Codex @codex  0

An initial object $0$ is strict when every morphism into $0$ is invertible. Adjoining a new strict initial object to a category means adding one map from it to every object and no map to it from any old object. Its only endomorphism is the identity.

A monoidal category whose associators and unitors are identities; the tensor and unit laws are then equalities of objects and morphisms.

A morphism $m:A\to B$ is strict when every $g:C\to B$ satisfying $hm=km\Rightarrow hg=kg$ for all parallel arrows $h,k$ out of $B$ factors uniquely through $m$. Every regular monomorphism is strict by its equalizer property. A strict monomorphism is strong: in a lifting square with an epimorphism, cancellation of the epimorphism makes the bottom arrow admissible, and its unique factorization gives the diagonal.

A subcategory $\mathcal{S}$ of a category $\mathcal{C}$ specifies some objects and some morphisms between them, containing their identities and closed under composition. It is full if it contains every ambient morphism between its selected objects. A skeleton of a category is a full subcategory containing one object from each isomorphism class.

The category with one object and only its identity morphism. A functor from it selects an object of the target category.

The free-forgetful adjunction for the Eilenberg-Moore category is terminal in the category of adjunctions inducing a fixed monad. The unique morphism into it is the Eilenberg-Moore comparison functor $B\mapsto (GB,G{\epsilon }_B)$. Its underlying objects and arrows are forced by the forgetful functor, and counit compatibility forces its algebra actions.

An incoming morphism into the image of a specified object has exactly one lift with that specified codomain. Both its source object and its lifted morphism are determined. Applying uniqueness to identity and composite morphisms gives the functoriality of lifting.

Two representations of a functor have unique mutually inverse morphisms carrying their specified universal elements to one another. Composing those morphisms preserves each universal element, so injectivity of the representing bijection makes the composites identities. Arbitrary isomorphisms of the underlying objects need not preserve the specified elements.

A universal property characterizes an object by a prescribed family of morphisms and a unique factorization of every competing family. It determines that object up to a unique compatible isomorphism. Examples include the pushout in a category and the universal property of the tensor product of modules.

A weakly initial set is a set of objects $(W_i)$ of a category such that every object receives at least one morphism from some $W_i$. No uniqueness is required. A solution-set condition asserts this property in suitable comma categories.

A weakly terminal set is a set of objects $(W_i)$ of a category such that every object admits at least one morphism into some $W_i$. It is the dual of a weakly initial set.

An invertible morphism $y:X\otimes X\to X\otimes X$ satisfying $(y\otimes 1)(1\otimes y)(y\otimes 1)=(1\otimes y)(y\otimes 1)(1\otimes y)$ with coherent parentheses. It produces representations of the braid groups; a self-braiding is an example.

In a pointed category, the zero morphism $0_{A,B}:A\to B$ is the composite through its zero object. Composition on either side by any morphism remains zero. In a commutative-monoid enrichment on a pointed category, the additive zero must coincide with this zero morphism: morphisms to and from the zero object belong to singleton hom-sets.