# 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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### Are strong monomorphisms coretrations in the category of graphs?

Consider the category of digraphs with strong homomorphisms as the morphisms. Here a strong homomorphism $f: G\longrightarrow H$, is a graph homomorphism which preserves and reflects adjacency of ...
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### Reference request: Categories enriched over $\textbf{Lat}$

I'm looking for some sources that discuss categories enriched over the category $\textbf{Lat}$ of lattices. Actually, more specifically, the category I'm studying is enriched over the category of (...
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### In what sense is metric space completion universal?

The completion of a metric space is unique up to metric monomorphism (usually called isometry). It is also the "obvious" way to make all Cauchy sequences convergent. Structures which are unique up ...
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### Cokernel of a module homomorphism

Let $A$ a $K$-algebra. Let $M$, $N$ $A$-modules and $f:M\rightarrow N$ a module homomorphism. The cokernel of $f$ is $Cokerf=N/Imf$ I define a homomorphism $\rho:N\rightarrow N/Imf$ by $\rho(n)=n+Imf$....
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### Is there a relationship between the pullback in differential geometry and that in category theory?

1. Is there a relationship between the pullback in differential geometry and the pullback in category theory? [2. Is there a relationship between the pushforward/pushout in differential geometry ...
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### Diagonal Functor an Isofibration?

Let $C$ be a category and let $\Delta:C\rightarrow C\times C, \Delta=(id_c,id_c)$ be the diagonal functor. Recal that an isofibration is a functor p: E→B such that for any object $e\in E$ and any ...
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### Algebra of Endomorphisms of a functor $\mathcal{F}$

Recently, I was reading a paper and stumbled upon something for which I don't find the formal definition unfortunately in any textbook that have in my hands. It's the notion of the endomorphism ...
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### “Alternatives” to Natural Transformations

I would like someone to either (1) point out the mistake in what follows or (2) confirm what is said is correct. This would be accomplished by addressing the part in yellow only. The rest of the ...
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### Are exponential objects examples of (co)universal objects which are not (co)limits?

This is in some sense a follow-up to previous questions I have had asking about the relationship between products and exponential objects. Products can be written as, and in often are defined to be, ...
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### Is the tensor product of BAOs a kind of extended BAO?

I've been reading "Boolean algebras with operators. Part I." (Jonsson, Tarski) where, given a subalgebra of a Boolean Algebra, they define its perfect extension. As far as I understand it can be ...
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### Categorical Interpretation of Strongest/Weakest Topology

One way to define the product topology is as the weakest topology for which all projection maps are continuous. Strongest/weakest topologies satisfying a given property are ubiquitous in topology and ...
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### Is the assignment of a root system to a semisimple Lie algebra functorial?

As described here, we have a category of root systems, where a morphism from a root system $\Phi$ in a Euclidean space $E$ to a root system $\Phi'$ in $E'$ is given by a linear map $f: E \to E'$ such ...
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### Why care about relation liftings under (covariant) power-set functor? (coalgebraic logic)

Moss' motivation to use the notion of relation lifting was in their importance when applied to membership relation, which leads to define the semantics of $\nabla$ operator. Now in the whole process, ...
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### Circular definition in slice category?

I am reading Aluffi (Algebra Chapter 0) there he introduces the slice category in a kind of excercise: When thinking about it I got confused about the "nature" of the $Z$ (and $A$). Since they are ...
I am reading the paper of Andrej Bauer that proves there is a realizability topos in which there is an injection from the internal Baire Space $\mathbb{N}^\mathbb{N}$ to the natural numbers \$\mathbb{N}...