# Element (category theory)

Within category theory, an object \( X \) in a category is said to have a **generalized element** whenever there exists a morphism \( x : A \to X \) whose codomain is \( X \); the domain \( A \) is referred to as the shape, or stage of definition, of that element.<sup>[1](https://www.lesswrong.com/w/generalized-element)</sup> The notion exists because objects of an arbitrary category have no elements in any set-theoretic sense, while maps into an object play the role of elements once a shape is chosen<sup>[2](https://www.voutsadakis.com/TEACH/LECTURES/CATEGORY/Chapter4.pdf)</sup>. Strictly, "generalized element" is nothing more than a synonym of "map", but thinking of maps as elements is often useful<sup>[2](https://www.voutsadakis.com/TEACH/LECTURES/CATEGORY/Chapter4.pdf)</sup>.

| Key fact | Detail |
|---|---|
| Definition | A generalized element of an object \( X \) is a morphism \( x : A \to X \); \( A \) is its shape or stage of definition<sup>[1](https://www.lesswrong.com/w/generalized-element)</sup> |
| Ordinary elements | In Set, elements of \( X \) correspond to functions from a one-element set \( I = \{\*\} \) to \( X \)<sup>[1](https://www.lesswrong.com/w/generalized-element)</sup> |
| Global element | If the category has a terminal object 1, an element defined at stage 1 is a global element (global section)<sup>[3](https://users-math.au.dk/kock/sdg99.pdf)</sup> |
| Action by composition | For \( f : B \to C \) and \( x \in_A B \), one writes \( f(x) \) for the composite \( f \circ x \), an A-element of \( C \)<sup>[4](https://archive.bukohome.com/content/wiktionary_en_all_maxi_2024-05/A/generalized_element)</sup> |
| Yoneda content | \( \mathrm{Hom}(-, A) \) sends \( X \) to the set of X-shaped generalized elements of \( A \); the Yoneda embedding is fully faithful, so no information about \( A \) is lost<sup>[5](https://math.jhu.edu/~eriehl/context.pdf)</sup> |
| Equality test | Parallel morphisms \( f, g : A \Rightarrow B \) are equal if they induce the same functions on generalized elements<sup>[5](https://math.jhu.edu/~eriehl/context.pdf)</sup> |
| Algebraic geometry | The functor of points \( h_X : \mathrm{Sch}^{\mathrm{opp}} \to \mathrm{Sets} \), \( T \mapsto \mathrm{Mor}(T, X) \), describes a scheme through its generalized elements<sup>[6](https://stacks.math.columbia.edu/tag/01J5)</sup> |

## From set elements to generalized elements

For a set \( X \), there is a natural correspondence between elements of \( X \) and functions from a fixed one-element set \( I = \{\*\} \) to \( X \). An element \( x \) of \( X \) defines the constant function \( f(i) = x \); conversely, a function \( f : I \to X \) yields the element \( f(\*) \)<sup>[1](https://www.lesswrong.com/w/generalized-element)</sup>. This observation, that an [element of a set](https://www.edgepedia.org/element-of-a-set) \( A \) is the same thing as a map \( 1 \to A \), inspires the general definition: a map \( S \to A \) is a generalized element of \( A \) of shape \( S \)<sup>[2](https://www.voutsadakis.com/TEACH/LECTURES/CATEGORY/Chapter4.pdf)</sup>.

The domain carries meaning, and different shapes give different kinds of element. When \( A \) is a set, a generalized element of shape 1 is an ordinary element, and a generalized element of shape \( \mathbb{N} \) is a sequence in \( A \). In the category of topological spaces, the shape-1 elements (for the one-point space) are the points, while the generalized elements of shape \( S^1 \), the circle, are loops. In categories of geometric objects one might equally say "figures of shape \( S \)"<sup>[2](https://www.voutsadakis.com/TEACH/LECTURES/CATEGORY/Chapter4.pdf)</sup>.

Morphisms act on generalized elements by composition. For \( f : B \to C \) and a generalized element \( x \in_A B \), the composite \( f \circ x \) is written \( f(x) \) and is an A-element of \( C \); at each stage \( A \), \( f \) takes A-elements of \( B \) to A-elements of \( C \)<sup>[4](https://archive.bukohome.com/content/wiktionary_en_all_maxi_2024-05/A/generalized_element)</sup>. Functoriality expresses this uniformly: for an object \( S \) of a category, the functor \( H_S \) sends each object to its set of S-shaped generalized elements, and any map \( A \to B \) transforms S-elements of \( A \) into S-elements of \( B \); taking \( \mathrm{Top} \) and \( S = S^1 \), any continuous map transforms loops into loops<sup>[2](https://www.voutsadakis.com/TEACH/LECTURES/CATEGORY/Chapter4.pdf)</sup>.

