Category theory Study Guide

📖 Core Concepts Category – collection of objects and morphisms (arrows) with a source & target, composition, and identities. Hom‑class \(\operatorname{Hom}(a,b)\) – all morphisms from \(a\) to \(b\). Monomorphism – left‑cancellable: \(f\circ g1 = f\circ g2 \Rightarrow g1=g2\). Epimorphism – right‑cancellable: \(h1\circ f = h2\circ f \Rightarrow h1=h2\). Isomorphism – invertible morphism; there exists \(g\) with \(f\circ g = 1b,\; g\circ f = 1a\). Functor – maps objects and morphisms between categories, preserving identities and composition. Covariant: \(F(f\colon x\to y)=F(f)\colon F(x)\to F(y)\). Contravariant: reverses arrows, \(F(f)\colon F(y)\to F(x)\). Natural Transformation \(\eta\colon F\Rightarrow G\) – a morphism \(\etaX\colon F(X)\to G(X)\) for each object \(X\) satisfying \(\etaY\circ F(f)=G(f)\circ\etaX\). Universal Property – characterises an object uniquely (up to unique isomorphism) by a universal mapping condition. Limit / Colimit – categorical versions of “product‑like” and “sum‑like” constructions defined by universal cones/cocones. Equivalence of Categories – functors \(F\) and \(G\) with natural isomorphisms \(G\!\circ\!F\cong 1{\mathcal C}\) and \(F\!\circ\!G\cong 1{\mathcal D}\). Adjoint Functors – \(F\dashv G\) iff \(\operatorname{Hom}{\mathcal D}(F(X),Y)\cong\operatorname{Hom}{\mathcal C}(X,G(Y))\) naturally in \(X,Y\). Yoneda Lemma – \(\operatorname{Nat}(\operatorname{Hom}(A,-),F)\cong F(A)\) for any functor \(F\colon\mathcal C\to\mathbf{Set}\). --- 📌 Must Remember Identity law: \(1y\circ f = f = f\circ 1x\). Associativity: \((h\circ g)\circ f = h\circ(g\circ f)\). Mono ⇔ left‑cancellable, Epi ⇔ right‑cancellable. Retraction ⇒ epi; Section ⇒ mono. Isomorphism ⇔ (mono + retraction) ⇔ (epi + section). Functor laws: \(F(1x)=1{F(x)}\) and \(F(g\circ f)=F(g)\circ F(f)\). Naturality square: \(\etaY\circ F(f)=G(f)\circ\etaX\). Equivalence criteria: functor is full, faithful, and essentially surjective. Adjunction bijection is natural in both arguments. Yoneda: knowing all maps from an object determines the object up to isomorphism. --- 🔄 Key Processes Checking a morphism is mono/epi: For mono, assume \(f\circ g1 = f\circ g2\); try to deduce \(g1=g2\). For epi, assume \(h1\circ f = h2\circ f\); try to deduce \(h1=h2\). Verifying a functor: Map every object \(x\mapsto F(x)\). Map each arrow \(f\) and test the two functor laws. Constructing a natural transformation: Define components \(\etaX\). For every arrow \(f\colon X\to Y\), draw the commuting square and check \(\etaY\circ F(f)=G(f)\circ\etaX\). Proving a universal property (limit example): Show an object \(L\) with projection morphisms \(\pii\) forms a cone. For any other cone \((C,\phii)\), produce a unique arrow \(u\colon C\to L\) with \(\pii\circ u=\phii\). Establishing an equivalence: Exhibit functor \(F\) that is full, faithful, essentially surjective. Construct quasi‑inverse \(G\) and natural isomorphisms \(G\!F\cong1\) and \(F\!G\cong1\). --- 🔍 Key Comparisons Monomorphism vs. Epimorphism Mono: left‑cancellable; think “injective‑like”. Epi: right‑cancellable; think “surjective‑like”. Section vs. Retraction Section (left inverse) ⇒ mono. Retraction (right inverse) ⇒ epi. Covariant vs. Contravariant Functor Covariant preserves arrow direction. Contravariant reverses arrow direction (equivalently a covariant functor from \(\mathcal C^{op}\)). Limit vs. Colimit Limit: universal cone into a diagram. Colimit: universal cocone out of a diagram (dual). Isomorphism vs. Equivalence of Categories Isomorphism: identical objects & morphisms up to strict equality. Equivalence: same “content” up to invertible functors and natural isomorphisms. --- ⚠️ Common Misunderstandings Mono = injective function – true in Set, but “mono” is defined purely categorically; in other categories it may behave