “SS-valued presheaf” is a contravariant functor with codomain a given category SS; in modern category theory the “default” value of SS for a presheaf is usually Set.)

But one says presheaf instead of (set-valued) contravariant functor when one is interested in studying its sheafification, or even if one is just intersted in regarding the category of functors with its structure of a topos: the presheaf topos.

Quivers

A quiver is just a directed graph (pseudograph, to be explicit).

But one says quiver instead of directed graph when one is interested in studying quiver representations: functors from the free category on that graph to the category of finite-dimensional vector spaces.

Fields (physics)

A field (in physics) is just a section of a fiber bundle.

But in mathematical physics one says field instead of section of a fiber bundle to indicate that one is going to consider a Lagrangian density on the corresponding jet bundle of the given fiber bundle (then called the field bundle ) and study the induced classical or quantum field theory.

Random variables and estimators

Both random variables and estimators are almost always just real valued measurable maps. But sometimes the former takes more general values in some Polish space instead.

But in probability theory a random variable is interpreted as a map from a sample space to a space of states which represents the observed outcomes or outcomes predicted by some model. In contrast in statistics an estimator guesses or estimates a certain parameter associated to some random model.

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Last revised on May 6, 2019 at 03:31:58. See the history of this page for a list of all contributions to it.