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What are...products and coproducts?

Goal. Explaining basic concepts of category theory in an intuitive way. This time. What are...products and coproducts? Or: Vector spaces rock! Disclaimer. Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references. Disclaimer. The distinction between “large classes” and “small classes (sets)” turns out is crucial for many categorical considerations, but somehow makes the language more cumbersome without too much gain imho. So I will strategically ignore all set-theoretical issues. Slides. http://www.dtubbenhauer.com/youtube.html Website with exercises. http://www.dtubbenhauer.com/lecture-c... Product. https://en.wikipedia.org/wiki/Product...) https://ncatlab.org/nlab/show/cartesi... https://mathworld.wolfram.com/Categor... Coproduct. https://en.wikipedia.org/wiki/Coproduct https://ncatlab.org/nlab/show/coproduct https://mathworld.wolfram.com/Coprodu... Direct sum. https://en.wikipedia.org/wiki/Direct_sum https://math.stackexchange.com/questi... Pictures from https://www.cs.toronto.edu/~sme/prese... Pictures used. https://en.wikipedia.org/wiki/Product... Some books I am using (I sometimes steal some pictures from there). https://en.wikipedia.org/wiki/Categor... https://www.cambridge.org/core/books/... http://www.tac.mta.ca/tac/reprints/ar... https://math.mit.edu/~dspivak/teachin... https://math.jhu.edu/~eriehl/context.pdf https://github.com/hmemcpy/milewski-c... Nlab. https://ncatlab.org/nlab/show/HomePage TheCatsters.    / @thecatsters   Mathematica. https://wildcatsformma.wordpress.com/ #categorytheory #categoricalalgebra #mathematics