Questions tagged [category-theory]

Categories are structures containing objects and arrows between them. Many mathematical structures can be viewed as objects of a category, with structure preserving morphisms as arrows. Reformulating properties of mathematical objects in the general language of category can help one see connections between seemingly different areas of mathematics.

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Mac Lane Chapter 7 Section 2 Exercise 1

Let $\mathcal{C}$ be a monodical category, with the monodical product written $\otimes$, the associator denoted $\alpha$, and the left/right unitors denoted $\iota^\ell,\iota^r$ respectively. Mac ...
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Are all unions in a topos with complete subobject lattices secretly colimits? On a logical analogue of the AB5 axiom

To clarify, here “topos” always means an elementary topos; I do not assume sheaves on a site, where I already knew my question to have a positive answer. It is known but not so immediate from the ...
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Majority vote EM category

I am working with the multiset monad. I want the EM category to have a structure map that is majority vote. That is to say that the map $f: X \rightarrow M(X)$ takes the set element with the highest ...
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Need Help Proving that $\mathbb{R}$ is a cogenerator of $\textbf{Vect}$

I am trying to prove that $\mathbb{R}$ is a co-generator of $\textbf{Vect}$. Recall that $B'$ is a co-generator of a category $\mathscr{B}$ is the collection of arrow $f:B \rightarrow B'$ is mono, for ...
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Need Help To Prove Fullness of a Functor

I am working on proving the following proposition A functor $F:\mathscr{A} \rightarrow \mathscr{B}$ is an equivalence of categories if and only if it is faithfull, full and essentially surjective. ...
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Studying modal logic using category theory

When reading about modal logic, namely general frames and algebras, I've been seeing a lot of potential functors and/or universal properties. Like constructing an algebra with Boolean operators from a ...
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Nontrivial advantages of thinking of groups as groupoids with one object?

There are already a number of questions here asking "what do people mean when they say a group is a groupoid with one object?". A natural question to ask is "are there any interesting ...
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What is the difference between direct sum $\bigoplus_{i\in I} M_i$ and restricted product $\prod'_{i \in I}M_i$ of abelian group?

What is the difference between direct sum and restricted product of abelian group? I recently heard that direct limit of direct product is not a direct sum but a restricted product. Until now, I ...
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Forgetful functor $V: \underline{\mathbf{PSet}} \rightarrow \underline{\mathbf{Set}}$ is not full

This might be a trivial question, but I don't understand why the trivial functor $V: \underline{\mathbf{PSet}} \rightarrow \underline{\mathbf{Set}}$ is not full. ($\underline{\mathbf{PSet}}$ is the ...
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On the definition of monomorphisms in arbitrary categories

I am used to working with the following definition of monomorphisms. Definition 1. In a category $\mathcal{C}$ we say that a morphism $f\in\operatorname{Hom}_{\mathcal{C}}(X,Y)$ is a monomorphism if ...
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Number of isomorphisms between categories

Let $\mathcal{C}_1$ be a category with objects $A=\{1,2,3\}$ and $B=\{4,5,6\}$. The number of isomorphisms between them is $3!=6$ and morphisms $27$. Can we extend this notion of number of ...
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Can individual topological space be considered as category?

Of course, I am aware of Top (category of topological spaces). My question is about something different - can any topological space be considered as category? E.g. its objects may be the open sets of ...
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Stable categories are tensored over spectra

It's a folklore result by Lurie that for a (presentable) stable $\infty$-category $\mathscr{C}$ the mapping space functor $\mathsf{Map}:\mathscr{C}^{\mathrm{op}} \times \mathscr{C} \to \mathsf{An}$ ...
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Commutativity of second square when first square and outer rectangle commute

The general question I reach is with the diagram below $\require{AMScd}$ \begin{CD} A @>f>> B @>\pi>> C \\ @| @VbVV @VcVV \\ A' @>f'>> B' @>\pi'>> C' \\ \end{CD} ...
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