# 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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### Is TOP a small category?

A quick question... is the category of topological spaces and continuous maps a small category? If so how do we know and if not how do we know not?
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### What is a composition in category theory?

I'm just beginning to learn category theory. So far, the basic examples (like Set) are making sense. But I'm having a little trouble getting my head around the fundamentals. Suppose I try to define a ...
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### When the empty family of arrows to an object is epimorphic, that object must be initial?

Is it true that when the empty family of arrows to an object $E$ in some category is epimorphic, that object $E$ must be the initial object $0$? This is a claim on page 433 (eq. 22) of Mac Lane and ...
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### Projective Module equivalence [closed]

I'm trying to prove that given an $A-module$: $P$ The functor $Hom(P,-)$ is exact $\implies$ $\exists Q (A-module): P\oplus Q$ is free Can anyone guide me with some hints? It's the first time I'...
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### Can we somehow use the functor $\mathbf{Set}(\mathbb{N},-)$ to define $\mathbb{N}$?

Hom functors can be used define coproducts in terms of products. In particular: $$\mathbf{Set}(A \sqcup B,X) \cong \mathbf{Set}(A,X) \times \mathbf{Set}(B,X)$$ To oversimplify a little: "a function ...
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### Characterization of split mono- and split epimorphisms of algebras

Given a concrete category $(\mathcal{A}, U : \mathcal{A} \to \mathsf{Set})$ of universal algebras, obviously if a morphism $f$ in $\mathcal{A}$ is split mono / split epi, then so is $U(f)$. What is ...
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### Pullback of a function along identity?

If $f:A\rightarrow B$ is a function, then $A\times _BB= \left\{ (a,b)\mid f(a)=b\right\}$. Isn't this the preimage of the image of $f$? I.e is $A\times_BB\cong f^\ast f_\ast(A)$? If working with ...
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### In an algebraic category a morphism is a regular epi iff it is surjective

According to the nLab (see the 4th point under "Examples") in an "algebraic category" a morphism is a regular epi if and only, if it is surjective. Here a morphism $e$ is said to be surjective, if its ...
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### Examples for the fact that a pullback of an epimorphism is not necessarily an epimorphism.

I'm reading in Borceux' book Basic Category Theory about pullbacks. It turns out that the pullback of an epimorphism is not necessarily an epimorphism. On the linked page, Borceux gives a ...
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### Enough projectives and $F$ preserves limits implies $G$ preserves epi's.

Exercise: Let $\mathcal{C}, \mathcal{D}$ be categories, $G : \mathcal{C} \to \mathcal{D}$ and $F : \mathcal{D} \to \mathcal{C}$ an adjunction $F \dashv G$. Suppose $\mathcal{D}$ has enough projectives ...
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### Strange question about free abelian group

I was given the following question for homework, but it makes no sense to me. Let $F$ be a free abelian group over a set $S$ with respect to the function $\varphi \colon S \to F$. Identify the set ...
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### Does the homotopy category functor $\mathsf{Top}_\ast\rightarrow \mathsf{hTop}_\ast$ create products?

Does the homotopy category functor $\mathsf{Top}_\ast\rightarrow \mathsf{hTop}_\ast$ create products? I know it preserves products, but it seems to actually create them.
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### How to recover a $K$-algebra from endomorphism algebra of forgetful functor?

I'm trying to work out how a finite-dimensional $K$-algebra $B$ can be recovered from the algebra of endomorphisms of the forgetful functor $\omega$ from the category of finitely generated left $B$-...
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### Functor is of the form Set(-,A)

Let $F: Set^{op} \rightarrow Set$ be a functor such that for corresponding functor $\overline{F}: Set \rightarrow Set^{op}$ we have $\overline{F} \dashv F$. With corresponding functor I mean that $F$ ...
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Let $F$ be a monad on some category $\mathsf{C}$ and $G$ be a comonad on the same category. Assume further that they "commute" (see below): $FG \cong GF$. Then, for lack of a better name, one can ...
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### Is there an extremal epi which is not epi, although binary products exist?

It is known, that in a category $\mathcal{A}$: strong epis are epi, if $\mathcal{A}$ has binary products extremal epis are epi, if $\mathcal{A}$ has equalizers all strong epis are extremal all ...
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### Groupoid objects in the category of algebras

Can anyone give me some references where I could read about groupoid objects in the category of algebras? References about groupoid objects in other categories would also be welcome.
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### “Spanning” a category

What does it mean for objects to "span" a category? I know what a span is but I'm not exactly sure what the phrase means. Does it just mean the category consists of the specified objects? An example ...
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### Proving cardinality of coproduct presentation is unique without choice?

The definition of an extensive category immediately implies that given two coproduct decompositions indexed by sets of equal cardinality, if the coproduct objects are isomorphic compatibly with their ...
I would like to know if there is a name for an object $X$ in a (finitely complete and cocomplete) category $\mathcal{C}$ which has the following property: $X$ is non-empty and for every sub-object ...