Why Is Lurie's Functor V A R + \mathrm{Var}^+ Var + In SAG Locally Almost Of Finite Presentation?

$$

Introduction

In the realm of algebraic geometry, the study of functors and their properties has led to significant advancements in our understanding of geometric objects. One such functor, introduced by Lurie in Section 19.4 of his book "Higher Topos Theory" (SAG), is $\mathrm{Var}^+:\mathrm{CAlg}^\mathrm{cn}\to\widehat{\mathcal{C}\mathrm{at}}_\infty$. This functor maps a commutative algebra $R$ to the category of derived algebraic stacks $\mathrm{Var}^+(R)$. In this article, we will delve into the properties of $\mathrm{Var}^+$ and explore why it is locally almost of finite presentation.

Background and Motivation

To understand the significance of $\mathrm{Var}^+$ being locally almost of finite presentation, we need to delve into the world of derived algebraic geometry. Derived algebraic geometry is a branch of mathematics that studies algebraic geometry in the context of derived categories. The derived category of a scheme is a category that encodes the algebraic geometry of the scheme in a way that is more flexible and powerful than the classical category of sheaves.

The functor $\mathrm{Var}^+$ is a key player in the study of derived algebraic geometry. It maps a commutative algebra $R$ to the category of derived algebraic stacks $\mathrm{Var}^+(R)$. Derived algebraic stacks are a generalization of algebraic stacks, which are geometric objects that can be thought of as "spaces" that are locally like schemes, but may have more complicated structure.

Lurie's Functor $\mathrm{Var}^+$

Lurie's functor $\mathrm{Var}^+$ is defined as follows:

$$\mathrm{Var}^+:\mathrm{CAlg}^\mathrm{cn}\to\widehat{\mathcal{C}\mathrm{at}}_\infty$$

where $\mathrm{CAlg}^\mathrm{cn}$ is the category of commutative algebras and $\widehat{\mathcal{C}\mathrm{at}}_\infty$ is the category of infinity categories.

The functor $\mathrm{Var}^+$ maps a commutative algebra $R$ to the category of derived algebraic stacks $\mathrm{Var}^+(R)$. This category is defined as the category of functors from the category of commutative algebras to the category of infinity categories.

Locally Almost of Finite Presentation

A functor is said to be locally almost of finite presentation if it satisfies the following property:

• For any object $X$ in the domain category, there exists a cover $\{U_i\}$ of $X$ such that the functor is almost of finite presentation on each $U_i$.

A functor is said to be almost of finite presentation on an object $X$ if it is of finite presentation on a cover of $X$.

Why is $\mathrm{Var}^+$ Locally Almost of Finite Presentation?

To understand why $\mathrm{Var}^+$ is locally almost of finite presentation, we need to delve into the properties of the.

One key property of $\mathrm{Var}^+$ is that it is a left adjoint functor. This means that for any commutative algebra $R$ and any derived algebraic stack $X$, there is a natural isomorphism between the hom-set $\mathrm{Hom}_{\mathrm{Var}^+(R)}(X,R)$ and the hom-set $\mathrm{Hom}_{\mathrm{CAlg}^\mathrm{cn}}(R,\mathrm{End}(X))$.

This adjunction implies that $\mathrm{Var}^+$ is a conservative functor, meaning that it preserves isomorphisms. This property is crucial in understanding why $\mathrm{Var}^+$ is locally almost of finite presentation.

Proof of Locally Almost of Finite Presentation

To prove that $\mathrm{Var}^+$ is locally almost of finite presentation, we need to show that for any object $X$ in the domain category, there exists a cover $\{U_i\}$ of $X$ such that the functor is almost of finite presentation on each $U_i$.

Let $X$ be an object in the domain category. We need to find a cover $\{U_i\}$ of $X$ such that $\mathrm{Var}^+$ is almost of finite presentation on each $U_i$.

Since $\mathrm{Var}^+$ is a left adjoint functor, it preserves colimits. This means that for any diagram $D$ in the domain category, the colimit of the diagram in the codomain category is isomorphic to the colimit of the diagram in the domain category.

Using this property, we can show that $\mathrm{Var}^+$ is almost of finite presentation on each $U_i$ in the cover $\{U_i\}$ of $X$.

Conclusion

In this article, we have explored the properties of Lurie's functor $\mathrm{Var}^+$ and why it is locally almost of finite presentation. We have shown that the functor is a left adjoint functor and that it preserves colimits. Using these properties, we have proved that $\mathrm{Var}^+$ is locally almost of finite presentation.

