nLab concept with an attitude

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 $n\mapsto a_n$ converges if and only if the sequence $n\mapsto \sum_{i\lt n}a_i$ converges.

Lists

A list is just a tuple, but calling it a list indicates that one is interested in the operation of concatenation of lists.

Presheaves and copresheaves

A presheaf is just a contravariant functor, analogous to how a copresheaf is just a covariant functor.

(More specifically, an “$S$-valued presheaf” is a contravariant functor with codomain a given category $S$; in modern category theory the “default” value of $S$ 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.

Sieves

A sieve is just a subfunctor of a representable functor, but calling them “sieves” serves to indicate that (mostly) one is interested in regarding these as covers of a Grothendieck topology or coverage, making the ambient category into a site.

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.

Causal set

A causal set is just a partially ordered set.

But one says causal set in order to indicate that one means to think of the elements of the poset as spacetime events and of the order-relation as given by causality.

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 $S$ with a group action $f \colon G \times S \to S$.

However, in dynamical systems theory, one takes the group $G$ to represent time, the set $S$ to represent the space of the dynamical system, and the group action $f$ 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 $X$.

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

Curves

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

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

Flows of time

A flow of time is just a set endowed with a strict partial order.

However, in temporal logic, a (point-based) model is a pair consisting of a flow of time and a valuation.

Further examples

Module objects

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

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 $S$ is just a predicate, a function with domain $S$ and codomain the class of truth values $\Omega$.

Q-categories

Q-categories are just coreflective subcategories in the setup in which they are used as a data for introducing generalizations of sheaves, monopresheaves (separated presheaves) and epipresheaves.

References

[Urs Schreiber:] In June 2018, while lecturing at Nesin Math Village (Categories and Toposes) my co-lecturer Murad Özaydın at some point said that “A quiver is a digraph with attitude.” Later it occurred to me that it may be entertaining and maybe even useful (for newcomers) to make explicit more examples of this kind of situation, so I coined the general term of “concept with an attitude” in Oct 2018.