Abstract Nonsense

Finite Limits and Colimits as Products/Coproducts and Equalizers/Coequalizers

Point of Post: In this post we prove the result that a category has all finite products/coproducts and equalizers/coequalizers if and only if it has all limits/colimits.

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Motivation

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In this post we prove a pretty neat fact that, at least for finite sets (we shall state a more general result, but we don’t care about it as much) limits/colimits are the same thing as equalizers/coequalizers and products/coproducts. This is often cool because it reduces having to prove some result holding for all finite limits/colimits to proving some problem holds for equalizers/coequalizers and finite products/coproducts. In particular, it will tell us that a category admits finite limits/colimits if and only if it admits finite products/coproducts.

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Finite Limits/Colimits ad Products/Coproducts and Equalizers/Coequalizers

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There is really nothing to discuss here, so I’ll just bang this one out:

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Theorem: Every finite limit/colimit can be written as the equalizers/coequalizer of a finite product/coproduct of maps.

Proof: Let $\mathcal{I}$ be a finite category (finite amount of objects AND finite amount of arrows) and $F:\mathcal{I}\to\mathcal{C}$ a diagram. Consider the two arrows

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$\displaystyle \prod_{i\in\text{Obj}(\mathcal{I})}F(i)\overset{\displaystyle \overset{\Phi}{\longrightarrow}}{\underset{\Psi}{\longrightarrow}}\prod_{\alpha\in\text{Mor}(\mathcal{I}}F(\text{codomain }\alpha)$

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(where, in case you’ve forgotten, $\text{Mor}(\mathcal{C})$ stands for all the morphisms in $\mathcal{C}$ and if $X\xrightarrow{\alpha}Y$ then $\text{codomain }\alpha=Y$) defined projection wise by $\pi_\alpha\circ\Phi=\pi_{\text{codomain }\alpha}$ and $\pi_\alpha\circ\Psi=F(\alpha)\circ\pi_{\text{domain }\alpha}$ where $\pi_{\text{codomain }\alpha}$ is  the projection arrow $\displaystyle \prod_{i\in\text{Obj}(\mathcal{C}}F(i)\to F(\text{codomain }\alpha)$ and $\pi_{\text{domain }\alpha}$ is defined similarly (of course you can guess what domain means here).

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Let $(E,e)$ be an equalizer of these two arrows. What we want to show then is that $(E,\pi_i\circ e)$ is actually a limit for $D$. To prove this we need to take any possible arrow $\displaystyle X\xrightarrow{\gamma}\prod_i F(i)$. Observe then that the set of arrows $\{\pi_i\circ\gamma\}$ is a cone for $F$ if and only if $\Phi\circ \gamma=\Psi\circ\gamma$ (check this).  From this then we may conclud ethat $(E,\pi_i\circ e)$ is a cone. Moreover, we see that any cone is going to have to factorize through $(E,\pi_i\circ e)$ since the cone functions of that cone will have to equalize $\Phi$ and $\Psi$. $\blacksquare$

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Using the exact same set up (dualizing) one can prove that

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Theorem: Every finite colimit can be written as the coequalizer of a finite coproduct.

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Corollary: If a category admits all equalizers (resp. coequalizers) and finite products (resp. coproducts) then the category admits all finite limits (resp. colimits).

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Corollary: An abelian category admits all finite limits and colimits.

Proof: Recall that an abelian category, by definition, includes all kernels an cokernels. And, since $\text{Equ}(f,g)=\ker(f-g)$ and $\text{Coeq}(f,g)=\text{coker}(f-g)$ we also know that it admits all equalizers and coequalizers. Since abelian categories also admit all finite products and coproducts the conclusion follows. $\blacksquare$

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References:

[1] Mac, Lane Saunders. Categories for the Working Mathematician. New York: Springer-Verlag, 1994. Print.

[2] Adámek, Jirí, Horst Herrlich, and George E. Strecker. Abstract and Concrete Categories: the Joy of Cats. New York: John Wiley & Sons, 1990. Print.

[3] Berrick, A. J., and M. E. Keating. Categories and Modules with K-theory in View. Cambridge, UK: Cambridge UP, 2000. Print.

[4] Freyd, Peter J. Abelian Categories. New York: Harper & Row, 1964. Print.

[5] Mitchell, Barry. Theory of Categories. New York: Academic, 1965. Print.

[6] Herrlich, Horst, and George E. Strecker. Category Theory: An Introduction. Lemgo: Heldermann, 2007. Print.