Abstract
A category \(\mathcal{C}\) is formed by a class of objects \(\mathop{\mathrm{Ob}}\nolimits \mathcal{C}\) and a class of disjoint sets \(\mathop{\mathrm{Hom}}\nolimits (X,Y ) =\mathop{ \mathrm{Hom}}\nolimits _{\mathcal{C}}(X,Y )\), one set for each ordered pair of objects \(X,Y \in \mathop{\mathrm{Ob}}\nolimits \mathcal{C}\). Elements of the set \(\mathop{\mathrm{Hom}}\nolimits _{\mathcal{C}}(X,Y )\) are called morphisms from X to Y in the category \(\mathcal{C}\). We will depict them by arrows φ: X → Y and refer to the objects X, Y as the source (or domain) and target (or codomain ) of φ respectively. Morphisms \(\varphi,\psi \in \mathop{\mathrm{Mor}}\nolimits \mathcal{C}\) are called composable if the source of φ coincides with the target of ψ. For every ordered triple of objects \(X,Y,Z \in \mathop{\mathrm{Ob}}\nolimits \mathcal{C}\), the composition map
is defined. It is associative, meaning that (η ∘φ) ∘ψ = η ∘ (φ ∘ψ) for all composable pairs η, φ and φ, ψ. Finally, for every \(X \in \mathop{\mathrm{Ob}}\nolimits \mathcal{C}\), there exists an identity endomorphism
such that φ ∘ IdX = φ and IdX ∘ψ = ψ for all morphisms φ: X → Y, ψ: Z → X in \(\mathcal{C}\). It is actually unique for every \(X \in \mathop{\mathrm{Ob}}\nolimits \mathcal{C}\), because IdX ′ = IdX ′∘ IdX ′ ′ = IdX ′ ′ for every two such endomorphisms \(\mathrm{Id}_{X}^{{\prime}},\mathrm{Id}_{X}^{{\prime\prime}}\in \mathop{\mathrm{Hom}}\nolimits (X,X)\).
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Notes
- 1.
For formal logical reasons, the collection of all sets (even all finite sets) is not itself a set. Similarly, the collections of all rings, groups, topological spaces, etc., are not sets, but something larger, namely classes. The notion of class enlarges the notion of set and makes it possible to formulate correct statements about the classes of all sets, groups, rings, vector spaces, etc. To the extent that we have omitted a discussion of rigorous set theory, we shall refer the reader to a basic course in mathematical logic for the rigorous theory of classes and their interaction with sets. For our purposes, it is enough to know that such a theory exists and that it allows us to deal with the categories of sets, algebras, topological spaces, etc.
- 2.
An R- module is called finitely presented if it is isomorphic to the quotient module of a free R- module of finite rank by a finitely generated R- submodule of relations.
- 3.
That is, a partially ordered set; see Sect. 1.4.1 of Algebra I.
- 4.
A set X is called a topological space if a set \(\mathcal{U}(X)\) of subsets in X is chosen such that \(\varnothing,X \in \mathcal{U}(X)\), \(U \cap W \in \mathcal{U}(X)\) for all \(U,W \in \mathcal{U}(X)\), and \(\bigcup _{\nu }U_{\nu } \in \mathcal{U}(X)\) for every set of \(U_{\nu } \in \mathcal{U}(X)\). The elements \(U \in \mathcal{U}(X)\) and their complements X∖U are called, respectively, the open and closed subsets of X.
- 5.
Also known as the path algebra of \(\mathcal{C}\).
- 6.
That is, φ: X → Y such that \(x_{1}\leqslant x_{2} \Rightarrow \varphi (x_{1})\leqslant \varphi (x_{2})\).
- 7.
Or more precisely, a covariant functor.
- 8.
One map per each ordered pair of objects \(X,Y \in \mathop{\mathrm{Ob}}\nolimits \mathcal{C}\).
- 9.
In the same way as an endomorphism means a map of a set to itself, an endofunctor means a functor from a category to itself.
- 10.
Such as \(\mathcal{T}\!\mathit{op}\), \(\mathcal{G}\mathit{rp}\), \(\mathcal{C}\mathit{mr}\).
- 11.
That is, injective and order-preserving.
- 12.
In other words, can S 1 be homeomorphic to the geometric realization of a semisimplicial set X such that both sets X 0, X 1 consist of three elements and all the other X i equal to \(\varnothing \)?
- 13.
Where all the sets X n are considered with the discrete topology and all the simplices \(\Delta ^{n} \subset \mathbb{R}^{n+1}\) with the standard topology induced from the ambient spaces \(\mathbb{R}^{n+1}\).
- 14.
That is, the space obtained from \(\Delta ^{n}\) by collapsing its boundary to one point. For example, the 2-sphere S 2 is obtained in this way from the triangle.
