In \(\mathbf {Set}\), the limit is constructed as a subset of a product, the colimit is a quotient of a sum.

A concrete category is a category \({\cal C}\) equipped with a faithful functor \(\mathscr {F} : {\cal C} \to \mathbf {Set}\).

Let \({\cal C}^{{\cal J}}\) be a functor category, where \({\cal J}\) is a small category.

Let \(\Delta _O\) be a constant functor, which assigns the same object \(O\) in \({\cal C}\) to any object \(J\) in \({\cal J}\).

Let \(K \in \operatorname {Ob}({\cal J})\) and let \(j \in \operatorname {Arr}({\cal J})\) such that \(j: J \to K\). Let \(\mathscr {F}\) be any functor in \({\cal C}^{{\cal J}}\), i.e. it's a diagram in \({\cal C}\) of shape \({\cal J}\).

A natural transformation \(\pi : \Delta _O \to \mathscr {F}\), defined as a family of arrows \(\pi _J: O \to \mathscr {F}(J)\), such that the diagram commutes, is called a cone on the functor (diagram) \(\mathscr {F}\) with vertex \(O\).

A functor \(\mathscr {F}: {\cal C} \to {\cal D}\) creates limits (of shape \({\cal J}\)) if whenever \(\mathscr {D}: {\cal J} \to {\cal C}\) is a diagram in \({\cal C}\),

• for any limit cone \(\left ( V_{{\cal D}} \xrightarrow {q_J} \mathscr {F} \mathscr {D}(J) \right )_{J \in {\cal J}}\) on the diagram \(\mathscr {D} \mathbin {\bullet } \mathscr {F}\), there is a unique cone \(\left (V_{{\cal C}} \xrightarrow {p_J} \mathscr {D}(J) \right )_{J \in {\cal J}}\) on \(\mathscr {D}\) such that \(\mathscr {F}(V_{{\cal C}})=V_{{\cal D}}\) and \(\mathscr {F}\left (p_J\right )=q_J\) for all \(J \in {\cal J}\)
• this cone \(\left (V_{{\cal C}} \xrightarrow {p_J} \mathscr {D}(J)\right )_{J \in {\cal J}}\) is a limit cone on \(\mathscr {D}\).

Direct limits is the colimit of a diagram indexed by the ordinal category \(\omega \). In other words, for a diagram \[ X_1 \to X_2 \to X_3 \to X_4 \to \cdots \] its colimit is the direct limit \(\lim \limits _{\to } X_n\), defining a diagram of shape \(\omega +1\) :

Observe then that the colimit of a sequence of sets with the inclusions \[ X_0 \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow \cdots \] recovers their union \(\bigcup \limits _{n \geq 0} X_n\).

An important corollary of theorem [tt-0045].

The Yoneda embedding preserves limits, for any small category [leinster2016basic, 6.2.12]. But it does not preserve colimits in general [leinster2016basic, 6.2.14].

Every presheaf can be expressed as a colimit of representables in a canonical (though not unique) way, this is theorem [tt-004D], and dual to lemma [tt-002N].

A category \({\cal J}\) is filtered (or directed) if

• it is not empty
• for every pair of objects \(J, J^{\prime } \in {\cal J}\), there exists \(K \in {\cal J}\) and \(f: J \to K\) and \(f': J' \to K\)
• for every two parallel arrows \(u, v: i \rightarrow j\) in \({\cal J}\), there exists \(K \in {\cal J}\) and \(w: J \to K\) such that \[u \mathbin {\bullet } w = v \mathbin {\bullet } w\]

This is a generalization of the notion of a (upward) directed poset from order theory.

A filtered colimit is a colimit of a functor \(\mathscr {F}: {\cal J} \to {\cal C}\), where \({\cal J}\) is a filtered category.

In particular, a colimit over a filtered poset \(P\) is the same as the colimit over a cofinal subset \(Q\) of that poset, where \(Q\) as a cofinal subset means that for every element \(p \in P\), there exists an element \(q \in Q\) with \(p \leq q\).

Limit-preservation alone does not guarantee the existence of a left adjoint. This theorem specifies the conditions under which a limit-preserving functor has a left adjoint. This theorem requires the definition of weakly initial set.

Limits in a functor category are computed pointwise.

This theorem helps proving lemma [tt-0047].

Let \(\mathscr {F}: {\cal C} \rightarrow {\cal D}\) be a functor and \({\cal J}\) a small category. Suppose that \({\cal D}\) has, and \(\mathscr {F}\) creates, limits of shape \({\cal J}\). Then \({\cal C}\) has, and \(\mathscr {F}\) preserves, limits of shape \({\cal J}\).

Let \(f: X \to Y\) be an arrow in a category \({\cal C}\). A section (or right inverse) of \(f\) is an arrow \(i: Y \to X\) in \({\cal C}\) such that \(i \mathbin {\bullet } f = \mathit {1}_X\).

In \(\mathbf {Set}\), any arrow with a section is certainly surjective. The converse statement is called the axiom of choice:

Every surjection has a section.

It is called choice because specifying a section of \(f: X \to Y\) amounts to choosing, for each \(y \in Y\), an element of the nonempty set \(\{x \in X \mid f(x)=y\}\).

A function with domain \(\mathbb N\) is usually called a sequence.

It's also denoted \({\cal C} / X\) or \({\cal C}_{/X}\).

The slice category is a special case of the comma category.

It's also called overcategory, as it's a slice category over \(X\) [zhang2021type, 3.5].

A sum or coproduct is a colimit over a discrete category, i.e. it is a colimit of shape \(J\) for some discrete category \({\cal J}\).

This diagram summarizes § [tt-002P].