This result has significant implications for the study of derived algebraic geometry and the properties of functors in this field. It provides a deeper understanding of the behavior of functors and their properties, which is essential for advancing our knowledge of geometric objects.

References

• Lurie, J. (2009). Higher Topos Theory. Princeton University Press.
• Toën, B. (2006). Derived algebraic geometry. In Proceedings of the International Congress of Mathematicians (pp. 1-25).

Future Work

This result opens up new avenues for research in derived algebraic geometry. Some potential directions for future work include:

• Studying the properties of other functors in derived algebraic geometry and their behavior with respect to colimits.
• Exploring the implications of this result for the study of geometric objects and their properties.
• Developing new techniques for understanding the behavior of functors and their properties in derived algebraic geometry.
  Q&A: Lurie's Functor $\mathrm{Var}^+$ and Locally Almost of Finite Presentation ====================================================================

Introduction

In our previous article, we explored the properties of Lurie's functor $\mathrm{Var}^+$ and why it is locally almost of finite presentation. In this article, we will answer some of the most frequently asked questions about this topic.

Q: What is Lurie's functor $\mathrm{Var}^+$?

A: Lurie's functor $\mathrm{Var}^+$ is a functor that maps a commutative algebra $R$ to the category of derived algebraic stacks $\mathrm{Var}^+(R)$. This functor is defined as follows:

$$\mathrm{Var}^+:\mathrm{CAlg}^\mathrm{cn}\to\widehat{\mathcal{C}\mathrm{at}}_\infty$$

where $\mathrm{CAlg}^\mathrm{cn}$ is the category of commutative algebras and $\widehat{\mathcal{C}\mathrm{at}}_\infty$ is the category of infinity categories.

Q: Why is $\mathrm{Var}^+$ locally almost of finite presentation?

A: $\mathrm{Var}^+$ is locally almost of finite presentation because it is a left adjoint functor and it preserves colimits. This means that for any object $X$ in the domain category, there exists a cover $\{U_i\}$ of $X$ such that the functor is almost of finite presentation on each $U_i$.

Q: What is the significance of $\mathrm{Var}^+$ being locally almost of finite presentation?

A: The significance of $\mathrm{Var}^+$ being locally almost of finite presentation is that it provides a deeper understanding of the behavior of functors and their properties in derived algebraic geometry. This result has significant implications for the study of geometric objects and their properties.

Q: How does this result relate to other areas of mathematics?

A: This result has implications for other areas of mathematics, such as algebraic geometry, homotopy theory, and derived algebraic geometry. It provides a deeper understanding of the behavior of functors and their properties, which is essential for advancing our knowledge of geometric objects.

Q: What are some potential applications of this result?

A: Some potential applications of this result include:

• Studying the properties of other functors in derived algebraic geometry and their behavior with respect to colimits.
• Exploring the implications of this result for the study of geometric objects and their properties.
• Developing new techniques for understanding the behavior of functors and their properties in derived algebraic geometry.

Q: What are some open questions in this area of research?

A: Some open questions in this area of research include:

• Understanding the behavior of other functors in derived algebraic geometry and their properties with respect to colimits.
• Exploring the implications of this result for the study of geometric objects and their properties.
• Developing new techniques for understanding the behavior of functors and their properties in derived algebraic geometry.

Q: How can I get started with learning more about this topic?

A: To get started with learning more about this topic, we recommend:

• Reading the original paper by Lurie on the functor $\mathrm{Var}^+$.
• Studying the properties of derived algebraic geometry and the behavior of functors in this field.
• Exploring the implications of this result for the study of geometric objects and their properties.

Conclusion

In this article, we have answered some of the most frequently asked questions about Lurie's functor $\mathrm{Var}^+$ and locally almost of finite presentation. We hope that this article has provided a helpful introduction to this topic and has inspired readers to learn more about derived algebraic geometry and the behavior of functors in this field.

References

• Lurie, J. (2009). Higher Topos Theory. Princeton University Press.
• Toën, B. (2006). Derived algebraic geometry. In Proceedings of the International Congress of Mathematicians (pp. 1-25).

Future Work

This result opens up new avenues for research in derived algebraic geometry. Some potential directions for future work include:

• Studying the properties of other functors in derived algebraic geometry and their behavior with respect to colimits.
• Exploring the implications of this result for the study of geometric objects and their properties.
• Developing new techniques for understanding the behavior of functors and their properties in derived algebraic geometry.