Category theory becomes usable when a definition summarizes constructions you can already perform. Products, quotient maps and free objects give functors and universal properties something to act on. Riehl develops the language with mathematical context and exercises; Mac Lane gives a compact graduate account extending through adjunctions and monads. Begin with a small collection of algebraic examples and enlarge it as the abstractions demand.
What to work on
Check functoriality and naturality in examples of sets or modules.
Identify a representing object from a universal property.
Construct products, equalizers and their universal cones.
Describe an adjunction by a natural hom-set bijection and its unit and counit.
A starting text
Category Theory in Context — Emily Riehl. Read sections 1.3-1.4 on functoriality and naturality; 2.1-2.3 on representability and Yoneda; 3.1-3.2 on limits in categories and sets; and 4.1-4.2 on adjunctions and universal arrows. Review 1.1 only if category axioms are unfamiliar.
Choosing what to read
For every functor, state its action on both objects and arrows and check composition. For every natural transformation, draw a square with a specific map instead of saying that compatibility is obvious. Derive a product or free-object construction from its universal property, then test it in sets and modules. Approach Yoneda by writing out the correspondence in a concrete representable example. An unfamiliar higher-category reference can wait until these ordinary categorical calculations are dependable.
Learn to write naturality squares and verify universal properties concretely.
How it reads
Contextual development through functors, representability, limits and adjunctions.
What it expects
Proof-based algebra and examples of sets, groups and modules.
What to keep in mind
Ordinary categories here do not provide an automatic bridge to infinity-categories.
Practice
Exercises provided; public solutions not verified. Exercises embedded in sections 1.3-1.4, 2.2-2.3, 3.1-3.2 and 4.1-4.2; keep each naturality square in the written solution.
Complete public PDF.
Edition, reading range and source
Read sections 1.3-1.4 on functoriality and naturality; 2.1-2.3 on representability and Yoneda; 3.1-3.2 on limits in categories and sets; and 4.1-4.2 on adjunctions and universal arrows. Review 1.1 only if category axioms are unfamiliar.
Functor, Yoneda, limit and adjunction exercises. Exercises embedded in sections 1.3-1.4, 2.2-2.3, 3.1-3.2 and 4.1-4.2; keep each naturality square in the written solution.. Exercises provided; public solutions not verified.
Try these before moving on
Try a proof or calculation before opening the hint. These solutions address the checks below, not the exercises in the linked textbooks.
Prove the universal property of the set-theoretic product X times Y and show that any two products of X and Y are uniquely isomorphic as cones.
A hint
The projections force the two coordinates of the mediating map. Then apply uniqueness twice to the composites between two proposed products.
Work through the solution
For sets X,Y, let P=X×Y with coordinate projections pX,pY. For any set Z and maps f:Z→X, g:Z→Y, define h(z)=(f(z),g(z)). Then pXh=f and pYh=g. Any other such map has these same two coordinates at every z, so equals h. This also handles empty sets whenever the given maps exist.
Suppose Q with projections qX,qY also has this property. There is a unique u:P→Q commuting with projections, and a unique v:Q→P commuting with projections. The composite vu and idP have identical composites with pX,pY, so uniqueness for P gives vu=idP. Likewise uv=idQ. Thus u is an isomorphism, and uniqueness in the universal property makes it the only isomorphism commuting with these projections. The qualification 'as cones' matters: the underlying sets can have other bijections.
Set Theory — Review Cartesian products and composition of maps.
Already comfortable with this? Skip the first sections if you can write naturality squares without guessing arrow directions. Skip this route if you can prove uniqueness of universal objects and construct the free-group adjunction; reserve monads and Kan extensions for a specific later need.
Emily Riehl and Dominic Verity · Monograph · Advanced
A substantial treatment of higher categories for mathematically mature readers. A later step after ordinary category theory, not a shortcut around its foundations.
Steve Awodey · Textbook · Intermediate · English · Purchase / institutional access
Purchase or institutional access required; previews are not the complete book.
A relatively concise introduction through examples, universal constructions, Yoneda and adjunctions. The linked edition must not be confused with later editions.
Preparation: Proof-based algebra; examples of mathematical structures.
Steven Roman · Textbook · Intermediate · English · Purchase / institutional access
Purchase or library subscription required; previews are not the complete book.
A compact route through functors, natural transformations, limits and adjunctions. Use concrete algebraic examples before studying higher categorical models.
Preparation: Proof-based algebra; sets and functions.
Saunders Mac Lane · Textbook · Graduate · English · Purchase / institutional access
Purchase or institutional access required; previews are not the complete book.
Develops categories, functors, natural transformations, universal constructions, adjunctions and monads. Use concrete examples alongside the general theory.
Preparation: Proof-based algebra; examples of mathematical structures.
C. H. Dowker · Long lecture notes · Graduate · English · PDF
Complete institutional PDF, authorized for individual use only. Copyright remains with TIFR; retypesetting may retain errors. This is not a redistribution licence.
A sustained sheaf-theoretic course linking topology and homological methods. Useful for readers moving beyond categorical vocabulary to local-to-global constructions.
Vladimir Hinich · Research paper · Graduate · English · PDF
Public full text; checked against the title and contents.已核对题名和目录，可公开阅读全文。
Revisits simplicial localization in infinity-categorical language and studies the passage from Quillen adjunctions to localized categories.以无穷范畴语言重新讨论单纯局部化，并研究Quillen伴随如何传到局部化后的范畴。
Preparation: Model categories; simplicial categories; adjunctions.模型范畴、单纯范畴与伴随。
Use: Compare simplicial localization with the induced adjunction from a Quillen pair.比较单纯局部化与Quillen对所诱导的伴随。
Peter T. Johnstone · Research monograph · Research · English · Purchase / institutional access
Purchase or library subscription required; previews are not the complete book.
A substantial reference in topos theory, categorical logic and related structures. Choose a route through the compendium rather than treating it as a first category-theory text.