This article is about the mathematical term. For other uses, see Monomorphic (disambiguation) and Polymorphism (disambiguation).
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation .
In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism. That is, an arrow f : X → Y such that for all objects Z and all morphisms g1, g2: Z → X,
pullback of monomorphism with itself
Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below.
In the setting of posets intersections are idempotent: the intersection of anything with itself is itself. Monomorphisms generalize this property to arbitrary categories. A morphism is a monomorphism if it is idempotent with respect to pullbacks.
The categorical dual of a monomorphism is an epimorphism, that is, a monomorphism in a category C is an epimorphism in the dual category Cop. Every section is a monomorphism, and every retraction is an epimorphism.
context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation...
normal monomorphism or conormal epimorphism is a particularly well-behaved type of morphism. A normal category is a category in which every monomorphism is...
morphism f : X → Y is called a monomorphism if f ∘ g1 = f ∘ g2 implies g1 = g2 for all morphisms g1, g2 : Z → X. A monomorphism can be called a mono for short...
split monomorphism is always a monomorphism, for both meanings of monomorphism. For sets and vector spaces, every monomorphism is a split monomorphism, but...
Injective ring homomorphisms are identical to monomorphisms in the category of rings: If f : R → S is a monomorphism that is not injective, then it sends some...
epimorphism. The dual of an epimorphism is a monomorphism (i.e. an epimorphism in a category C is a monomorphism in the dual category Cop). Many authors in...
monomorphism is a monomorphism i in an abelian category C such that for a morphism f in C, the composition f i {\displaystyle fi} is a monomorphism only...
any morphism f, the composite fg is a monomorphism only if f is a monomorphism. If g is an essential monomorphism with domain X and an injective codomain...
homomorphism is also called a monomorphism. However, in the more general context of category theory, the definition of a monomorphism differs from that of an...
Look up monomorphic or monomorphism in Wiktionary, the free dictionary. Monomorphic or Monomorphism may refer to: Monomorphism, an injective homomorphism...
(from A to B) has kernel {0}; that is, if and only if that map is a monomorphism (injective, or one-to-one). Consider the dual sequence B → C → 0. The...
In the mathematical field of graph theory, a graph homomorphism is a mapping between two graphs that respects their structure. More concretely, it is a...
field F and let T: V → W be a linear map. T is said to be injective or a monomorphism if any of the following equivalent conditions are true: T is one-to-one...
one thinks of a monomorphism in terms of its image. An equivalence class of monomorphisms is determined by the image of each monomorphism in the class;...
epimorphism, and every section is a monomorphism. Furthermore, the following three statements are equivalent: f is a monomorphism and a retraction; f is an epimorphism...
dimorphisms seen in the order Charadriiformes. However, cases of sexual monomorphism, where there are no distinguishing physical features besides external...
epimorphisms and q is a monomorphism, then n is an epimorphism. If the rows in the commutative diagram are exact and m and p are monomorphisms and l is an epimorphism...
A unimorph or monomorph is a cantilever that consists of one active layer and one inactive layer. In the case where active layer is piezoelectric, deformation...
embeddings, and that all embeddings are monomorphisms. Other typical requirements are: any extremal monomorphism is an embedding and embeddings are stable...
221-223. Dunham, A. E., and V. H. W. Rudolf. "Evolution of sexual size monomorphism: the influence of passive mate guarding." Journal of evolutionary biology...
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