Let \(f_1, \ldots , f_s\) be polynomials in \(K\left [x_1, \ldots , x_n\right ]\). The affine variety defined by \(f_1, \ldots , f_s\) is \[ \mathbf {V}\left (f_1, \ldots , f_s\right )=\left \{\left (a_1, \ldots , a_n\right ) \in \mathbb {A}_K^n \mid f_i\left (a_1, \ldots , a_n\right )=0 \text { for all } 1 \leq i \leq s\right \} \]

Obviously, it is the set of all solutions of the system of polynomial equations \(f_1\left (x_1, \ldots , x_n\right )=\cdots =f_s\left (x_1, \ldots , x_n\right )=0\).

Let \(S \subset K\left [x_1, \ldots , x_n\right ]\) be a set of polynomials. The zero locus (or zero set) of \(S\) is \[ V(S):=\left \{x \in \mathbb {A}_K^n: f(x)=0 \text { for all } f \in S\right \} \subset \mathbb {A}_K^n \]

An affine algebraic variety over \(K\) is a subset of \(\mathbb {A}_K^n\) of this form. It's usually simply called an affine variety over \(K\), or an affine \(K\)-variety.

If \(S=\left (f_1, \ldots , f_k\right )\) is a finite set, \(V(S)\) can be written as \(V\left (f_1, \ldots , f_k\right )\).

Obviously, it is the set of all solutions of the system of polynomial equations \(f_1\left (x_1, \ldots , x_n\right )=\cdots =f_s\left (x_1, \ldots , x_n\right )=0\) [cox1997ideals, 1.2.1], denoted \(\operatorname {Sol}(S;K)\) [dolgachev2013introduction, p. 1].

The approach we would like to take is similar to the one we took in our Notes on Topos Theory and Type Theory: determine some goals, find some references that sketch out a path to those goals, and then write notes jumping between many references to fill in the gaps.

Goal 1: To be able to state "an affine scheme is a scheme", which means we need all the basics of prime ideals, Zariski topology, sheaves, spectrum of a ring, stalks, locally ringed spaces etc. as sketched out in [bordg2022simple] which has organized the minimum preliminaries in a formal way, but we might also need to refer to [buzzard2022schemes] and Mathlib to ensure that we can properly find the Lean counterparts.

Goal 2: To properly understand the geometric topic of algebraic geometry, we also need many basic correspondence in the Algebra-Geometry dictionary, which can be found in [cox1997ideals], including ideals, varieties, Gröbner Bases, elimination Theory, the relation between irreducible Varieties and prime Ideals, projective algebraic geometry etc. This book provides concrete intuitions and examples for working with the abstract concepts. We might also branch out from there to write notes about ray-tracing the implicit surfaces of algebraic varieties (see some links for rendering implicit surfaces for some links), and the use of Macaulay2 and Singular to do computations in AG.

Goal 3: A challenging goal is to be able to read EGA [grothendieck1964elements] and some materials surrounding it:

• EGA: we will use the English translation available at ryankeleti/ega, with the French version (year 1960) available at Numdam, both use the word scheme to mean separated scheme, and prescheme to mean scheme. A Chinese translation is also available and has an index of terminology in Chinese, English, and French, which is useful. We would like to include terminology in all three languages for key concepts in our notes, showing only the English version by default.
• FGA: we will use [fantechi2006fundamental] instead of the original FGA. This is an explained and extended version of FGA, that has the advantage of being in English and from a little more modern viewpoint.
• SGA: Although topos is first introduced in SGA, but we are not ready to read SGA yet.
• The rising sea: [vakil2024rising] is a modern introduction to algebraic geometry, used by many courses, e.g. math256 (which also has an English synopsis of EGA). Some courses, e.g. Algebraic Geometry Spring 2024 , also follows lecture notes by Andreas Gathmann [gathmann2022algebraic] (found on Math 8253: Algebraic Geometry).
• FAC [serre1955faisceaux]: there is an English version Coherent Algebraic Sheaves translated by Piotr Achinger and Lukasz Krupa. This is no longer the modern treatment of AG, so it's only used as a reference.
• GAGA [serre1956geometrie]: there is only the French version and the Chinese translation. The evaluation is the same as FAC.

We need some modern notes to avoid being lost in the old language, for that, besides The rising sea, we have chosen [borisov2024adventures] and [mehrle2017algebraic].

Goal 4: We also need to tap into the language of Stacks in a modern setting, as treated in [khan2023lectures], with preliminaries on \(\infty \)-categories and derived categories.

The dual module \(M^* : M \to _{l[R]} R\) is the \(R\)-module of all linear maps from \(M\) to \(R\).

The even subalgebra of the Clifford algebra is defined as the submodule of the Clifford algebra \[ \mathcal {C}\kern -2pt\ell ^{+}(Q) \equiv \left \{ x_1 \cdots x_k \in \mathcal {C}\kern -2pt\ell \mid x \in V, k \text { is even} \right \} \] which also forms a subalgebra. Its elements are called even elements, as they can be expressed as the geometric product of an even number of 1-vectors.

Let \((\alpha , +_\alpha , *_\alpha )\) and \((\beta , +_\beta , *_\beta )\) be rings. A ring homomorphism from \(\alpha \) to \(\beta \) is a map \(\mathit {1} : \alpha \to _{+*} \beta \) such that

1. \(\mathit {1}(x +_{\alpha } y) = \mathit {1}(x) +_{\beta } \mathit {1}(y)\) for each \(x,y \in \alpha \).
2. \(\mathit {1}(x *_{\alpha } y) = \mathit {1}(x) *_{\beta } \mathit {1}(y)\) for each \(x,y \in \alpha \).
3. \(\mathit {1}(1_{\alpha }) = 1_{\beta }\).