differently. Epi = surjective function – same caveat; in Set epi ⇔ surjective, but not universally. All functors preserve limits – only continuous functors do; general functors need not. Natural isomorphism = pointwise isomorphism – each component must be an isomorphism and satisfy the naturality condition; pointwise isomorphism alone is insufficient. Adjoint = inverse – adjoints are weaker; they give a bijection of hom‑sets, not a two‑sided inverse on objects. --- 🧠 Mental Models / Intuition Categories are “worlds” of objects with arrows; think of a city map where intersections are objects and roads are morphisms. Mono: “arrow that you can’t hide behind” – if two different routes arrive at the same place after the mono, they must have been the same route. Epi: “arrow that you can’t hide after” – if two routes start the same after the epi, they were the same route. Functor: “translator” that respects the grammar (composition & identities) of the source language. Natural transformation: a “smooth deformation” between two functors, same at every object, with no “jumps” (commuting squares). Universal property: “best possible” solution; like a shortest‑path hub that any other candidate must factor through uniquely. --- 🚩 Exceptions & Edge Cases In Poset categories, every morphism is both mono and epi, yet not necessarily an isomorphism. A retraction need not be a section; it can be epi without being mono unless the category is balanced. Dual statements hold only when the opposite category exists; some constructions (e.g., limits) may not exist in a given category, even if the dual colimit does. Yoneda Lemma requires the target category to be Set; the statement changes for other codomains. --- 📍 When to Use Which Identify a mono when you need to prove uniqueness of a factorisation into a morphism. Identify an epi when you need uniqueness of a factorisation out of a morphism. Use a functor to transport structures (e.g., “take underlying set” functor) while preserving algebraic relations. Apply a natural transformation to compare two constructions (e.g., homology vs. cohomology functors). Invoke a universal property when you need the canonical object (product, pullback, free object). Choose limits for “pulling together” data (products, equalizers); choose colimits for “gluing together” (coproducts, coequalizers). Check for equivalence when two categories appear different but you suspect they encode the same mathematics (e.g., finite‑dimensional vector spaces ↔ matrices). Seek an adjoint when you have a “free/forgetful” pair (e.g., free group functor left adjoint to the forgetful functor). --- 👀 Patterns to Recognize Universal cone diagrams always have a single object with arrows into every object of the diagram. Naturality squares appear repeatedly: same shape, different labels – if the square commutes, the transformation is natural. Dual pairs (limit ↔ colimit, mono ↔ epi, section ↔ retraction) often arise together in exam questions. Adjunction bijection: look for a pair of hom‑set expressions that look like \(\operatorname{Hom}(F(X),Y)\) and \(\operatorname{Hom}(X,G(Y))\). --- 🗂️ Exam Traps Mistaking “mono” for “injective” in non‑Set categories – leads to false “counter‑examples”. Assuming every functor preserves limits – only continuous functors do; a generic functor may send a product to a non‑product. Confusing section with split mono – a section guarantees a left inverse, but not every mono splits. Reading “natural isomorphism” as “isomorphism of functors” – you must verify naturality, not just componentwise isomorphisms. Over‑applying Yoneda – the lemma only talks about functors to Set; using it with functors to other categories is invalid. Equivalence vs. isomorphism – an equivalence does not require objects to be equal; ignoring the “essentially surjective” nuance can miss a subtle distinction. ---