# projective

## English

### Etymology

Formed by the suffixation of prōiect- (the perfect passive participial stem of the Classical Latin prōiciō, whence the English verb project) with the English -ive; however, compare the post-Classical Latin prōiectīvus (relating to purging).

### Pronunciation

projective (comparative more projective, superlative most projective)

1. projecting outward
2. of, relating to, or caused by a projection
3. (geometry) Of or related to projective geometry:
1. (now usually in set phrases, of a property of a geometric object, figure, etc.) Invariant under projective transformations.
• 1912 November, Edwin B. Wilson, Gillbert B. Lewis, “The Space-Time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics”, in Proceedings of the American Academy of Arts and Sciences, volume 45, number 11, page 503:
Hence the non-Euclidean angle is measured by one-half the logarithm of the cross-ratio of four rays. Although the Euclidean point of view has been adopted for simplicity, the final result, depending as it does only on the cross-ratio, is projective; it is therefore independent of the particular assumptions that the rays α and β are perpendicular and that the initial line bisects the angle between them.
2. (of a geometric object, figure, etc.) Defined in or inhabiting a projective space.
4. (algebra) In a technical sense, general (but not necessarily so general as to be free); involving such objects:
1. (module theory, of a (left) module ${\displaystyle M}$ over a ring ${\displaystyle R}$) Such that there is another (left) ${\displaystyle R}$-module ${\displaystyle N}$ with ${\displaystyle M\oplus N}$ a free ${\displaystyle R}$-module. Equivalently, projective in the category of modules (see below).
2. (category theory, most generally, of an object ${\displaystyle P}$ in a category) Such that, given an epimorphism ${\displaystyle e:E\to X}$ and morphism ${\displaystyle f:P\to X}$, ${\displaystyle f}$ factors through ${\displaystyle e}$; that is, there exists a morphism ${\displaystyle {\tilde {f}}:P\to E}$ with ${\displaystyle e\circ {\tilde {f}}=f}$.
3. (group theory, of a profinite group ${\displaystyle G}$) Such that every epimorphism from a profinite group onto ${\displaystyle G}$ has a right inverse which is a homomorphism.
4. () Such that every object in the resolution is projective.

### Noun

projective (plural projectives)

1. An assessment test that presents subjects with some sort of stimulus to which they react by projecting or imagining details.
• 1984, Lawrence A. Pervin, Personality: Theory and Research, →ISBN, page 322:
The projectives suggested considerable difficulty with women and a conflict between sexual preoccupation and hostility.
• 2009, Paul J. Frick, Christopher T. Barry, Randy W. Kamphaus, Clinical Assessment of Child and Adolescent Personality and Behavior, →ISBN:
For example, using projectives as a psychometric technique allows one to compare a person's score with those from a normative group, or with those from some relevant clinic group, or with some other clinically important criterion (e.g., response to treatment).
• 2012, Linda C. Wing, Bernard R. Gifford, Policy Issues in Employment Testing, →ISBN, page 174:
The unimpressive evidence for validity and operational problems related to projectives led Reilly and Chao to a pessimistic conclusion regarding projectives.
• 2015, Paul Hackett, Qualitative Research Methods in Consumer Psychology, →ISBN:
With its origins based in the field of psychology, projectives (also referred to as projective exercises or projective techniques) when used in qualitative research are fun "assignments" most often implemented during focus groups. Their goal is to elicit deeper, more visceral feelings from respondents -- about brands, products, concepts, advertising, and so on -- viewpoints that may go unmentioned when using more direct lines of inquiry.
2. A projective member of a category.
• 1965, Theory of Categories, →ISBN, page 109:
By 2.2 we see that this is a full, contravariant imbedding, and by 2.3 the image of A in (A, G) is a generating set of small projectives.
• 1999, Maurice Auslander, Idun Reiten, Sverre O. Smalø, Selected Works of Maurice Auslander - Part 1, →ISBN, page 490:
In particular our assumptions hold if B is an abelian category with enough projectives.
• 2012, M. Scott Osborne, Basic Homological Algebra, →ISBN, page 187:
The idea behind “cheating with projectives” in a pre-Abelian category with a separating class of projectives is this: Make the arrows do the work that elements do in concrete categories.
3. A statement about a conditional or potential state of affairs, as opposed to one about a situation that actually exists or existed.
• 1995, Keith E. Nelson, Zita Reger, Children's Language - Volume 8, →ISBN:
There was no basis for expecting differences in the frequency of projectives or turnabouts as a function of partner.
• 2007, Ronald James Williams, John C. Beckman, Williams' Hebrew Syntax, →ISBN, page 78:
The volitive moods (also called volitives, volitional forms, modals, or projectives) are the imperative, jussive, and cohortative.
• 2012, Jean Curthoys, Victor Dudman, Victor Dudman's Grammar and Semantics, →ISBN, page 74:
This implies they contain more information than projectives. For if language is a code, then every element of that code – here, every word, every form of a word – would register a distinct semantic ingredient.