Category Theory

Table of Contents

MEGA resource:

Notation

Definition

A category category_theory_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png consists of the three mathematical entities:

  1. A class category_theory_74ff30f9ff5118454842055e67e232e042505551.png, whose elements are the objects
  2. A class category_theory_f63aa3345ddf704438871bc70a3404e5476a88a7.png, whose elements are called morphisms or maps or arrows. Each morphism category_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png has a source object category_theory_e46729bc781c25bbc7120ee2892cc1c0215af7da.png and target object category_theory_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png
  3. A binary operation category_theory_3d4fd6de0d71bddfdc5feee738a7a6b8796b7657.png, called composition of morphisms, such that for any three objects category_theory_e46729bc781c25bbc7120ee2892cc1c0215af7da.png, category_theory_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png, and category_theory_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png, we have

    category_theory_5b90c48e8797ac6bcadf8696bd9ac8c89888a2d8.png

    The composition of category_theory_ed0ef953cd667827a5d71c78f56d7ab5cbdf0774.png and category_theory_367f9d82a92d85aa3507138864d4b1ee8f7c3ae4.png is governed by two axioms:

    1. Associativity:

      category_theory_f7360c6bbfff38749b869ccd2db60769bd2bc223.png

    2. Identity map:

      category_theory_28dfdce12720ce7b9834198f533293b95caf489c.png

Category of Sets

The category category_theory_93b30420948e881992ebf19e3f647a791e2da8a9.png of category_theory_4b7402e5f4df193424babbee35569b6538d86964.png for any partially ordered set category_theory_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png, where

  • a category_theory_d6446aeac785d45fff78443e70397eb36339d62d.png is a function category_theory_38b26a58b0289d21826e6f1d449b0f012d96ff62.png from a set category_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png to category_theory_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png
  • a morphism category_theory_b0efcc631496421ff5d918b95b88fc022d8abeb6.png of category_theory_4b7402e5f4df193424babbee35569b6538d86964.png category_theory_38b26a58b0289d21826e6f1d449b0f012d96ff62.png and category_theory_43f154f6fd6e8a8823997e70dd74dcc346fc1312.png is a function

    category_theory_3386b3db7af5554ed50bf2307fe5b6571775773f.png

Let category_theory_dc51e00dd9204b32a62671c09bed4f1f9a4cb58f.png, i.e. category_theory_e46729bc781c25bbc7120ee2892cc1c0215af7da.png be the singleton set, then, for any set category_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png there is an isomorphism of sets

category_theory_2c742e31350236f16703c8987821bc0618b14c4d.png

Just to make the point clear here, for the set category_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, category_theory_ab724b4857fc6736ee643574326cbd15474fd7fe.png where category_theory_9e1a724d7f21d8f7d2d5450dac9699c7059fbbcf.png is defined by the mapping:

category_theory_cbf02012accdd47353e4fdf67dc12c05c1ddc506.png

Products

Let category_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and category_theory_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png be sets.

The product of category_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and category_theory_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png, denoted category_theory_efff620f53f3019406002c15cb48473f684ad92a.png, is defined as the set of ordered pairs category_theory_bcaca16349a57fa4db86decededc11f33eed0a02.png where category_theory_4c91032750887d8dca229b906727af58e9268232.png and category_theory_fa981de37e65c7147e4b39c3e635c864cbb63cf9.png. That is,

category_theory_421864d7a8ce4f3a7a441a1b4013950cc61fbcb4.png

There are two natural projections category_theory_f8760928a635131b3d3913061baac003d0918ca2.png and category_theory_4c04c79aa8af8e7cf5251407808e892f42ae286e.png.

products.png

Let category_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and category_theory_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png be sets.

For any set category_theory_e46729bc781c25bbc7120ee2892cc1c0215af7da.png and functions category_theory_a63a2b2551d660706dd0eb9cdedfce46306408e3.png and category_theory_a74a37e8d1935e3d5a78ab9f6d3d23d4aabe67b7.png there exist a unique function category_theory_8e1de05d5057ca2a23f9be009c968e3f35d52e21.png such that the following diagram commutes:

universal-property-of-products.png

We might write the unique function as

category_theory_533010b6dbfb13ef8114fbd681080a2bbe3c5f78.png

Suppose we are given category_theory_09a8c99a5e7e916be687cfd3c64d9f5cc16047c5.png as above. To provide a function category_theory_696748d42e846289d082452ffabac56b61d52c8d.png is equivalent of providing an element category_theory_0a07c86c7b51a07cbd7698207d5743a240b265b1.png for each category_theory_059a13dab69c3d1523accbe2cb781457a950d07b.png.

