> **Disclaimer.** Unlike “functor” or “limit,” the word “concept” does **not** have an official, canonical definition inside category theory. Rather, it is a meta-level term that we use to talk about **the structure** of our mathematics, **the abstract ideas** we manipulate, and **the objects of thought** we attempt to unify. Different authors use “concept” in different but overlapping ways. Below is one systematic attempt to be as precise as possible, without stepping afoul of the usual framework of category theory. Below, we'll attempt to make precise *what we mean when we use the term "**concept**"*. Our take is an opinionated one, and readers are unlikely to immediately see the connection between their usage of the term and ours. It is worth stating explicitly that our goal is to account for common usage of the term. Our claim is at its core a rehashing / re-emphasis of what is (by now) an aging claim: "All concepts are Kan extensions." Yet unlike others, we do not wish to narrow or restrict the scope of the claim one iota, if we can get away with it. We also want to emphasize the even more fundamental relationship it implies: "All concepts are functors." We submit that these claims should be read as *broadly* as possible; that the run of the mill concepts you know and love (and use in your day-to-day), should be understood in terms of their *structural* properties and practical usage / semantics. Concepts are not exactly static things. The concepts most people seem to think of as worth having, at least, are the sorts of things (structures) that we can "hold up against the world", that we can enhance and adorn with related knowledge, that we can refine, that we value at one point and discard at another, and that we aim to share, communicate, and "hone in on" in conversation with others (dialectic) and with ourselves (thought). Concepts involve identifying a relating structures to other structures, like the structure of our thoughts to the structure of the world, or the structure of my thoughts to the structure of yours. We share concepts by extending concepts to account for others. We define one concept in terms of others. And concepts might seem useless when first acquired, but through proper usage, and combination / composition with other concepts, they can become tools for acquiring other concepts. This is because all concepts can be used (inter alia) as metaphors, and metaphors themselves operate by mapping structural features from one context onto structural features of another. This common knowledge about concepts suggests that concepts (their structural properties / relationships) are suitably thought of in terms of *functors*. We suggest that concepts are definable in terms of, and imply the existence of, extensions (approximations of other concepts / functors, indexed by some fixed context). We will observe that the two canonical ways to extend a functor along another functor are recognizable discursive modes for extending concepts along other concepts: (1) a "left" (free, colimit-like) mode, and (2) a "right" (constrained, limit-like) mode. If you do not yet share these views, it may be that we do not yet share the same concepts; or that we are using terms differently; or indeed that there are errors in our reasoning or judgement! We merely ask that you attempt to extend the "concepts are functors; concepts involve Kan extensions" metaphor as far as you can, and see when and where it breaks. --- ## 1. Toward a Definition of “Concept” In mathematical practice, we often call something a “concept” if: 1. It **admits of a precise definition** that can be expressed in formal language (e.g. first-order logic, type theory, or whatever foundational system we are using). 2. It **unifies or subsumes a range of examples** or instances, by capturing the essential structures that appear repeatedly in different contexts. Examples include “group,” “topological space,” “functor,” “limit,” “adjoint,” “Kan extension,” and so on. Each is a “concept” in the sense that it: - Has a formal definition in category theory (or some other formal setting). - Is recognized as a canonical _type_ of structure that arises across mathematics. - Possesses a universal or guiding property that tells us **why** it is the right notion to capture phenomena of a certain sort. Hence, a rough but workable definition could be: > **Definition (Concept, informal).