# 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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### Tensor structure on category of symmetric monoidal categories, $\mathsf{SymMonCat}$.

Let $\mathsf{SymMonCat}$ denote the category with objects symmetric monoidal categories and morphisms lax symmetric monoidal functors. Can $\mathsf{SymMonCat}$ be endowed with a "tensor" ...
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### Why isn't the Disjoint Union in Set a *product* in addition to being a coproduct?

So I'm understanding that in $Set$ that the cartesian product is a categorical product, and further get why the disjoint union is a categorical coproduct, but why is it also not a product? I want to ...
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### Proving Segal's Category $\Gamma \simeq \mathbf{FinSet}_*^{op}$

I'm trying to show that Segal's category $\Gamma$ is equivalent to $\mathbf{FinSet}_*^{op}$, the opposite of the category of finite pointed sets with basepoint preserving morphisms. I'm intuitively ...
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### Dual of a statement involving adjoint functors

Let $F:\mathcal{C}\rightarrow\mathcal{D}$ be a functor. The following conditions are equivalent: $F$ is full and faithful and has a full and faithful left adjoint $G$. $F$ has a left adjoint $G$ and ...
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### Is a continuous function with this property injective?

Suppose that I have a continuous $f$ in a topological space with the following property: for every continuous $g$ and $h$ such that $f \circ g = f \circ h$ $\Rightarrow$ $g=h$ It is true that a ...
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### Pairs of notation: Which one is the co-thing?

Many notaions come in pairs, to mention the most classical example, sine/cosine are related by the the fact that $\sin(\alpha)=\cos(\beta)$ if $\alpha+\beta=\pi/2$, i.e., the angles are complementary. ...
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### morphism is epic?

If we have three morphisms f, g and h between objects of a category. Suppose gf = h. If g and h are epic, can we conclude that so is f ? Any help would be appreciated!
In my abstract algebra textbook, when introducing category it says that morphisms should satisfy several properties and two of them are: For every object $A$ of $C$, there exists (at least) one ...