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

Series

A series is just a sequence.

But one says series instead of sequence when one is interested in studying its partial sums. In particular it means something different to say that a series converges than to say that a sequence converges. The series na nn\mapsto a_n converges if and only if the sequence n i<na in\mapsto \sum_{i\lt n}a_i converges.

Presheaves and copresheaves

A presheaf is just a contravariant functor, analogous to how a copresheaf is just a covariant 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 here 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 interested in regarding the category of functors with its structure of a topos: the presheaf topos.

Young diagrams

A Young diagram is a partition that wants to become a Young tableau.

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.

Persistence modules

A persistence module is just a sequence of linear maps (or a zigzag of these, for zigzag persistence modules), but one says persistence module to indicate that one is interested in the persistence diagrams encoding this sequence.

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. Though 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.

Tensor networks

A tensor network is just a string diagram (i.e. Penrose notation).

But in discussion of the AdS/CFT correspondence in solid state physics one says tensor network when regarding string diagrams as encoding quantum states in the Hilbert space which is the tensor product of all the external vertices.

Dynamical systems

A dynamical system is just a set SS with a group action f:G×SSf \colon G \times S \to S.

However, in dynamical systems theory, one takes the group GG to represent time, the set SS to represent the space of the dynamical system, and the group action ff to represent the laws of motion of this dynamical system.

Abstract rewriting systems

An abstract re-writing system is just a relation on some set XX.

However, calling this relation an abstract rewriting system indicates that one is interested in studying the behaviour of chains of related elements xx 1x 2x \to x_1 \to x_2 \to \cdots (thought of as successive stages of rewriting xx), for instance to see if they are confluent.

Curves

A curve in nn-dimensional Cartesian space is just a smooth function r: nr:\mathbb{R} \to \mathbb{R}^n.

However, in differential geometry, one takes the function rr as defining a parameterization of a smooth curve.

Further examples

Module objects

A module object in a monoidal category CC is just an action object over a monoid object in CC.

Typically one says “module object” to amplify a linear algebraic nature or intended meaning of such actions (which, for instance, would be lacking for a group action in Set, but be present for an action of the corresponding group algebra in Vect).

Subsets and predicates

In some foundations of mathematics, a subset of a set SS is just a predicate, a function with domain SS and codomain the class of truth values Ω\Omega.



Last revised on November 25, 2022 at 06:39:09. See the history of this page for a list of all contributions to it.