- 15.
This means that every point x ∈ U is mapped to the fiber p −1(x) over x.
- 16.
Algebraic manifolds will be defined and studied in Chap. 12
- 17.
Equivalently, the locally constant functions.
- 18.
In which every subset U ⊂ X is declared to be open.
- 19.
Note that they are distinct, because \(\vert X\vert \geqslant 2\).
- 20.
Or functorial.
- 21.
Whenever the words “canonical isomorphism” have been used previously in this course, it was precisely in this explicit sense.
- 22.
See Sect. 9.2.3 on p. 193.
- 23.
Sending a vector to the sequence of its coordinates in the chosen basis.
- 24.
Meaning that there is a unique pair of quasi-inverse equivalences between them.
- 25.
That is, all maps \(G:\mathop{ \mathrm{Hom}}\nolimits _{\mathcal{C}}(X,Y ) \rightarrow \mathop{\mathrm{Hom}}\nolimits _{\mathcal{D}}(G(X),G(Y ))\) are bijective.
- 26.
See Example 9.6 on p. 190.
- 27.
See Example 9.9 on p. 194.
- 28.
Intuitively, a simplicial map of triangulated spaces | X | → | Y | is a continuous map respecting the triangulations. The latter property is formalized as a natural transformation of presheaves X → Y. In particular, this forces a simplicial map to send every k-simplex of the triangulation on | X | to a k-simplex of the triangulation on | Y | via an affine map, and to respect the incidences between the simplices.
- 29.
In particular, in the category \(\mathcal{A}\mathit{b} = \mathcal{M}\mathit{od}_{\mathbb{Z}}\).
- 30.
See Sect. 13.1.1 of Algebra I.
- 31.
That is, \(\mathbb{Z}\)- modules.
- 32.
Compare with Sect. 6.3 on p. 139 and Sect. 6.3.4 on p. 144.
- 33.
See Example 9.7 on p. 191.
- 34.
Every such continuous map is called a singular simplex of Y.
- 35.
See formula (9.15) on p. 203.
- 36.
See the same formula (9.15).
- 37.
See Sect. 9.3 on p. 195.
- 38.
Or the projective limit.
- 39.
Or the injective limit.
- 40.
See Example 9.8 on p. 192.
- 41.
Recall that \(\Delta _{Y } =\{ (y,y) \in Y \times Y \mid y \in Y \}\); compare with Sect. 1.2.2 of Algebra I.
- 42.
That is, the intersection of all equivalences R ⊂ Y × Y containing \(\mathop{\mathrm{im}}\nolimits (\varphi \times \psi )\); see Sect. 1.2.2 of Algebra I.
- 43.
In the categories of groups and commutative rings, the pushforwards are traditionally called the amalgamated and tensor products respectively.
- 44.
Compare with Example 9.2 on p. 186.
- 45.
Filtered diagrams are also called direct or inductive systems of morphisms. Cofiltered diagrams are also called inverse or projective systems of morphisms.
- 46.
This means that 1 ∈ S and st ∈ S for all s, t ∈ S; see Sect. 4.1.1 of Algebra I.
- 47.
Or ring of fractions with numerators in K and denominators in S; see Sect. 4.1.1 of Algebra I.
- 48.
The horizontal arrows in (9.32) are the canonical morphisms from the limit to the nodes of the diagram.
- 49.
Whose horizontal arrows are the canonical morphisms from the nodes of the diagram to the colimit.
- 50.
Namely, the (co) kernel of a homomorphism is the (co)equalizer of this homomorphism and the zero homomorphism.
- 51.
That is, the completion of \(\mathbb{Z}\) with respect to the p-adic distance | x, y | p = ∥ x − y ∥ p, where the p-adic norm ∥ z ∥ p of an integer z = p sm, gcd(m, p) = 1, equals p −s.
- 52.
That is, 1 ∈ S and st ∈ S for all s, t ∈ S.
- 53.
See Problem 7.8 of Algebra I.
- 54.
By Corollary 9.5 on p. 218, it is enough to show that filtered colimits commute with kernels.
- 55.
This means that the first two functors are not right exact, and the third is not left exact.
- 56.
A module P with these properties is called projective.
- 57.
A module I possessing these properties is called injective.
- 58.
That is, \(r\varphi (x)\stackrel{\mathrm{def}}{=}\varphi (xr)\) for all r, x ∈ R, \(\varphi: R \rightarrow \mathbb{Q}/\mathbb{Z}\).
- 59.
See Sect. 9.2 on p. 189.
- 60.
That is, an equivalence can be established by means of exact quasi-inverse functors.
References
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Gorodentsev, A.L. (2017). Categories and Functors. In: Algebra II. Springer, Cham. https://doi.org/10.1007/978-3-319-50853-5_9
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