# 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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### Alternative Definition of Contravariant Functor

Given two categories, $C$ and $D$, a covariant functor is usually defined as a regular functor $C \to D$, whereas a contravariant functor is usually defined as a regular functor $C^{op} \to D$. ...
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### The completeness of a category of additive functors between additive categories

In what follows $\textbf{preadditive}$: a category $\mathscr{C}$ is preadditive when $\forall\ A,B,\ \mathscr{C}(A,B)$ is an abelian group and the morphisms composition is a group homomorphism on ...
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### Does $\mbox{Hom}(\bullet,A)$ functor preserves pullbacks?

I know that $\mbox{Hom}(A,\bullet)$ functor preserves pullback (Hom-functor preserves pullbacks) but what could we say about the contravariant functor $\mbox{Hom}(\bullet,A)$?
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### Two definitions of double categories?

A double category can be defined as a category object in $\mathbf{Cat}$ the category of small categories. We can also define a double category as four categories satisfying some compatibility ...
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### Is there a connection between the concepts of limits in ordinals, functions and categories?

In set theory there is the concept of a limit ordinal: Nonzero ordinals that are the supermum of all ordinals below them. In functional analysis there are the concepts of limits of functions (and ...
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### Is there any book similar to “Halmos Naive Set Theory” in Category Theory?

When I wanted to learn set theory in high school, I found Halmos Naive Set Theory book very readable and understandable. But now, at university, I have been searching for a similar book in category ...
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### Understanding the tensor-hom adjunction intuitively

I'm currently trying to teach myself some category theory. Recently, I learned that the tensor product is left adjoint to the hom functor in suitable categories, e.g. vector spaces with linear maps, i....
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### What do we lose if we allow the Hom sets not to be pairwise disjoint?

In Category Theory the $hom(A, B)$ sets are generally formulated to be pairwise disjoint. We know that set theorists do not agree to this day on whether functions are also defined by their codomain or ...
### A scheme is affine iff the natural map $X\to \operatorname{Spec}\Gamma(X)$ is an isomorphism
We know that the functor $\operatorname{Spec}: \mathsf{Rings}^{\text{op}}\to \mathsf{Schemes}$ is right adjoint to the global section functor $\Gamma: \mathsf{Schemes}\to \mathsf{Rings}^{\text{op}}$. ...
This is part of an early exercise in Freyd's abelian categories. Let $\mathscr{G}$ be the category of abelian groups. The group of integers is distinguished, up to isomorphism, by the facts that: ...