We need such a function category_theory_4cd082f9db7ea80260b7e962dd0e38070da0d1a2.png for which

category_theory_e1af193d4f57d6570fc613c845e07f2c3e4c41d6.png

An element of category_theory_efff620f53f3019406002c15cb48473f684ad92a.png is an ordered pair category_theory_e0cd6d07f279306623c45a74124b9526adbeef2f.png, and so we can use

category_theory_8312e190d612a2b64ed4e3c90ad1f07271d25fbd.png

Hence, it's necessary and sufficient to define

category_theory_ddbab18ec3696cd8a0adc7a967dc2a680f7f84e5.png

Coproducts

Let category_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and category_theory_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png be sets.

The coproduct of category_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and category_theory_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png, denoted category_theory_8c13651932f678b904923f541f37dce116a5e5bd.png is defined as the disjoint union of category_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and category_theory_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png, i.e. the set for which an element is either an element of category_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png or an element of category_theory_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png.

If something is an element of both category_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and category_theory_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png then we include both copies, and distinguish between them, in category_theory_8c13651932f678b904923f541f37dce116a5e5bd.png.

There are two natural inclusion functions

category_theory_c465c9973704acb9b2d33d48637ed4f1c4e3bf85.png

coproduct-inclusions.png

Finite limits in Set

Pullbacks

Suppose the given diagram of sets and functions below

pullback-of-sets.png

Its fiber product is the set

category_theory_3f4237eedcd5bb15d3e6b1f56bd9dd34094e1d09.png

There are obvious projections

category_theory_64b44a80af01d121b3e23163cd4f207abf5eb3e9.png

Note that if

category_theory_399fafac568c72d67362169623ab3ec5b19bd0d2.png

then the diagram pullback-of-sets-2.png commutes.

Given the setup in the diagram above, we define the pullback of category_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and category_theory_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png over category_theory_ca2bcc80309e331f71dd2c1e1f4788cbc048c242.png to be any set category_theory_f53cf2c3985c80987bd9707fd5a0b28594db886f.png for which we have an isomorphism

category_theory_36e4c2cf938db6628d907cc0408dab570ac15bef.png

The corner symbol denotes that category_theory_f53cf2c3985c80987bd9707fd5a0b28594db886f.png is the pullback.

Sometimes you'll see the fiber product category_theory_ecddc19439096f746fbe625844a6f90d486916fc.png denoted category_theory_908de0f96db157e7313921ccc70c436d17b40a24.png.

Suppose given the diagram of sets and functions as below.

universal-property-of-pullbacks-1.png

For any set category_theory_e46729bc781c25bbc7120ee2892cc1c0215af7da.png and commutative solid arrow diagram as below (i.e. functions category_theory_a63a2b2551d660706dd0eb9cdedfce46306408e3.png and category_theory_a74a37e8d1935e3d5a78ab9f6d3d23d4aabe67b7.png such that category_theory_c5611d3f78e407890071bdd122678c13c9fb9910.png),

universal-property-of-pullbacks-2.png

there exists a unique arrow category_theory_d3e1c3ac35b2c00f3c05d86f8bd5b8a343e8a922.png making everything commmute,

category_theory_6638510ed22e0f99011321dedb4ffc708dd0c022.png

Category of Fuzzy Sets

The category of fuzzy subsets is denoted

category_theory_4318d9871131bc14d6e57af139f2a7b364389f4b.png

The objects of category_theory_c9dac3d8c7645fbcf8461e182d255264c7484369.png are all pairs category_theory_7a8cf5a4341ca48e9b0c44a6c7ab985680ba2b8d.png where

  • category_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is a set
  • category_theory_131a39c1c0ef31b754ad0ad6c9f7068836579066.png is a function from category_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png to the unit interval

The maps of category_theory_c9dac3d8c7645fbcf8461e182d255264c7484369.png are defined by

category_theory_5d5758f5ebed05285ef570d53a3262950462e8b3.png

where category_theory_1b6695c19c6474b77e05182f94fdb38c647e5070.png denotes the category of sets.

  • With composition simply being the composition of functions