In topology, a space is said to be simply connected if it is path-connected and every loop (closed path) in the space can be continuously contracted to a single point. When the term "at infinity" is used, it generally refers to the behavior of the space as we consider points that are "far away" or tend toward infinity.

A simplicial set is a fundamental concept in algebraic topology and category theory that generalizes the notion of a topological space. It is a combinatorial structure used to study objects in homotopy theory and other areas of mathematics. ### Definition A **simplicial set** consists of: 1. **Sets of n-simplices**: For each non-negative integer \( n \), there is a set \( S_n \) which consists of n-simplices.

In algebraic topology, the concept of "products" generally refers to ways of combining topological spaces or algebraic structures (such as groups or simplicial complexes) to derive new spaces or groups. There are several key notions of products that are important in this field: 1. **Product of Topological Spaces**: Given two topological spaces \( X \) and \( Y \), their product is defined as the Cartesian product \( X \times Y \) together with the product topology.

In topology, a space is said to be **semi-locally simply connected** if, for every point in the space, there exists a neighborhood around that point in which every loop (i.e., a continuous map from the unit circle \( S^1 \) to the space) can be contracted to a point within that neighborhood, provided the loop is sufficiently small.

The term "presentation complex" can refer to different concepts depending on the context in which it is used. However, in the field of immunology, it specifically refers to a group of proteins known as Major Histocompatibility Complex (MHC) molecules that are crucial for the immune system's ability to recognize foreign substances.

Pontryagin cohomology is a concept that arises in algebraic topology and is closely related to the study of topological spaces and their properties through the use of cohomological techniques. Specifically, Pontryagin cohomology is a type of characteristic class theory that is used primarily in the context of topological groups and differentiable manifolds.

Size theory is a concept used in various fields, including mathematics, physics, and philosophy, but it can vary significantly based on context. Here are some interpretations of "size theory" in different disciplines: 1. **Mathematics**: In mathematical contexts, size theory can refer to concepts related to the measure and dimension of sets, particularly in geometry and topology. It may deal with how different dimensions and sizes of objects can be understood and compared.

The term "size homotopy group" does not appear to be a standard term in algebraic topology or related fields as of my last knowledge update in October 2023.

In group theory, a branch of abstract algebra, a **peripheral subgroup** is a specific type of subgroup that has particular significance in the study of group actions and the structure of groups. A subgroup \( H \) of a group \( G \) is called a *peripheral subgroup* if it meets certain criteria within the context of a relatively small subgroup of \( G \) that is critical to the structure of \( G \).

A "shriek map" seems to refer to a concept in different contexts, but it is not widely recognized as a standard term in disciplines like geography, computer science, or social sciences.

In mathematics, a **sheaf** is a fundamental concept in the fields of topology and algebraic geometry that provides a way to systematically track local data attached to the open sets of a topological space. The idea is to gather local information and then piece it together to understand global properties.

In algebraic geometry and related fields, an **orientation sheaf** is a concept that arises in the context of differentiable manifolds and schemes. It provides a way to systematically keep track of the "orientation" of a geometrical object, which is vital in various mathematical and physical applications, such as integration, intersection theory, and the study of moduli spaces.

The term "N-skeleton" could refer to different concepts depending on the context, but it generally relates to certain structures in mathematics, particularly in geometry, topology, or combinatorics. Here are a few interpretations: 1. **Simplicial Complexes**: In the context of algebraic topology, the "N-skeleton" of a simplicial complex is the subcomplex consisting of all simplices of dimension less than or equal to \(N\).

Microbundle is a lightweight and zero-configuration JavaScript bundler designed to help developers create and bundle JavaScript libraries easily. It is particularly optimized for building libraries that may be shared via npm and used in various environments, including browser and Node.js environments. Key features of Microbundle include: 1. **Zero Configuration**: Microbundle is designed to work out of the box with minimal configuration. It uses sensible defaults while allowing customization if needed.

Metaplectic structures are concepts arising in the context of symplectic geometry and representation theory. They are particularly associated with the study of the metaplectic group, which is a double cover of the symplectic group.

The Nottingham Group refers to a research consortium or collective of researchers based at the University of Nottingham, primarily focusing on various fields such as health, education, and social sciences. The group often collaborates on projects related to public health, clinical research, and innovations in education, among other areas. One of the notable subsets associated with the Nottingham Group is the **Nottingham University Hospitals NHS Trust**, which collaborates with the university for clinical research and advancements in healthcare.

The term "mapping spectrum" can refer to different concepts depending on the context in which it is used. Below are a few interpretations in various fields: 1. **Mathematics and Functional Analysis**: In functional analysis, the mapping spectrum can refer to the set of values (spectrum) that a linear operator can take when mapping from one function space to another. The spectrum may include points related to eigenvalues as well as continuous spectrum.

A locally constant sheaf is a concept from the field of sheaf theory, which is a branch of mathematics primarily used in algebraic topology, differential geometry, and algebraic geometry. To understand what a locally constant sheaf is, let's break it down into a few components. ### Sheaves 1. **Sheaf**: A sheaf on a topological space assigns data (like sets, groups, or rings) to open sets in a way that is "local".