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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About the minimal equivalence relation identifying some points.

I am solving a problem where I have a set $X$ together with a subset of elements that I want to identify. To do this I consider the minimal equivalence relation identifying these points. I have a ...
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Graphic intuition for generalizing to weighted limits

One of the ways to define a limit of a functor $F:\mathsf C\longrightarrow\mathsf D$ is a representation of $\mathsf{Nat}(\Delta-,F)$. Along the journey of generalization to the enriched setting, one ...
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What is the dual category of topological spaces? [duplicate]

What is the dual category of topological spaces $Top$? I know that the order theoretic dual of a topological space is a closed set system rather than an open set system. However, this doesn't answer ...
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Is every finite category identifiable with a directed multigraph? (and vice versa?) [duplicate]

What seems implicit in this talk on youtube, is the claim that every directed multigraph (with loops) can be identified with a finite category and vice versa, if we consider the paths of the directed ...
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Show that for each of the following graphs G there exists up to isomorphism precisely one category A with G(A) = G.

I was working through the exercises in Abstract and Concrete Categories: The Joy of Cats (http://katmat.math.uni-bremen.de/acc/acc.pdf) and I was stuck on exercise 3A.(d). It seems to me that the ...
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Let $\mathbf{A}$ be a category, then $\mathcal{P}_\mathbf{A} (\mathrm{dom}(f))\implies\mathcal{Q}_\mathbf{A}(f)$

This is a very trivial question in category theory, and the textbook I'm working from has this as a supposedly trivial example of the dual property for categories, and unfortunately, I can't seem to ...
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Homotopy Colimit of Truncations

Let $\mathcal{A}$ be an additive category with countable coproducts. I am just starting to learn about homotopy colimits and I am struggling with the following example that I am very interested in ...
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In category theory: Do we need products to define exponentials?

In the HoTT book the type of functions $A\to C$ construction is described first and the product type $A\times B$ construction later, using function types in its definition. So my obvious naive ...
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Does a binormal category always admit an additive structure?

Let $\mathsf{C}$ be a category. We call $\mathsf{C}$ binormal if it has a null object, has all equilizers and coequilizers, all monomorphisms are kernels and all epimorphisms are cokernels (whereby a ...
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Existential axioms for category theory

There are some existential axioms in set theory, for example, axiom schema of specification. It's my understanding that category theory isn't based essentially on set theoretic foundation. If so, I ...
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“External” Lawvere-Tierney Topologies?

Suppose I have a map $j : \text{Sub}(1) \to \text{Sub}(1)$ from subterminal objects of a topos to themselves which satisfies analogous axioms to those of a Lawvere-Tierney topology, namely $j(1) = 1$, ...
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How to verify commutativity of a diagram?

Let $C$ be a category, and let all objects $X_i, Y_j$ belong to $ob(C)$, and morphisms $f_{ij}, h_i, g_{ij}$ be morphisms between them in $C$. Let us have a diagram then: How do we verify it's ...
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Sheafification by the small object argument

Is the sheafification functor constructed as a double application of the plus construction a special case of the small object argument? I thought it might be since I can't think of any other general ...
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F-algebras in 2-categories

Given a functor $F : C \to C$, one can usually study the $F$-algebras: morphisms $\alpha : F X \to X$. Where can I read about its generalisation to 2-categories? I think that one can consider now "lax"...
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Checking if two diagrams have isomorphic colimits

Consider the infinite diagrams: $$C_1\to C_2 \to \cdots \to C_n \to \cdots$$ $$D_1\to D_2 \to \cdots \to D_n \to \cdots$$ in some category, and suppose both colimits exist. How do I check if ...
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Ind- and pro-objects, reference request

Can someone point me to a good exposition of ind- and pro-objects, the intuition behind, and how one "in practice" works with them (i.e. prove things)? The nlab page is nice (especially for the ...
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When are two ind-objects isomorphic?

Two diagrams may be different, but they may still have the same isomorphic limits (or colimits). Ind-objects are, so to speak, formal colimits of diagrams, even if the actual limit may not exist. ...
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Elements in commutative diagram

The same way I define a function, by explicitly including the image of an element:  \begin{aligned} \mathbb{R} & & \overset{\exp}{\longrightarrow} & & \mathbb{R} \\ x & & \...
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Characterization of internal groupoids via pullbacks

The most intuitive way (for me) to define an internal groupoid is as an internal category with extra structure, namely an involution on the object of morphisms which "produces inverses". In Borceux ...