# Tagged Questions

Various structures are studied in category theory using properties of objects and morphisms between them. Many constructions are special cases of categorical limits and colimits (e.g. products in various categories). The notions of functor and natural transformations are very important in category ...

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### Exact adjoint functors of triangulated categories

Let $T$ and $S$ be triangulated categories. Let $F:T\rightarrow S$ and $G:S\rightarrow T$ be two adjoint functors. Assume that one of them is exact (i.e. sends exact triangles to exact triangles and ...
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### Why does the name “epimorphism” refer to a surjective homorphism?

The wikipedia page talks about epimorphisms with category theory in mind, but I have no experience with this and ask this question from a group theory point of view (answers from any point of view are ...
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### Non-injective monomorphisms

I am reading Borceux, vol. 1, I found this example at page 27: we consider the category whose object are the pairs $\langle X,x\rangle$ where $X$ is a topological space and $x$ a point of $X$ (base ...
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### Is there some kind of deep relationship between substitution and recursion?

Define $\mathbb{N}$ as the initial object in the following category: Objects. Sets $X$ equipped with a function $S : X \rightarrow X$ and an element $0:X$. Morphisms. Functions that preserves ...
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### Representable bifunctors

Is there a notion of representability for functors in the form $F:C^{op} \times C \to Set$? Can anyone please give me a reference? Thanks.
1answer
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### Regular coimages in a finitely complete and finitely cocomplete category

According to the nLab, in a finitely complete and finitely cocomplete category $\mathcal{A}$, for every morphism $f : A \to B$ a regular coimage exists and is given by the coequalizer of $f$'s kernel ...
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### Free cocompletion of thin categories

Let $C$ be a thin category. Is there a cocomplete thin category $C\to C'$ such that for every functor $C\to D$ into a cocomplete thin category there is a unique, up to isomorphism, cocontinous functor ...
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### The bigger picture the Five Lemma fits into

The Five Lemma is a statement in category theory about certain conditions under which certain maps in exact sequences are isomorphisms. It has a few relatives like the 4 lemmas and maybe the Nine ...