## Stages of definition and global elements

In the terminology of categorical logic, the domain of the map is the stage of definition of the element. If \( r \) is an element of \( R \) defined at stage \( X \), one writes \( r \in_X R \). Given a map \( \alpha : Y \to X \), the composite \( r \circ \alpha \) is the element \( r \) considered at the later stage \( Y \), and it is a standard abuse of notation to write \( r \) for it, omitting the change-of-stage map<sup>[3](https://users-math.au.dk/kock/sdg99.pdf)</sup>.

If the category \( \mathbf{E} \) has a terminal object 1, an element defined at stage 1 is a **global element** of \( R \), also called a global section. The name reflects that, via the unique map \( Y \to 1 \), such an element gives an element of \( R \) at every stage \( Y \)<sup>[3](https://users-math.au.dk/kock/sdg99.pdf)</sup>. When \( \mathbf{E} = \mathrm{Set} \), there is a bijective correspondence between the set \( R \) and its set of global elements, so the generalized notion is a genuine extension of the ordinary one<sup>[3](https://users-math.au.dk/kock/sdg99.pdf)</sup>. In a general category the two notions need not coincide: the distinction between well-pointed categories and concrete categories is precisely the distinction between elements and global elements, which do not coincide in general<sup>[7](https://ncatlab.org/nlab/show/well-pointed%2Btopos)</sup>.

There is a technique for treating any given element as global: replace the category \( \mathbf{E} \) by the comma category \( \mathbf{E}/X \) of objects over \( X \), which works provided the pullback functor preserves the structure in question<sup>[3](https://users-math.au.dk/kock/sdg99.pdf)</sup>.

## Structure inherited at each stage

[Algebraic structure](https://www.edgepedia.org/algebraic-structure) on an object transfers to its generalized elements. If \( R \) is a ring object in a category \( \mathbf{E} \) with finite products, the ring object structure gives, for each \( X \in \mathbf{E} \), a ring structure on the set \( \mathrm{hom}_{\mathbf{E}}(X, R) \). The additive neutral element of \( R \) is a global element \( 1 \to R \), which yields \( 0 \in_X R \) at every stage, and each change of stage \( \alpha : Y \to X \) induces a ring homomorphism \( \mathrm{hom}_{\mathbf{E}}(X, R) \to \mathrm{hom}_{\mathbf{E}}(Y, R) \)<sup>[3](https://users-math.au.dk/kock/sdg99.pdf)</sup>.

Products behave as one expects. There is a natural bijective correspondence between elements of \( A \times B \) and pairs of elements of \( A \) and \( B \) with a common stage of definition, obtained by composing with the projections<sup>[3](https://users-math.au.dk/kock/sdg99.pdf)</sup>.

The vocabulary also supports categorical definitions phrased elementwise: a morphism \( f \) is mono when it is injective on the generalized elements of its domain<sup>[8](https://categorytheory.gitlab.io/glossary.html)</sup>.

## The Yoneda lemma and the philosophy of generalized elements

The contravariant represented functor \( \mathrm{Hom}(-, A) : \mathbf{C}^{\mathrm{op}} \to \mathrm{Set} \) sends an object \( X \) to the set of X-shaped generalized elements of \( A \). Because the Yoneda embedding \( \mathbf{C} \hookrightarrow \mathrm{Set}^{\mathbf{C}^{\mathrm{op}}} \) is fully faithful, no information about \( A \) is lost by considering instead this functor of generalized elements<sup>[5](https://math.jhu.edu/~eriehl/context.pdf)</sup>.

This underwrites what [Emily Riehl](https://www.edgepedia.org/emily-riehl), professor of mathematics at [Johns Hopkins University](https://www.edgepedia.org/johns-hopkins-university), describes in *Category Theory in Context* as the philosophy of generalized elements, first articulated by Lawvere<sup>[5](https://math.jhu.edu/~eriehl/context.pdf)</sup>. Two practical consequences follow. To show that a parallel pair of morphisms \( f, g : A \Rightarrow B \) are equal, it suffices to show that \( f \) and \( g \) induce the same functions on generalized elements. To show that a cone with summit \( A \) is a limit cone, it suffices to show, for generalized elements of any shape, that the induced functions define a limit cone in Set<sup>[5](https://math.jhu.edu/~eriehl/context.pdf)</sup>. The same principle appears as an exercise in Awodey's *Category Theory*: two parallel morphisms \( f, g : A \to B \) are identical if and only if \( f \circ x = g \circ x \) for each generalized element \( x : X \to A \)<sup>[9](https://math.stackexchange.com/questions/63069/why-do-all-objects-have-enough-generalized-elements)</sup>.

Within the internal type theory of a category \( \mathbf{C} \), one may regard the generalized elements of \( X \) at stage \( U \) as the same thing as terms of type \( X \) taken in the context \( u : U \)<sup>[10](https://ncatlab.org/nlab/show/generalized%20element)</sup>; this is the syntactic expression of that notion.

