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In each of these categories of modules equipped with a cocartesian monoidal structure, finite products and coproducts coincide (in the sense that the product and coproduct of finitely many objects are isomorphic). Or more formally, if
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with finite products (a "finite product category") can be thought of as a cartesian monoidal category. In any cartesian monoidal category, the
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Cartesian monoidal categories have a number of special and important properties, such as the existence of
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or not) becomes a cocartesian monoidal category with the
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as "deleting data". These maps make any object into a
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735:. If, in addition, the category in question has a
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554:{\displaystyle \coprod _{j\in {1,\ldots ,n}}X_{j}}
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611:{\displaystyle \prod _{j\in {1,\ldots ,n}}X_{j}}
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697:}) is a product diagram for the objects
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165:Cartesian categories with an internal
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44:adding citations to reliable sources
284:Cocartesian monoidal categories:
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400:is the "canonical" map from the
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274:bicategory of small categories
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328:More generally, the category
160:cocartesian monoidal category
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175:Cartesian closed categories
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355:as tensor product and the
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662:such that the pair (
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305:trivial vector space
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877:Monoidal categories
618:together with maps
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96:January 2017
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38:Please help
33:verification
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737:zero object
426:isomorphism
349:commutative
317:, with the
169:that is an
167:Hom functor
120:mathematics
866:Categories
834:References
818:for more.
336:of (left)
181:Properties
66:newspapers
587:…
577:∈
570:∏
530:…
520:∈
513:∐
473:…
463:∈
456:⨁
430:biproduct
276:with the
262:with the
152:coproduct
822:See also
688:, {
666:, {
375:โ ... โ
359:as unit.
325:as unit.
307:as unit.
303:and the
247:Examples
241:comonoid
224:for any
140:category
849:at the
384:โ
340:over a
338:modules
80:scholar
792:, the
785:δ
424:is an
313:, the
291:, the
272:, the
258:, the
226:object
148:Dually
138:. Any
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297:field
87:JSTOR
73:books
805:and
743:and
640:and
561:and
342:ring
289:Vect
126:, a
59:news
854:Lab
334:Mod
270:Cat
256:Set
118:In
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