** > A _concept_ in mathematics is a formally definable type of structure or property that (1) can be specified in a foundational system (e.g. via axioms or universal properties), and (2) acts as a unifying notion across multiple examples or domains of application. In category theory specifically, the fundamental “concepts” are objects, morphisms, functors, natural transformations, limits, colimits, adjunctions, monoidal categories, higher categories, and so on. Each _captures_ or _codifies_ structures that reappear in many mathematical areas. --- ## 2. How the Notion of “Functor” Relates to Concepts A **functor** $F \;:\; \mathcal{C} \;\longrightarrow\; \mathcal{D}$ _encodes_ or _relates_ concepts, because functors tell us **how** a certain structure in $\mathcal{C}$ is mapped into, or realized in, $\mathcal{D}$. In effect: - A functor is a “structure-preserving map” of categories. - It is a “concept” precisely because it has a **formal definition** (viz. functors preserve composition and identity) and unifies all attempts to “translate” one category’s data into another. Functors can be used to “transfer concepts” from one setting to another. For example, if something is definable or has a property in $\mathcal{C}$, and if we have a functor $F : \mathcal{C}\to \mathcal{D}$, we can look at the image of that “concept” in $\mathcal{D}$ and see whether the functor preserves or reflects that property. Hence the notion of “functor” is both: 1. A concept in itself (with a precise definition in category theory). 2. A mechanism by which _other_ concepts are systematically transported, tested, or compared between categories. --- ## 3. How the Notions of “Diagram” and “(Co)cones” Relate to Concepts ### **Diagrams** A **diagram** in a category $\mathcal{C}$ of shape $\mathcal{J}$ is a functor $D : \mathcal{J}\to \mathcal{C}$. This is a “concept” capturing the idea “I have many objects and morphisms arranged in the pattern (shape) given by $\mathcal{J}$.” - Formally, it is the concept “representation of $\mathcal{J}$ inside $\mathcal{C}$.” - Diagrammatic reasoning in category theory uses such functors as building blocks for discussing limits, colimits, Kan extensions, etc. ### **Cones and Cocones** A **cone** to a diagram $D$ is _another_ concept that systematically organizes morphisms from a single “apex” object into the objects of $D$ in a way respecting the diagram’s arrows. A **cocone** is the dual notion, with morphisms into a single apex from the diagram’s objects. Together, “diagrams” and “(co)cones” are the fundamental “concepts” behind **limits** and **colimits**. Every step here is formal, definable, and unifying, so each is indeed a recognized concept in category theory. --- ## 4. How the Notions of “Limit” and “Colimit” Relate **Limits** and **colimits** are two sides of the same coin (“dual” concepts). Each has a precise definition in terms of universal cones or cocones on a diagram. They unify an enormous range of classical constructions in mathematics: - **Products, pullbacks, equalizers** are all _limits_ of certain shapes. - **Coproducts, pushouts, coequalizers** are all _colimits_ of certain shapes. Limits and colimits are therefore “concepts” that reveal the universal-unifying thread among many apparently distinct constructions. In short, they are “the concept that solves the problem: ‘Given a diagram $D$, find a universal object capturing how to land on $D$ (limit) or emanate from $D$ (colimit).’” --- ## 5. How Left and Right Kan Extensions Relate A **Kan extension** (left or right) is a _further_ concept capturing “the most general way of extending (or restricting) a functor along another functor, subject to a universal property.” In practice: - **Right Kan extensions** are precisely the _limits_ of the diagrams, when the index category is suitably chosen (e.g. $\mathcal{J} \to 1$). - **Left Kan extensions** are precisely the _colimits_ of diagrams, when the index category is suitably chosen (e.g. $\mathcal{J} \to 1$). Hence, _Kan extensions_ unify the conceptual frameworks of all (co)limits. You can view Kan extensions as a more general “universal mapping property” describing how one functor extends or coextends along another. Many advanced developments in higher category theory treat _Kan extension_ as the “primary concept,” from which (co)limits are a special case. --- ## Putting It All Together - “**Concept**” (in an informal, meta-categorical sense): A definitively stated, unifying, mathematical notion that arises across many examples and contexts, usually definable by a universal property or a small set of axioms. - **Functors**: A vehicle by which other concepts are transferred or expressed between categories. - **Diagram, (Co)cones**: Foundational concepts for expressing data $(\mathcal{J}\to\mathcal{C})$ and “landing on” or “emanating from” that data (via cones or cocones). - **(Co)limits**: Unifying concepts for many classical constructions, each definable as a universal (co)cone over a diagram. Limits and colimits _are_ special cases of right and left Kan extensions, respectively. - **Kan extensions**: Possibly the most general concept among these, describing how a functor is “extended” (left) or “co-extended” (right) along another functor, with limits and colimits as prime examples. In that sense, we see a hierarchy of concepts: 1. **Diagram** + **(Co)cone** 2. **(Co)limit** (as a universal (co)cone) 3. **Kan extension** (as the general mechanism from which (co)limits spring) 4. **Functor** (the “language” in which all these are stated and with which they interrelate) Each step is both _the notion itself_ and _the unifier_ of a wide array of particular cases. That is precisely what we mean by saying they are _concepts_ in category theory.