## Generalized elements in algebraic geometry

The functor of points of a scheme \( X \) is \( h_X : \mathrm{Sch}^{\mathrm{opp}} \to \mathrm{Sets} \), \( T \mapsto \mathrm{Mor}(T, X) \); a substantial part of scheme theory consists of describing the internal geometry of \( X \) in terms of this functor<sup>[6](https://stacks.math.columbia.edu/tag/01J5)</sup>. Here the generalized elements are T-valued points, and the Stacks project makes this concrete: for a local ring \( R \), maps \( \mathrm{Spec}(R) \to X \) are in bijection with pairs \( (x, \varphi) \), where \( x \in X \) is a point and \( \varphi : \mathcal{O}_{X,x} \to R \) is a local ring homomorphism.<sup>[6](https://stacks.math.columbia.edu/tag/01J5)</sup>

The shape of the probe matters. Field-valued points do not detect nilpotents: comparing the dual numbers \( D = \mathrm{Spec}\, k[\varepsilon]/(\varepsilon^2) \) with the reduced point \( P = \mathrm{Spec}\, k \), any \( k \)-algebra map to a field \( K \) sends \( \varepsilon \) to an element with \( a^2 = 0 \), hence to 0, so \( D(K) \cong P(K) \) for every field extension \( K/k \); yet \( D \) and \( P \) are not isomorphic, because the structure sheaf of \( D \) has the nonzero nilpotent \( \varepsilon \). Reading a scheme requires not only ordinary points but nonreduced probes<sup>[11](https://math.jp/textbooks/1CRoRjUbux7ZBI2ijFLX/gcRBgXf83cQC1carn7Si)</sup>.

Historically, functors of points for schemes are discussed in EGA I, chapter I, section 3.4, written in 1960, and at a 1973 Buffalo seminar Grothendieck urged that the functor of points definition should replace the one previously expounded by Dieudonné and himself<sup>[12](https://mathoverflow.net/questions/228336/history-of-the-functor-of-points)</sup>. Lawvere later wrote that he had learned the functorial point of view from Gabriel in 1966 at the Strasbourg-Heidelberg-Oberwolfach seminar<sup>[12](https://mathoverflow.net/questions/228336/history-of-the-functor-of-points)</sup>.

## Related notions

In an [abelian category](https://www.edgepedia.org/abelian-category), Mac Lane's definition (VIII.4, also Gelfand-Manin, II.5) takes an element of an object \( W \) to be an equivalence class \( [X, h] \) of pairs \( (X, h) \) with \( h : X \to W \) a generalized element, where \( [X, h] = [Y, k] \) when there is an object \( Z \) with epimorphisms \( u : Z \to X \) and \( v : Z \to Y \) such that \( h \circ u = k \circ v \)<sup>[13](https://ncatlab.org/nlab/show/element%2Bin%2Ban%2Babelian%2Bcategory)</sup>. The term "generic element" appears as a hyponym of generalized element<sup>[4](https://archive.bukohome.com/content/wiktionary_en_all_maxi_2024-05/A/generalized_element)</sup>, and in topos theory a refinement of the domain of definition is allowed while the actual generalized elements are remembered<sup>[13](https://ncatlab.org/nlab/show/element%2Bin%2Ban%2Babelian%2Bcategory)</sup>.

## References

1. [Generalized element, LessWrong (Luke Sciarappa et al., 2016)](https://www.lesswrong.com/w/generalized-element)
2. [George Voutsadakis, Category Theory lecture notes, Chapter 4: Representables and the Yoneda Lemma (LSSU, July 2020)](https://www.voutsadakis.com/TEACH/LECTURES/CATEGORY/Chapter4.pdf)
3. [Anders Kock, Synthetic Differential Geometry, Part II, Section II.1: Generalized elements](https://users-math.au.dk/kock/sdg99.pdf)
4. [generalized element, English Wiktionary (2024 dump)](https://archive.bukohome.com/content/wiktionary_en_all_maxi_2024-05/A/generalized_element)
5. [Emily Riehl, Category Theory in Context, Section 3.4](https://math.jhu.edu/~eriehl/context.pdf)
6. [The Stacks Project, Section 26.13: Points of schemes](https://stacks.math.columbia.edu/tag/01J5)
7. [well-pointed topos, nLab](https://ncatlab.org/nlab/show/well-pointed%2Btopos)
8. [Category Theory: a concise course, Glossary](https://categorytheory.gitlab.io/glossary.html)
9. [Understanding Limits and Colimits by Generalized Elements, Mathematics Stack Exchange](https://math.stackexchange.com/questions/63069/why-do-all-objects-have-enough-generalized-elements)
10. [generalized element, nLab](https://ncatlab.org/nlab/show/generalized%20element)
11. [代数幾何学I, 6-2: schemeの点とresidue field, Mathpedia](https://math.jp/textbooks/1CRoRjUbux7ZBI2ijFLX/gcRBgXf83cQC1carn7Si)
12. [History of the functor of points, MathOverflow](https://mathoverflow.net/questions/228336/history-of-the-functor-of-points)
13. [element in an abelian category, nLab](https://ncatlab.org/nlab/show/element%2Bin%2Ban%2Babelian%2Bcategory)

---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Algebraic geometry › Schemes, stacks, and morphisms › Schemes and stacks: overview and history*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license