🧮Topos Theory Unit 3 – Limits and Colimits

Key Concepts

• Limits and colimits generalize the notions of products, coproducts, equalizers, and coequalizers in category theory
• Limits represent the idea of a universal object that satisfies a certain property with respect to a diagram
• Colimits dual to limits, representing the idea of a universal object that is "built up" from a diagram
• Limits and colimits play a crucial role in the study of topoi, as they allow for the construction of new objects and the analysis of relationships between objects
• The existence of limits and colimits in a category is an important property that can be used to characterize the structure and behavior of the category
  • For example, a category with all finite limits is called complete, while a category with all finite colimits is called cocomplete
• Limits and colimits can be used to define important constructions in topos theory, such as the subobject classifier and the power object
• The preservation of limits and colimits by functors is a key concept in the study of topoi and their relationships to other categories

Definitions and Terminology

• Diagram a functor $D: J \to C$ from a small category $J$ to a category $C$, representing a collection of objects and morphisms in $C$
• Cone a natural transformation $\alpha: \Delta_c \Rightarrow D$ from a constant functor $\Delta_c: J \to C$ to a diagram $D$, where $c$ is an object in $C$
  • Intuitively, a cone is a collection of morphisms from an object $c$ to each object in the diagram $D$, commuting with the morphisms in $D$
• Limit a universal cone, i.e., a cone $\alpha: \Delta_{\lim D} \Rightarrow D$ such that for any other cone $\beta: \Delta_c \Rightarrow D$, there exists a unique morphism $f: c \to \lim D$ satisfying $\alpha \circ \Delta_f = \beta$
• Cocone a natural transformation $\alpha: D \Rightarrow \Delta_c$ from a diagram $D$ to a constant functor $\Delta_c$
• Colimit a universal cocone, i.e., a cocone $\alpha: D \Rightarrow \Delta_{\operatorname{colim} D}$ such that for any other cocone $\beta: D \Rightarrow \Delta_c$, there exists a unique morphism $f: \operatorname{colim} D \to c$ satisfying $\Delta_f \circ \alpha = \beta$
• Complete category a category that has all small limits, i.e., limits for all diagrams indexed by small categories
• Cocomplete category a category that has all small colimits

Categorical Foundations

• Limits and colimits are defined in the context of category theory, which provides a general framework for studying mathematical structures and their relationships
• The notion of a functor is essential for defining diagrams, cones, and cocones, as it allows for the comparison of objects and morphisms across different categories
• Natural transformations play a crucial role in the definition of limits and colimits, as they capture the idea of a "morphism between functors" and allow for the expression of universal properties
• The concept of universality is central to the definition of limits and colimits, as it characterizes the unique relationship between the limit (or colimit) object and other objects in the category
• Limits and colimits can be seen as a generalization of various constructions in category theory, such as:
  • Products and coproducts (limits and colimits of discrete diagrams)
  • Equalizers and coequalizers (limits and colimits of parallel pair diagrams)
  • Pullbacks and pushouts (limits and colimits of span and cospan diagrams)
• The existence of limits and colimits in a category can be related to other categorical properties, such as the existence of adjoint functors and the preservation of certain structures by functors

Limits: Theory and Examples

• A limit of a diagram $D: J \to C$ is an object $\lim D$ in $C$ together with a universal cone $\alpha: \Delta_{\lim D} \Rightarrow D$
• The universal property of limits ensures that the limit object is unique up to unique isomorphism
• Examples of limits in various categories include:
  • Products in the category of sets (cartesian products)
  • Pullbacks in the category of sets (subsets of cartesian products satisfying certain conditions)
  • Equalizers in the category of groups (subgroups defined by equalities)
  • Inverse limits in the category of topological spaces (topological spaces defined by compatible families of maps)
• Limits can be constructed using the concept of a limit cone, which is a cone satisfying a universal property with respect to all other cones over the same diagram
• The existence of limits in a category can be characterized by the existence of certain adjoint functors, such as the right adjoint to the diagonal functor $\Delta: C \to C^J$
• In topos theory, the existence of finite limits is one of the defining properties of a topos, and it allows for the construction of important objects such as the subobject classifier

