# Tagged Questions

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### What is the “opposite” of a forgetful functor?

Consider a category $C$ and a monoid $M$. Consider a functor $F:C\to M$. It maps the objects of $C$ into the only object of $M$. But I don't want it to map every morphism of $C$ into the identity on ...
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### Collections of Homomorphic (defined) structures via $f$

Long ago I read a text about a collection of algebraic sturctures all homomorphic (or isomorphic) via a unique homomorphism An Example similar to the construction I found was this: Lets take define ...
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### Categories without identities

What's the name of "categories without identities", i.e. of digraphs with just an associative binary operation on its "matching" arrows (disregarding identities)?
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### Injective Morphisms, Monomorphisms and Left Invertible Morphisms in Abelian Categories

Let $\mathcal{C}$ be an abelian category. A morphism $f:X \rightarrow Y$ is called injective if its kernel is zero. $f$ is called monomorphism if whenever $f \circ g=0$, for $g:Z \rightarrow X$, then ...
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### Quivers with a binary operation on the arrows

A set with an arbitrary binary operation is called a magma. A set of dots with a set of arrows between them is called a quiver. A category is a quiver with a binary operation on the arrows obeying ...
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### Do the terms “quiver” and “meta graph” refer to the same concept?

Do the terms "quiver" and "metagraph" refer to the same concept? Or is there a distinction I am missing. My sources are Quiver - http://ncatlab.org/nlab/show/quiver Metagraph - ...
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### Names of certain morphisms in Pos

Pos is the category of small posets and monotone maps. I call a morphism $f:\mathfrak{A}\rightarrow\mathfrak{B}$ of Pos monovalued iff it maps every atom of $\mathfrak{A}$ either into an atom of ...
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### Meaning of commutative diagram

What is the meaning of a commutative diagram in mathematics? For example, if a map translate an object, then rotate it around the origin and then translate it again, is this a commutative ...
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### What would be a more suggestive name for a “Comma Category”?

Comma categories are pretty expressive construction but apparently many mathematicians including their inventor, Dr. Lawvere, dislike the term for its non-informativeness. I was wondering if anyone ...
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### Do function with the following property have special name?

I'm writing "a structure preserving surjection" way too much when I need to refer a function of the following property:  Y \subseteq Z, X \subseteq Z. g: Z \to A, g \text{ is some fixed ...
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### What are “Lazard” sheaves?

Early in Categories, Allegories, by Freyd and Scedrov (p.12, in the section on basic examples) there appears the following example: Let $\mathcal{LH}$ be the category whose objects are topological ...
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### Graph of a Rel-morphism

Let $F=(f;A;B)$ is a morphism of the category $\mathbf{Rel}$ (the category whose objects are sets and morphisms are defined as binary relations). How to name and how to denote $f$ when we know $F$? ...