Colimits: Theory and Examples

• A colimit of a diagram $D: J \to C$ is an object $\operatorname{colim} D$ in $C$ together with a universal cocone $\alpha: D \Rightarrow \Delta_{\operatorname{colim} D}$
• The universal property of colimits ensures that the colimit object is unique up to unique isomorphism
• Examples of colimits in various categories include:
  • Coproducts in the category of sets (disjoint unions)
  • Pushouts in the category of sets (quotient sets defined by equivalence relations)
  • Coequalizers in the category of groups (quotient groups defined by normal subgroups)
  • Direct limits in the category of topological spaces (topological spaces defined by compatible families of maps)
• Colimits can be constructed using the concept of a colimit cocone, which is a cocone satisfying a universal property with respect to all other cocones under the same diagram
• The existence of colimits in a category can be characterized by the existence of certain adjoint functors, such as the left adjoint to the diagonal functor $\Delta: C \to C^J$
• In topos theory, the existence of finite colimits is one of the defining properties of a topos, and it allows for the construction of important objects such as the coproduct and the coequalizer

Relationships Between Limits and Colimits

• Limits and colimits are dual notions in category theory, meaning that the definition of a limit can be obtained from the definition of a colimit by reversing the direction of all morphisms and replacing the diagram with its opposite
• The duality between limits and colimits can be expressed using the concept of an opposite category $C^{op}$, where the limits in $C$ correspond to the colimits in $C^{op}$ and vice versa
• In a complete and cocomplete category, the limit and colimit functors form an adjoint pair, with the limit functor being the right adjoint and the colimit functor being the left adjoint
• The preservation of limits and colimits by functors is an important property that can be used to study the relationships between different categories
  • For example, a functor that preserves limits is called continuous, while a functor that preserves colimits is called cocontinuous
• In topos theory, the relationships between limits and colimits are used to define important properties such as the existence of exponential objects and the validity of the axiom of choice
• The interplay between limits and colimits can also be used to study the behavior of certain constructions, such as the sheafification of presheaves and the formation of quotient topoi

Applications in Topos Theory

• Limits and colimits are essential tools in the study of topoi, as they allow for the construction of new objects and the analysis of relationships between objects
• In a topos, the subobject classifier $\Omega$ can be defined as a limit of a certain diagram, which captures the idea of a "truth value object" and allows for the internalization of logic within the topos
• The power object $P(A)$ of an object $A$ in a topos can be defined as a limit of a diagram involving the subobject classifier, which generalizes the notion of the power set in set theory
• Exponential objects in a topos can be defined using limits, which allows for the internalization of function spaces and the study of higher-order logic within the topos
• Colimits in a topos can be used to construct quotient objects and to study the behavior of certain sheaf-theoretic constructions, such as the sheafification functor and the formation of sheaf subtopoi
• The preservation of limits and colimits by geometric morphisms between topoi is a key property that allows for the comparison of different topoi and the study of their relationships
• In the context of Grothendieck topoi, limits and colimits can be used to define the notion of a site and to study the relationships between different sites and their associated topoi

Common Challenges and Misconceptions

• One common challenge in understanding limits and colimits is the abstract nature of the definitions, which can make it difficult to develop intuition for these concepts
  • It is important to work through concrete examples in various categories to build a solid understanding of how limits and colimits behave in practice
• Another challenge is the dual nature of limits and colimits, which can lead to confusion when trying to distinguish between the two concepts
  • It is helpful to keep in mind the general idea that limits represent "universal objects that satisfy a property," while colimits represent "universal objects that are built up from a diagram"
• A common misconception is that limits and colimits always exist in a given category, which is not true in general
  • It is important to be aware of the specific conditions under which limits and colimits are guaranteed to exist, such as the completeness and cocompleteness of the category
• Another misconception is that limits and colimits are always unique, which is not strictly true
  • While the universal property ensures that limits and colimits are unique up to unique isomorphism, there may be multiple objects that satisfy the universal property
• In the context of topos theory, it is important to distinguish between the internal logic of a topos and the external logic used to reason about the topos
  • The existence of limits and colimits within a topos may have different implications than their existence in the external category of sets
• Finally, it is crucial to be aware of the role that size considerations play in the study of limits and colimits, particularly when dealing with large categories or categories with a proper class of objects
  • The distinction between small and large limits and colimits can have important consequences for the behavior of a category and its relationship to other categories